From c7306b80730bf72f6c70819c11d338d0dcde0166 Mon Sep 17 00:00:00 2001 From: PeterLuschny Date: Tue, 28 Feb 2023 09:43:35 +0100 Subject: [PATCH] gra --- IntroductionToTabl.ipynb | 3 +- README.md | 62 ++--- data/csv/Abel.csv | 6 +- data/csv/Baxter.csv | 6 +- data/csv/Bell.csv | 6 +- data/csv/Bessel.csv | 8 +- data/csv/Bessel2.csv | 8 +- data/csv/Binomial.csv | 6 +- data/csv/Catalan.csv | 6 +- data/csv/CatalanAer.csv | 6 +- data/csv/CentralCyc.csv | 52 ---- data/csv/CentralSet.csv | 6 +- data/csv/ChebyshevS.csv | 10 +- data/csv/ChebyshevT.csv | 6 +- data/csv/ChebyshevU.csv | 6 +- data/csv/ChristTree.csv | 6 +- data/csv/CompositionMax.csv | 6 +- data/csv/Compositions.csv | 54 ----- data/csv/Delannoy.csv | 6 +- data/csv/Euler.csv | 12 +- data/csv/EulerSec.csv | 10 +- data/csv/EulerTan.csv | 6 +- data/csv/Eulerian.csv | 6 +- data/csv/Eulerian2.csv | 6 +- data/csv/EulerianB.csv | 6 +- data/csv/FallingFact.csv | 6 +- data/csv/Fibonacci.csv | 6 +- data/csv/Fubini.csv | 6 +- data/csv/FussCatalan.csv | 6 +- data/csv/Gaussq2.csv | 6 +- data/csv/Genocchi.csv | 6 +- data/csv/Harmonic.csv | 6 +- data/csv/HermiteE.csv | 8 +- data/csv/HermiteH.csv | 6 +- data/csv/Laguerre.csv | 6 +- data/csv/Lah.csv | 8 +- data/csv/Lehmer.csv | 6 +- data/csv/Leibniz.csv | 6 +- data/csv/Levin.csv | 6 +- data/csv/Lozanic.csv | 6 +- data/csv/Motzkin.csv | 6 +- data/csv/MotzkinGF.csv | 6 +- data/csv/Narayana.csv | 6 +- data/csv/Nicomachus.csv | 6 +- data/csv/One.csv | 6 +- data/csv/OrderedCyc.csv | 52 ---- data/csv/Ordinals.csv | 6 +- data/csv/Partition.csv | 6 +- data/csv/PartitionMax.csv | 6 +- data/csv/Polygonal.csv | 10 +- data/csv/PowLaguerre.csv | 10 +- data/csv/Rencontres.csv | 6 +- data/csv/RisingFact.csv | 8 +- data/csv/Schroeder.csv | 6 +- data/csv/SchroederP.csv | 6 +- data/csv/Seidel.csv | 6 +- data/csv/SeidelBoust.csv | 6 +- data/csv/Sierpinski.csv | 6 +- data/csv/StirlingCyc.csv | 8 +- data/csv/StirlingCyc2.csv | 6 +- data/csv/StirlingCycB.csv | 6 +- data/csv/StirlingSet.csv | 6 +- data/csv/StirlingSet2.csv | 6 +- data/csv/StirlingSetB.csv | 6 +- data/csv/Sylvester.csv | 6 +- data/csv/SymPoly.csv | 94 ++++---- data/csv/TernaryTrees.csv | 6 +- data/csv/WardSet.csv | 6 +- data/md/Abel.md | 338 +++++++++++++------------- data/md/Baxter.md | 156 ++++++------ data/md/Bell.md | 156 ++++++------ data/md/Bessel.md | 338 +++++++++++++------------- data/md/Bessel2.md | 260 +++++++++++++------- data/md/Binomial.md | 440 +++++++++++++++++++-------------- data/md/Catalan.md | 364 +++++++++++++++++++++------- data/md/CatalanAer.md | 338 +++++++++++++------------- data/md/CatalanSqr.md | 416 ++++++++++++++++++++++++++++++++ data/md/CentralCycle.md | 208 ++++++++++++++++ data/md/CentralSet.md | 338 +++++++++++++------------- data/md/ChebyshevS.md | 336 ++++++++++++-------------- data/md/ChebyshevT.md | 158 ++++++------ data/md/ChebyshevU.md | 156 ++++++------ data/md/ChristTree.md | 440 +++++++++++++++++++-------------- data/md/Composition.md | 336 ++++++++++++-------------- data/md/CompositionMax.md | 156 ++++++------ data/md/Delannoy.md | 468 +++++++++++++++++++++++++++++------- data/md/Euler.md | 338 +++++++++++++------------- data/md/EulerSec.md | 336 ++++++++++++-------------- data/md/EulerTan.md | 156 ++++++------ data/md/Eulerian.md | 262 +++++++++++++------- data/md/Eulerian2.md | 158 ++++++------ data/md/EulerianB.md | 440 +++++++++++++++++++-------------- data/md/FallingFact.md | 260 +++++++++++++------- data/md/Fibonacci.md | 158 ++++++------ data/md/Fubini.md | 158 ++++++------ data/md/FussCatalan.md | 158 ++++++------ data/md/Gaussq2.md | 440 +++++++++++++++++++-------------- data/md/Genocchi.md | 156 ++++++------ data/md/Harmonic.md | 338 +++++++++++++------------- data/md/HermiteE.md | 338 +++++++++++++------------- data/md/HermiteH.md | 156 ++++++------ data/md/LabeledGraphs.md | 208 ++++++++++++++++ data/md/Laguerre.md | 336 ++++++++++++-------------- data/md/Lah.md | 338 +++++++++++++------------- data/md/Lehmer.md | 338 +++++++++++++------------- data/md/Leibniz.md | 156 ++++++------ data/md/Levin.md | 156 ++++++------ data/md/Lozanic.md | 440 +++++++++++++++++++-------------- data/md/Motzkin.md | 260 +++++++++++++------- data/md/MotzkinGF.md | 338 +++++++++++++------------- data/md/Narayana.md | 338 +++++++++++++------------- data/md/Nicomachus.md | 156 ++++++------ data/md/One.md | 440 +++++++++++++++++++-------------- data/md/OrderedCycle.md | 208 ++++++++++++++++ data/md/Ordinals.md | 158 ++++++------ data/md/Partition.md | 338 +++++++++++++------------- data/md/PartitionMax.md | 158 ++++++------ data/md/Polygonal.md | 158 ++++++------ data/md/PowLaguerre.md | 158 ++++++------ data/md/Rencontres.md | 336 ++++++++++++-------------- data/md/RisingFact.md | 336 ++++++++++++-------------- data/md/Schroeder.md | 338 +++++++++++++------------- data/md/SchroederL.md | 416 ++++++++++++++++++++++++++++++++ data/md/SchroederP.md | 338 +++++++++++++------------- data/md/Seidel.md | 156 ++++++------ data/md/SeidelBoust.md | 156 ++++++------ data/md/Sierpinski.md | 440 +++++++++++++++++++-------------- data/md/StirlingCyc.md | 338 +++++++++++++------------- data/md/StirlingCyc2.md | 156 ++++++------ data/md/StirlingCycB.md | 336 ++++++++++++-------------- data/md/StirlingSet.md | 338 +++++++++++++------------- data/md/StirlingSet2.md | 156 ++++++------ data/md/StirlingSetB.md | 336 ++++++++++++-------------- data/md/Sylvester.md | 156 ++++++------ data/md/SymPoly.md | 338 +++++++++++++------------- data/md/TernaryTrees.md | 156 ++++++------ data/md/WardSet.md | 158 ++++++------ data/md/Worpitzky.md | 260 +++++++++++++------- src/Abel.py | 2 +- src/Bessel.py | 2 +- src/CatalanAer.py | 2 +- src/CentralCyc.py | 2 +- src/CentralSet.py | 2 +- src/ChebyshevT.py | 2 +- src/Euler.py | 2 +- src/Eulerian.py | 2 +- src/Eulerian2.py | 2 +- src/Fibonacci.py | 2 +- src/Fubini.py | 2 +- src/FussCatalan.py | 2 +- src/Harmonic.py | 2 +- src/HermiteE.py | 2 +- src/LabeledGraphs.py | 41 ++++ src/Lah.py | 2 +- src/Lehmer.py | 2 +- src/MotzkinGF.py | 2 +- src/Narayana.py | 2 +- src/OrderedCyc.py | 2 +- src/Ordinals.py | 2 +- src/Partition.py | 2 +- src/PartitionMax.py | 4 +- src/Polygonal.py | 2 +- src/PowLaguerre.py | 2 +- src/Schroeder.py | 2 +- src/SchroederL.py | 2 +- src/SchroederP.py | 2 +- src/StirlingCyc.py | 2 +- src/StirlingSet.py | 2 +- src/SymPoly.py | 2 +- src/WardSet.py | 2 +- src/_tabldata.py | 22 +- src/_tablmake.py | 4 + src/_tabltabls.py | 15 +- src/_tabltraitcard.py | 96 +++++--- src/tabl.py | 184 +++++++++----- src/tablupdate.py | 34 +++ 176 files changed, 11433 insertions(+), 8930 deletions(-) delete mode 100644 data/csv/CentralCyc.csv delete mode 100644 data/csv/Compositions.csv delete mode 100644 data/csv/OrderedCyc.csv create mode 100644 data/md/CatalanSqr.md create mode 100644 data/md/CentralCycle.md create mode 100644 data/md/LabeledGraphs.md create mode 100644 data/md/OrderedCycle.md create mode 100644 data/md/SchroederL.md create mode 100644 src/LabeledGraphs.py create mode 100644 src/tablupdate.py diff --git a/IntroductionToTabl.ipynb b/IntroductionToTabl.ipynb index 0890cf4..69afb9f 100644 --- a/IntroductionToTabl.ipynb +++ b/IntroductionToTabl.ipynb @@ -157,10 +157,11 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ - "'stirling_set.mat' converts the triangle to a square matrix by filling T(n, k) for k > n with zeros." + "StirlingSet.mat converts the triangle to a square matrix by filling T(n, k) for k > n with zeros." ] }, { diff --git a/README.md b/README.md index 4fdd176..6971fb1 100644 --- a/README.md +++ b/README.md @@ -8,71 +8,73 @@ The notebook gives a first introduction for the user. | Table | Source | Traits | OEIS similars | | :--- | :--- | :--- | :--- | -| [Abel](https://github.com/PeterLuschny/tabl/blob/main/data/md/Abel.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Abel.py) | [traits](https://luschny.de/math/oeis/Abel.html) | [A061356, A137452, A139526](https://oeis.org/search?q=%7Cid%3AA061356%7Cid%3AA137452%7Cid%3AA139526) | +| [Abel](https://github.com/PeterLuschny/tabl/blob/main/data/md/Abel.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Abel.py) | [traits](https://luschny.de/math/oeis/Abel.html) | [A137452, A061356, A139526](https://oeis.org/search?q=%7Cid%3AA137452%7Cid%3AA061356%7Cid%3AA139526) | | [Baxter](https://github.com/PeterLuschny/tabl/blob/main/data/md/Baxter.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Baxter.py) | [traits](https://luschny.de/math/oeis/Baxter.html) | [A359363, A056939](https://oeis.org/search?q=%7Cid%3AA359363%7Cid%3AA056939) | | [Bell](https://github.com/PeterLuschny/tabl/blob/main/data/md/Bell.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Bell.py) | [traits](https://luschny.de/math/oeis/Bell.html) | [A011971, A011972, A123346](https://oeis.org/search?q=%7Cid%3AA011971%7Cid%3AA011972%7Cid%3AA123346) | -| [Bessel](https://github.com/PeterLuschny/tabl/blob/main/data/md/Bessel.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Bessel.py) | [traits](https://luschny.de/math/oeis/Bessel.html) | [A001497, A001498, A122850, A132062](https://oeis.org/search?q=%7Cid%3AA001497%7Cid%3AA001498%7Cid%3AA122850%7Cid%3AA132062) | +| [Bessel](https://github.com/PeterLuschny/tabl/blob/main/data/md/Bessel.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Bessel.py) | [traits](https://luschny.de/math/oeis/Bessel.html) | [A132062, A001497, A001498, A122850](https://oeis.org/search?q=%7Cid%3AA132062%7Cid%3AA001497%7Cid%3AA001498%7Cid%3AA122850) | | [Bessel2](https://github.com/PeterLuschny/tabl/blob/main/data/md/Bessel2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Bessel2.py) | [traits](https://luschny.de/math/oeis/Bessel2.html) | [A359760, A073278, A066325, A099174, A111924, A144299, A104556](https://oeis.org/search?q=%7Cid%3AA359760%7Cid%3AA073278%7Cid%3AA066325%7Cid%3AA099174%7Cid%3AA111924%7Cid%3AA144299%7Cid%3AA104556) | | [Binomial](https://github.com/PeterLuschny/tabl/blob/main/data/md/Binomial.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Binomial.py) | [traits](https://luschny.de/math/oeis/Binomial.html) | [A007318, A074909, A108086, A117440, A118433, A130595, A135278, A154926](https://oeis.org/search?q=%7Cid%3AA007318%7Cid%3AA074909%7Cid%3AA108086%7Cid%3AA117440%7Cid%3AA118433%7Cid%3AA130595%7Cid%3AA135278%7Cid%3AA154926) | | [Catalan](https://github.com/PeterLuschny/tabl/blob/main/data/md/Catalan.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Catalan.py) | [traits](https://luschny.de/math/oeis/Catalan.html) | [A128899, A039598](https://oeis.org/search?q=%7Cid%3AA128899%7Cid%3AA039598) | -| [CatalanAer](https://github.com/PeterLuschny/tabl/blob/main/data/md/CatalanAer.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CatalanAer.py) | [traits](https://luschny.de/math/oeis/CatalanAer.html) | [A052173, A053121, A112554, A322378](https://oeis.org/search?q=%7Cid%3AA052173%7Cid%3AA053121%7Cid%3AA112554%7Cid%3AA322378) | +| [CatalanAer](https://github.com/PeterLuschny/tabl/blob/main/data/md/CatalanAer.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CatalanAer.py) | [traits](https://luschny.de/math/oeis/CatalanAer.html) | [A053121, A052173, A112554, A322378](https://oeis.org/search?q=%7Cid%3AA053121%7Cid%3AA052173%7Cid%3AA112554%7Cid%3AA322378) | | [CatalanSqr](https://github.com/PeterLuschny/tabl/blob/main/data/md/CatalanSqr.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CatalanSqr.py) | [traits](https://luschny.de/math/oeis/CatalanSqr.html) | [A039599, A050155](https://oeis.org/search?q=%7Cid%3AA039599%7Cid%3AA050155) | -| [CentralCycle](https://github.com/PeterLuschny/tabl/blob/main/data/md/CentralCycle.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CentralCycle.py) | [traits](https://luschny.de/math/oeis/CentralCycle.html) | [A111999, A259456, A269940](https://oeis.org/search?q=%7Cid%3AA111999%7Cid%3AA259456%7Cid%3AA269940) | -| [CentralSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/CentralSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CentralSet.py) | [traits](https://luschny.de/math/oeis/CentralSet.html) | [A008957, A036969, A269945](https://oeis.org/search?q=%7Cid%3AA008957%7Cid%3AA036969%7Cid%3AA269945) | +| [CentralCycle](https://github.com/PeterLuschny/tabl/blob/main/data/md/CentralCycle.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CentralCycle.py) | [traits](https://luschny.de/math/oeis/CentralCycle.html) | [A269940, A111999, A259456](https://oeis.org/search?q=%7Cid%3AA269940%7Cid%3AA111999%7Cid%3AA259456) | +| [CentralSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/CentralSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CentralSet.py) | [traits](https://luschny.de/math/oeis/CentralSet.html) | [A269945, A008957, A036969](https://oeis.org/search?q=%7Cid%3AA269945%7Cid%3AA008957%7Cid%3AA036969) | | [ChebyshevS](https://github.com/PeterLuschny/tabl/blob/main/data/md/ChebyshevS.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/ChebyshevS.py) | [traits](https://luschny.de/math/oeis/ChebyshevS.html) | [A049310, A053119, A112552, A168561](https://oeis.org/search?q=%7Cid%3AA049310%7Cid%3AA053119%7Cid%3AA112552%7Cid%3AA168561) | -| [ChebyshevT](https://github.com/PeterLuschny/tabl/blob/main/data/md/ChebyshevT.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/ChebyshevT.py) | [traits](https://luschny.de/math/oeis/ChebyshevT.html) | [A039991, A053120, A081265](https://oeis.org/search?q=%7Cid%3AA039991%7Cid%3AA053120%7Cid%3AA081265) | +| [ChebyshevT](https://github.com/PeterLuschny/tabl/blob/main/data/md/ChebyshevT.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/ChebyshevT.py) | [traits](https://luschny.de/math/oeis/ChebyshevT.html) | [A053120, A039991, A081265](https://oeis.org/search?q=%7Cid%3AA053120%7Cid%3AA039991%7Cid%3AA081265) | | [ChebyshevU](https://github.com/PeterLuschny/tabl/blob/main/data/md/ChebyshevU.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/ChebyshevU.py) | [traits](https://luschny.de/math/oeis/ChebyshevU.html) | [A053117, A053118, A115322](https://oeis.org/search?q=%7Cid%3AA053117%7Cid%3AA053118%7Cid%3AA115322) | | [Composition](https://github.com/PeterLuschny/tabl/blob/main/data/md/Composition.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Composition.py) | [traits](https://luschny.de/math/oeis/Composition.html) | [A048004](https://oeis.org/search?q=%7Cid%3AA048004) | | [CompositionMax](https://github.com/PeterLuschny/tabl/blob/main/data/md/CompositionMax.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/CompositionMax.py) | [traits](https://luschny.de/math/oeis/CompositionMax.html) | [A126198](https://oeis.org/search?q=%7Cid%3AA126198) | | [ChristTree](https://github.com/PeterLuschny/tabl/blob/main/data/md/ChristTree.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/ChristTree.py) | [traits](https://luschny.de/math/oeis/ChristTree.html) | [A106465, A106470](https://oeis.org/search?q=%7Cid%3AA106465%7Cid%3AA106470) | | [Delannoy](https://github.com/PeterLuschny/tabl/blob/main/data/md/Delannoy.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Delannoy.py) | [traits](https://luschny.de/math/oeis/Delannoy.html) | [A008288](https://oeis.org/search?q=%7Cid%3AA008288) | -| [Euler](https://github.com/PeterLuschny/tabl/blob/main/data/md/Euler.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Euler.py) | [traits](https://luschny.de/math/oeis/Euler.html) | [A109449, A247453](https://oeis.org/search?q=%7Cid%3AA109449%7Cid%3AA247453) | -| [Eulerian](https://github.com/PeterLuschny/tabl/blob/main/data/md/Eulerian.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Eulerian.py) | [traits](https://luschny.de/math/oeis/Eulerian.html) | [A008292, A123125, A173018](https://oeis.org/search?q=%7Cid%3AA008292%7Cid%3AA123125%7Cid%3AA173018) | -| [Eulerian2](https://github.com/PeterLuschny/tabl/blob/main/data/md/Eulerian2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Eulerian2.py) | [traits](https://luschny.de/math/oeis/Eulerian2.html) | [A008517, A112007, A163936, A340556](https://oeis.org/search?q=%7Cid%3AA008517%7Cid%3AA112007%7Cid%3AA163936%7Cid%3AA340556) | +| [Euler](https://github.com/PeterLuschny/tabl/blob/main/data/md/Euler.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Euler.py) | [traits](https://luschny.de/math/oeis/Euler.html) | [A247453, A109449](https://oeis.org/search?q=%7Cid%3AA247453%7Cid%3AA109449) | +| [Eulerian](https://github.com/PeterLuschny/tabl/blob/main/data/md/Eulerian.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Eulerian.py) | [traits](https://luschny.de/math/oeis/Eulerian.html) | [A173018, A008292, A123125](https://oeis.org/search?q=%7Cid%3AA173018%7Cid%3AA008292%7Cid%3AA123125) | +| [Eulerian2](https://github.com/PeterLuschny/tabl/blob/main/data/md/Eulerian2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Eulerian2.py) | [traits](https://luschny.de/math/oeis/Eulerian2.html) | [A340556, A008517, A112007, A163936](https://oeis.org/search?q=%7Cid%3AA340556%7Cid%3AA008517%7Cid%3AA112007%7Cid%3AA163936) | | [EulerianB](https://github.com/PeterLuschny/tabl/blob/main/data/md/EulerianB.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/EulerianB.py) | [traits](https://luschny.de/math/oeis/EulerianB.html) | [A060187, A138076](https://oeis.org/search?q=%7Cid%3AA060187%7Cid%3AA138076) | | [EulerSec](https://github.com/PeterLuschny/tabl/blob/main/data/md/EulerSec.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/EulerSec.py) | [traits](https://luschny.de/math/oeis/EulerSec.html) | [A119879, A081658, A153641](https://oeis.org/search?q=%7Cid%3AA119879%7Cid%3AA081658%7Cid%3AA153641) | | [EulerTan](https://github.com/PeterLuschny/tabl/blob/main/data/md/EulerTan.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/EulerTan.py) | [traits](https://luschny.de/math/oeis/EulerTan.html) | [A162660, A350972, A155585, A009006, A000182](https://oeis.org/search?q=%7Cid%3AA162660%7Cid%3AA350972%7Cid%3AA155585%7Cid%3AA009006%7Cid%3AA000182) | | [FallingFact](https://github.com/PeterLuschny/tabl/blob/main/data/md/FallingFact.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/FallingFact.py) | [traits](https://luschny.de/math/oeis/FallingFact.html) | [A008279, A068424, A094587, A173333, A181511](https://oeis.org/search?q=%7Cid%3AA008279%7Cid%3AA068424%7Cid%3AA094587%7Cid%3AA173333%7Cid%3AA181511) | -| [Fibonacci](https://github.com/PeterLuschny/tabl/blob/main/data/md/Fibonacci.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Fibonacci.py) | [traits](https://luschny.de/math/oeis/Fibonacci.html) | [A105809, A228074, A354267](https://oeis.org/search?q=%7Cid%3AA105809%7Cid%3AA228074%7Cid%3AA354267) | -| [Fubini](https://github.com/PeterLuschny/tabl/blob/main/data/md/Fubini.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Fubini.py) | [traits](https://luschny.de/math/oeis/Fubini.html) | [A019538, A090582, A131689, A278075](https://oeis.org/search?q=%7Cid%3AA019538%7Cid%3AA090582%7Cid%3AA131689%7Cid%3AA278075) | -| [FussCatalan](https://github.com/PeterLuschny/tabl/blob/main/data/md/FussCatalan.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/FussCatalan.py) | [traits](https://luschny.de/math/oeis/FussCatalan.html) | [A030237, A054445, A355173](https://oeis.org/search?q=%7Cid%3AA030237%7Cid%3AA054445%7Cid%3AA355173) | +| [Fibonacci](https://github.com/PeterLuschny/tabl/blob/main/data/md/Fibonacci.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Fibonacci.py) | [traits](https://luschny.de/math/oeis/Fibonacci.html) | [A354267, A105809, A228074](https://oeis.org/search?q=%7Cid%3AA354267%7Cid%3AA105809%7Cid%3AA228074) | +| [Fubini](https://github.com/PeterLuschny/tabl/blob/main/data/md/Fubini.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Fubini.py) | [traits](https://luschny.de/math/oeis/Fubini.html) | [A131689, A019538, A090582, A278075](https://oeis.org/search?q=%7Cid%3AA131689%7Cid%3AA019538%7Cid%3AA090582%7Cid%3AA278075) | +| [FussCatalan](https://github.com/PeterLuschny/tabl/blob/main/data/md/FussCatalan.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/FussCatalan.py) | [traits](https://luschny.de/math/oeis/FussCatalan.html) | [A355173, A030237, A054445](https://oeis.org/search?q=%7Cid%3AA355173%7Cid%3AA030237%7Cid%3AA054445) | | [Gaussq2](https://github.com/PeterLuschny/tabl/blob/main/data/md/Gaussq2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Gaussq2.py) | [traits](https://luschny.de/math/oeis/Gaussq2.html) | [A022166](https://oeis.org/search?q=%7Cid%3AA022166) | | [Genocchi](https://github.com/PeterLuschny/tabl/blob/main/data/md/Genocchi.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Genocchi.py) | [traits](https://luschny.de/math/oeis/Genocchi.html) | [A297703](https://oeis.org/search?q=%7Cid%3AA297703) | -| [Harmonic](https://github.com/PeterLuschny/tabl/blob/main/data/md/Harmonic.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Harmonic.py) | [traits](https://luschny.de/math/oeis/Harmonic.html) | [A109822, A358694](https://oeis.org/search?q=%7Cid%3AA109822%7Cid%3AA358694) | -| [HermiteE](https://github.com/PeterLuschny/tabl/blob/main/data/md/HermiteE.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/HermiteE.py) | [traits](https://luschny.de/math/oeis/HermiteE.html) | [A066325, A073278, A099174](https://oeis.org/search?q=%7Cid%3AA066325%7Cid%3AA073278%7Cid%3AA099174) | +| [Harmonic](https://github.com/PeterLuschny/tabl/blob/main/data/md/Harmonic.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Harmonic.py) | [traits](https://luschny.de/math/oeis/Harmonic.html) | [A358694, A109822](https://oeis.org/search?q=%7Cid%3AA358694%7Cid%3AA109822) | +| [HermiteE](https://github.com/PeterLuschny/tabl/blob/main/data/md/HermiteE.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/HermiteE.py) | [traits](https://luschny.de/math/oeis/HermiteE.html) | [A099174, A066325, A073278](https://oeis.org/search?q=%7Cid%3AA099174%7Cid%3AA066325%7Cid%3AA073278) | | [HermiteH](https://github.com/PeterLuschny/tabl/blob/main/data/md/HermiteH.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/HermiteH.py) | [traits](https://luschny.de/math/oeis/HermiteH.html) | [A060821](https://oeis.org/search?q=%7Cid%3AA060821) | +| [LabeledGraphs](https://github.com/PeterLuschny/tabl/blob/main/data/md/LabeledGraphs.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/LabeledGraphs.py) | [traits](https://luschny.de/math/oeis/LabeledGraphs.html) | [A360603](https://oeis.org/search?q=%7Cid%3AA360603) | | [Laguerre](https://github.com/PeterLuschny/tabl/blob/main/data/md/Laguerre.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Laguerre.py) | [traits](https://luschny.de/math/oeis/Laguerre.html) | [A021009, A021010, A144084](https://oeis.org/search?q=%7Cid%3AA021009%7Cid%3AA021010%7Cid%3AA144084) | -| [Lah](https://github.com/PeterLuschny/tabl/blob/main/data/md/Lah.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Lah.py) | [traits](https://luschny.de/math/oeis/Lah.html) | [A008297, A066667, A089231, A105278, A111596, A271703](https://oeis.org/search?q=%7Cid%3AA008297%7Cid%3AA066667%7Cid%3AA089231%7Cid%3AA105278%7Cid%3AA111596%7Cid%3AA271703) | -| [Lehmer](https://github.com/PeterLuschny/tabl/blob/main/data/md/Lehmer.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Lehmer.py) | [traits](https://luschny.de/math/oeis/Lehmer.html) | [A039621, A354794](https://oeis.org/search?q=%7Cid%3AA039621%7Cid%3AA354794) | +| [Lah](https://github.com/PeterLuschny/tabl/blob/main/data/md/Lah.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Lah.py) | [traits](https://luschny.de/math/oeis/Lah.html) | [A271703, A008297, A066667, A089231, A105278, A111596](https://oeis.org/search?q=%7Cid%3AA271703%7Cid%3AA008297%7Cid%3AA066667%7Cid%3AA089231%7Cid%3AA105278%7Cid%3AA111596) | +| [Lehmer](https://github.com/PeterLuschny/tabl/blob/main/data/md/Lehmer.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Lehmer.py) | [traits](https://luschny.de/math/oeis/Lehmer.html) | [A354794, A039621](https://oeis.org/search?q=%7Cid%3AA354794%7Cid%3AA039621) | | [Leibniz](https://github.com/PeterLuschny/tabl/blob/main/data/md/Leibniz.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Leibniz.py) | [traits](https://luschny.de/math/oeis/Leibniz.html) | [A003506](https://oeis.org/search?q=%7Cid%3AA003506) | | [Levin](https://github.com/PeterLuschny/tabl/blob/main/data/md/Levin.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Levin.py) | [traits](https://luschny.de/math/oeis/Levin.html) | [A356546](https://oeis.org/search?q=%7Cid%3AA356546) | | [Lozanic](https://github.com/PeterLuschny/tabl/blob/main/data/md/Lozanic.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Lozanic.py) | [traits](https://luschny.de/math/oeis/Lozanic.html) | [A034851](https://oeis.org/search?q=%7Cid%3AA034851) | | [Motzkin](https://github.com/PeterLuschny/tabl/blob/main/data/md/Motzkin.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Motzkin.py) | [traits](https://luschny.de/math/oeis/Motzkin.html) | [A359364](https://oeis.org/search?q=%7Cid%3AA359364) | -| [MotzkinGF](https://github.com/PeterLuschny/tabl/blob/main/data/md/MotzkinGF.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/MotzkinGF.py) | [traits](https://luschny.de/math/oeis/MotzkinGF.html) | [A026300, A064189, A009766](https://oeis.org/search?q=%7Cid%3AA026300%7Cid%3AA064189%7Cid%3AA009766) | -| [Narayana](https://github.com/PeterLuschny/tabl/blob/main/data/md/Narayana.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Narayana.py) | [traits](https://luschny.de/math/oeis/Narayana.html) | [A001263, A090181, A131198](https://oeis.org/search?q=%7Cid%3AA001263%7Cid%3AA090181%7Cid%3AA131198) | +| [MotzkinGF](https://github.com/PeterLuschny/tabl/blob/main/data/md/MotzkinGF.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/MotzkinGF.py) | [traits](https://luschny.de/math/oeis/MotzkinGF.html) | [A064189, A026300, A009766](https://oeis.org/search?q=%7Cid%3AA064189%7Cid%3AA026300%7Cid%3AA009766) | +| [Narayana](https://github.com/PeterLuschny/tabl/blob/main/data/md/Narayana.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Narayana.py) | [traits](https://luschny.de/math/oeis/Narayana.html) | [A090181, A001263, A131198](https://oeis.org/search?q=%7Cid%3AA090181%7Cid%3AA001263%7Cid%3AA131198) | | [Nicomachus](https://github.com/PeterLuschny/tabl/blob/main/data/md/Nicomachus.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Nicomachus.py) | [traits](https://luschny.de/math/oeis/Nicomachus.html) | [A036561, A081954, A175840](https://oeis.org/search?q=%7Cid%3AA036561%7Cid%3AA081954%7Cid%3AA175840) | | [One](https://github.com/PeterLuschny/tabl/blob/main/data/md/One.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/One.py) | [traits](https://luschny.de/math/oeis/One.html) | [A000012, A008836, A014077](https://oeis.org/search?q=%7Cid%3AA000012%7Cid%3AA008836%7Cid%3AA014077) | -| [Ordinals](https://github.com/PeterLuschny/tabl/blob/main/data/md/Ordinals.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Ordinals.py) | [traits](https://luschny.de/math/oeis/Ordinals.html) | [A002260, A002262, A004736, A025581](https://oeis.org/search?q=%7Cid%3AA002260%7Cid%3AA002262%7Cid%3AA004736%7Cid%3AA025581) | -| [OrderedCycle](https://github.com/PeterLuschny/tabl/blob/main/data/md/OrderedCycle.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/OrderedCycle.py) | [traits](https://luschny.de/math/oeis/OrderedCycle.html) | [A048594, A075181, A225479](https://oeis.org/search?q=%7Cid%3AA048594%7Cid%3AA075181%7Cid%3AA225479) | -| [Partition](https://github.com/PeterLuschny/tabl/blob/main/data/md/Partition.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Partition.py) | [traits](https://luschny.de/math/oeis/Partition.html) | [A008284, A058398, A072233](https://oeis.org/search?q=%7Cid%3AA008284%7Cid%3AA058398%7Cid%3AA072233) | -| [PartitionMax](https://github.com/PeterLuschny/tabl/blob/main/data/md/PartitionMax.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/PartitionMax.py) | [traits](https://luschny.de/math/oeis/PartitionMax.html) | [A008284, A058398, A072233](https://oeis.org/search?q=%7Cid%3AA008284%7Cid%3AA058398%7Cid%3AA072233) | -| [Polygonal](https://github.com/PeterLuschny/tabl/blob/main/data/md/Polygonal.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Polygonal.py) | [traits](https://luschny.de/math/oeis/Polygonal.html) | [A057145, A134394, A139600, A139601](https://oeis.org/search?q=%7Cid%3AA057145%7Cid%3AA134394%7Cid%3AA139600%7Cid%3AA139601) | -| [PowLaguerre](https://github.com/PeterLuschny/tabl/blob/main/data/md/PowLaguerre.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/PowLaguerre.py) | [traits](https://luschny.de/math/oeis/PowLaguerre.html) | [A021012, A196347](https://oeis.org/search?q=%7Cid%3AA021012%7Cid%3AA196347) | +| [Ordinals](https://github.com/PeterLuschny/tabl/blob/main/data/md/Ordinals.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Ordinals.py) | [traits](https://luschny.de/math/oeis/Ordinals.html) | [A002262, A002260, A004736, A025581](https://oeis.org/search?q=%7Cid%3AA002262%7Cid%3AA002260%7Cid%3AA004736%7Cid%3AA025581) | +| [OrderedCycle](https://github.com/PeterLuschny/tabl/blob/main/data/md/OrderedCycle.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/OrderedCycle.py) | [traits](https://luschny.de/math/oeis/OrderedCycle.html) | [A225479, A048594, A075181](https://oeis.org/search?q=%7Cid%3AA225479%7Cid%3AA048594%7Cid%3AA075181) | +| [Partition](https://github.com/PeterLuschny/tabl/blob/main/data/md/Partition.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Partition.py) | [traits](https://luschny.de/math/oeis/Partition.html) | [A072233, A008284, A058398](https://oeis.org/search?q=%7Cid%3AA072233%7Cid%3AA008284%7Cid%3AA058398) | +| [PartitionMax](https://github.com/PeterLuschny/tabl/blob/main/data/md/PartitionMax.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/PartitionMax.py) | [traits](https://luschny.de/math/oeis/PartitionMax.html) | [A026820](https://oeis.org/search?q=%7Cid%3AA026820) | +| [Polygonal](https://github.com/PeterLuschny/tabl/blob/main/data/md/Polygonal.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Polygonal.py) | [traits](https://luschny.de/math/oeis/Polygonal.html) | [A139600, A057145, A134394, A139601](https://oeis.org/search?q=%7Cid%3AA139600%7Cid%3AA057145%7Cid%3AA134394%7Cid%3AA139601) | +| [PowLaguerre](https://github.com/PeterLuschny/tabl/blob/main/data/md/PowLaguerre.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/PowLaguerre.py) | [traits](https://luschny.de/math/oeis/PowLaguerre.html) | [A196347, A021012](https://oeis.org/search?q=%7Cid%3AA196347%7Cid%3AA021012) | | [Rencontres](https://github.com/PeterLuschny/tabl/blob/main/data/md/Rencontres.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Rencontres.py) | [traits](https://luschny.de/math/oeis/Rencontres.html) | [A008290, A098825](https://oeis.org/search?q=%7Cid%3AA008290%7Cid%3AA098825) | | [RisingFact](https://github.com/PeterLuschny/tabl/blob/main/data/md/RisingFact.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/RisingFact.py) | [traits](https://luschny.de/math/oeis/RisingFact.html) | [A008279, A068424, A094587, A173333, A181511](https://oeis.org/search?q=%7Cid%3AA008279%7Cid%3AA068424%7Cid%3AA094587%7Cid%3AA173333%7Cid%3AA181511) | -| [Schroeder](https://github.com/PeterLuschny/tabl/blob/main/data/md/Schroeder.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Schroeder.py) | [traits](https://luschny.de/math/oeis/Schroeder.html) | [A033877, A080245, A080247, A122538, A106579](https://oeis.org/search?q=%7Cid%3AA033877%7Cid%3AA080245%7Cid%3AA080247%7Cid%3AA122538%7Cid%3AA106579) | -| [SchroederP](https://github.com/PeterLuschny/tabl/blob/main/data/md/SchroederP.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SchroederP.py) | [traits](https://luschny.de/math/oeis/SchroederP.html) | [A063007, A104684](https://oeis.org/search?q=%7Cid%3AA063007%7Cid%3AA104684) | +| [Schroeder](https://github.com/PeterLuschny/tabl/blob/main/data/md/Schroeder.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Schroeder.py) | [traits](https://luschny.de/math/oeis/Schroeder.html) | [A122538, A033877, A080245, A080247, A106579](https://oeis.org/search?q=%7Cid%3AA122538%7Cid%3AA033877%7Cid%3AA080245%7Cid%3AA080247%7Cid%3AA106579) | +| [SchroederL](https://github.com/PeterLuschny/tabl/blob/main/data/md/SchroederL.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SchroederL.py) | [traits](https://luschny.de/math/oeis/SchroederL.html) | [A172094](https://oeis.org/search?q=%7Cid%3AA172094) | +| [SchroederP](https://github.com/PeterLuschny/tabl/blob/main/data/md/SchroederP.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SchroederP.py) | [traits](https://luschny.de/math/oeis/SchroederP.html) | [A104684, A063007](https://oeis.org/search?q=%7Cid%3AA104684%7Cid%3AA063007) | | [Seidel](https://github.com/PeterLuschny/tabl/blob/main/data/md/Seidel.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Seidel.py) | [traits](https://luschny.de/math/oeis/Seidel.html) | [A008281, A008282, A010094](https://oeis.org/search?q=%7Cid%3AA008281%7Cid%3AA008282%7Cid%3AA010094) | | [SeidelBoust](https://github.com/PeterLuschny/tabl/blob/main/data/md/SeidelBoust.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SeidelBoust.py) | [traits](https://luschny.de/math/oeis/SeidelBoust.html) | [A008280, A108040, A236935, A239005](https://oeis.org/search?q=%7Cid%3AA008280%7Cid%3AA108040%7Cid%3AA236935%7Cid%3AA239005) | | [Sierpinski](https://github.com/PeterLuschny/tabl/blob/main/data/md/Sierpinski.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Sierpinski.py) | [traits](https://luschny.de/math/oeis/Sierpinski.html) | [A047999, A090971, A114700, A143200, A166282](https://oeis.org/search?q=%7Cid%3AA047999%7Cid%3AA090971%7Cid%3AA114700%7Cid%3AA143200%7Cid%3AA166282) | -| [StirlingCyc](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingCyc.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingCyc.py) | [traits](https://luschny.de/math/oeis/StirlingCyc.html) | [A008275, A008276, A048994, A054654, A094638, A130534, A132393](https://oeis.org/search?q=%7Cid%3AA008275%7Cid%3AA008276%7Cid%3AA048994%7Cid%3AA054654%7Cid%3AA094638%7Cid%3AA130534%7Cid%3AA132393) | +| [StirlingCyc](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingCyc.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingCyc.py) | [traits](https://luschny.de/math/oeis/StirlingCyc.html) | [A132393, A008275, A008276, A048994, A054654, A094638, A130534](https://oeis.org/search?q=%7Cid%3AA132393%7Cid%3AA008275%7Cid%3AA008276%7Cid%3AA048994%7Cid%3AA054654%7Cid%3AA094638%7Cid%3AA130534) | | [StirlingCyc2](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingCyc2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingCyc2.py) | [traits](https://luschny.de/math/oeis/StirlingCyc2.html) | [A358622, A008306, A106828](https://oeis.org/search?q=%7Cid%3AA358622%7Cid%3AA008306%7Cid%3AA106828) | | [StirlingCycB](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingCycB.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingCycB.py) | [traits](https://luschny.de/math/oeis/StirlingCycB.html) | [A028338, A039757, A039758, A109692](https://oeis.org/search?q=%7Cid%3AA028338%7Cid%3AA039757%7Cid%3AA039758%7Cid%3AA109692) | -| [StirlingSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingSet.py) | [traits](https://luschny.de/math/oeis/StirlingSet.html) | [A008277, A008278, A048993, A080417, A106800, A151511, A151512, A154959, A213735](https://oeis.org/search?q=%7Cid%3AA008277%7Cid%3AA008278%7Cid%3AA048993%7Cid%3AA080417%7Cid%3AA106800%7Cid%3AA151511%7Cid%3AA151512%7Cid%3AA154959%7Cid%3AA213735) | +| [StirlingSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingSet.py) | [traits](https://luschny.de/math/oeis/StirlingSet.html) | [A048993, A008277, A008278, A080417, A106800, A151511, A151512, A154959, A213735](https://oeis.org/search?q=%7Cid%3AA048993%7Cid%3AA008277%7Cid%3AA008278%7Cid%3AA080417%7Cid%3AA106800%7Cid%3AA151511%7Cid%3AA151512%7Cid%3AA154959%7Cid%3AA213735) | | [StirlingSet2](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingSet2.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingSet2.py) | [traits](https://luschny.de/math/oeis/StirlingSet2.html) | [A358623, A008299, A137375](https://oeis.org/search?q=%7Cid%3AA358623%7Cid%3AA008299%7Cid%3AA137375) | | [StirlingSetB](https://github.com/PeterLuschny/tabl/blob/main/data/md/StirlingSetB.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/StirlingSetB.py) | [traits](https://luschny.de/math/oeis/StirlingSetB.html) | [A154602](https://oeis.org/search?q=%7Cid%3AA154602) | | [Sylvester](https://github.com/PeterLuschny/tabl/blob/main/data/md/Sylvester.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Sylvester.py) | [traits](https://luschny.de/math/oeis/Sylvester.html) | [A341101](https://oeis.org/search?q=%7Cid%3AA341101) | -| [SymPoly](https://github.com/PeterLuschny/tabl/blob/main/data/md/SymPoly.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SymPoly.py) | [traits](https://luschny.de/math/oeis/SymPoly.html) | [A093905, A105954, A165674, A165675](https://oeis.org/search?q=%7Cid%3AA093905%7Cid%3AA105954%7Cid%3AA165674%7Cid%3AA165675) | +| [SymPoly](https://github.com/PeterLuschny/tabl/blob/main/data/md/SymPoly.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/SymPoly.py) | [traits](https://luschny.de/math/oeis/SymPoly.html) | [A165675, A093905, A105954, A165674](https://oeis.org/search?q=%7Cid%3AA165675%7Cid%3AA093905%7Cid%3AA105954%7Cid%3AA165674) | | [TernaryTrees](https://github.com/PeterLuschny/tabl/blob/main/data/md/TernaryTrees.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/TernaryTrees.py) | [traits](https://luschny.de/math/oeis/TernaryTrees.html) | [A355172](https://oeis.org/search?q=%7Cid%3AA355172) | -| [WardSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/WardSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/WardSet.py) | [traits](https://luschny.de/math/oeis/WardSet.html) | [A134991, A269939](https://oeis.org/search?q=%7Cid%3AA134991%7Cid%3AA269939) | +| [WardSet](https://github.com/PeterLuschny/tabl/blob/main/data/md/WardSet.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/WardSet.py) | [traits](https://luschny.de/math/oeis/WardSet.html) | [A269939, A134991](https://oeis.org/search?q=%7Cid%3AA269939%7Cid%3AA134991) | | [Worpitzky](https://github.com/PeterLuschny/tabl/blob/main/data/md/Worpitzky.md) | [source](https://github.com/PeterLuschny/tabl/blob/main/src/Worpitzky.py) | [traits](https://luschny.de/math/oeis/Worpitzky.html) | [A028246, A053440, A075263, A130850, A163626](https://oeis.org/search?q=%7Cid%3AA028246%7Cid%3AA053440%7Cid%3AA075263%7Cid%3AA130850%7Cid%3AA163626) | diff --git a/data/csv/Abel.csv b/data/csv/Abel.csv index 9175fb8..5d2b261 100644 --- a/data/csv/Abel.csv +++ b/data/csv/Abel.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Abel,A137452,Tabl,[1 0 1 0 2 1 0 9 6 1 0 64 48 12 1 0 625 500 150 20 1 0 7776 6480 ] Abel,A139526,RevTabl,[1 1 0 1 2 0 1 6 9 0 1 12 48 64 0 1 20 150 500 625 0 1 30 360 2160 ] Abel,A059297,InvTabl,[1 0 1 0 -2 1 0 3 -6 1 0 -4 24 -12 1 0 5 -80 90 -20 1 0 -6 240 -540 ] -Abel,,InvRevTabl,[] Abel,A059299,RevInvTabl,[1 1 0 1 -2 0 1 -6 3 0 1 -12 24 -4 0 1 -20 90 -80 5 0 1 -30 240 ] +Abel,,InvRevTabl,[] Abel,,AccTabl,[1 0 1 0 2 3 0 9 15 16 0 64 112 124 125 0 625 1125 1275 1295 1296 0 ] Abel,,RevAccTabl,[1 1 0 3 2 0 16 15 9 0 125 124 112 64 0 1296 1295 1275 1125 625 0 ] Abel,,AccRevTabl,[1 1 1 1 3 3 1 7 16 16 1 13 61 125 125 1 21 171 671 1296 1296 1 31 ] Abel,,AntiDiagTabl,[1 0 0 1 0 2 0 9 1 0 64 6 0 625 48 1 0 7776 500 12 0 117649 6480 ] -Abel,,DiffxTabl,[1 0 2 0 4 3 0 18 18 4 0 128 144 48 5 0 1250 1500 600 100 6 0 15552 ] Abel,,BinTabl,[1 0 1 0 2 5 0 9 24 46 0 64 176 348 593 0 625 1750 3525 6120 9726 0 ] Abel,,InvBinTabl,[1 0 1 0 2 -3 0 9 -12 10 0 64 -80 60 -15 0 625 -750 525 -80 -474 0 ] +Abel,,DiffxTabl,[1 0 2 0 4 3 0 18 18 4 0 128 144 48 5 0 1250 1500 600 100 6 0 15552 ] Abel,A000272,RowSum,[1 1 3 16 125 1296 16807 262144 4782969 100000000 2357947691 ] Abel,A274278,EvenSum,[1 0 1 6 49 520 6841 107744 1979713 41611392 985263601 25958682112 ] Abel,A195136,OddSum,[0 1 2 10 76 776 9966 154400 2803256 58388608 1372684090 ] @@ -28,9 +28,9 @@ Abel,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Abel,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Abel,,BinConv,[1 1 5 46 593 9726 192637 4457580 117769409 3492894070 114790042901 ] Abel,,InvBinConv,[1 1 -3 10 -15 -474 12565 -258572 5136705 -102255290 2019481101 ] -Abel,A225497,TransSqrs,[0 1 6 42 380 4320 59682 974848 18423288 396000000 9548713790 ] Abel,A089946,TransNat0,[0 1 4 24 200 2160 28812 458752 8503056 180000000 4287177620 ] Abel,,TransNat1,[1 2 7 40 325 3456 45619 720896 13286025 280000000 6645125311 ] +Abel,A225497,TransSqrs,[0 1 6 42 380 4320 59682 974848 18423288 396000000 9548713790 ] Abel,A002378,DiagRow1,[0 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] Abel,,DiagRow2,[0 9 48 150 360 735 1344 2268 3600 5445 7920 11154 15288 20475 ] Abel,,DiagRow3,[0 64 500 2160 6860 17920 40824 84000 159720 285120 483340 784784 ] diff --git a/data/csv/Baxter.csv b/data/csv/Baxter.csv index 4c9bb75..f144f2e 100644 --- a/data/csv/Baxter.csv +++ b/data/csv/Baxter.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Baxter,A359363,Tabl,[1 0 1 0 1 1 0 1 4 1 0 1 10 10 1 0 1 20 50 20 1 0 1 35 175 175 35 1 ] Baxter,A359363,RevTabl,[1 1 0 1 1 0 1 4 1 0 1 10 10 1 0 1 20 50 20 1 0 1 35 175 175 35 1 0 ] Baxter,,InvTabl,[1 0 1 0 -1 1 0 3 -4 1 0 -21 30 -10 1 0 289 -420 150 -20 1 0 -6931 ] -Baxter,,InvRevTabl,[] Baxter,,RevInvTabl,[1 1 0 1 -1 0 1 -4 3 0 1 -10 30 -21 0 1 -20 150 -420 289 0 1 -35 ] +Baxter,,InvRevTabl,[] Baxter,,AccTabl,[1 0 1 0 1 2 0 1 5 6 0 1 11 21 22 0 1 21 71 91 92 0 1 36 211 386 ] Baxter,,RevAccTabl,[1 1 0 2 1 0 6 5 1 0 22 21 11 1 0 92 91 71 21 1 0 422 421 386 211 ] Baxter,,AccRevTabl,[1 1 1 1 2 2 1 5 6 6 1 11 21 22 22 1 21 71 91 92 92 1 36 211 386 ] Baxter,,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 1 4 0 1 10 1 0 1 20 10 0 1 35 50 1 0 1 56 175 ] -Baxter,,DiffxTabl,[1 0 2 0 2 3 0 2 12 4 0 2 30 40 5 0 2 60 200 100 6 0 2 105 700 875 ] Baxter,,BinTabl,[1 0 1 0 1 3 0 1 6 16 0 1 12 43 105 0 1 22 113 344 806 0 1 37 283 ] Baxter,,InvBinTabl,[1 0 1 0 1 -1 0 1 2 -8 0 1 8 -17 17 0 1 18 -7 -64 206 0 1 33 73 ] +Baxter,,DiffxTabl,[1 0 2 0 2 3 0 2 12 4 0 2 30 40 5 0 2 60 200 100 6 0 2 105 700 875 ] Baxter,A001181,RowSum,[1 1 2 6 22 92 422 2074 10754 58202 326240 1882960 11140560 ] Baxter,,EvenSum,[1 0 1 4 11 40 211 1092 5377 28464 163120 950048 5570280 33537088 ] Baxter,,OddSum,[0 1 1 2 11 52 211 982 5377 29738 163120 932912 5570280 33792904 ] @@ -28,9 +28,9 @@ Baxter,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Baxter,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Baxter,,BinConv,[1 1 3 16 105 806 6867 63316 620433 6383026 68335124 756256788 ] Baxter,,InvBinConv,[1 1 -1 -8 17 206 -565 -8420 26369 445186 -1513092 -27773844 ] -Baxter,,TransSqrs,[0 1 5 26 147 876 5427 34630 226193 1505626 10180040 69742960 ] Baxter,A350265,TransNat0,[0 1 3 12 55 276 1477 8296 48393 291010 1794320 11297760 72413640 ] Baxter,,TransNat1,[1 2 5 18 77 368 1899 10370 59147 349212 2120560 13180720 83554200 ] +Baxter,,TransSqrs,[0 1 5 26 147 876 5427 34630 226193 1505626 10180040 69742960 ] Baxter,A000292,DiagRow1,[0 1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 ] Baxter,A006542,DiagRow2,[0 1 10 50 175 490 1176 2520 4950 9075 15730 26026 41405 63700 ] Baxter,A047819,DiagRow3,[0 1 20 175 980 4116 14112 41580 108900 259545 572572 1184183 ] diff --git a/data/csv/Bell.csv b/data/csv/Bell.csv index 69daf34..4b598ca 100644 --- a/data/csv/Bell.csv +++ b/data/csv/Bell.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Bell,A011971,Tabl,[1 1 2 2 3 5 5 7 10 15 15 20 27 37 52 52 67 87 114 151 203 203 255 ] Bell,A123346,RevTabl,[1 2 1 5 3 2 15 10 7 5 52 37 27 20 15 203 151 114 87 67 52 877 674 ] Bell,,InvTabl,[] -Bell,,InvRevTabl,[] Bell,,RevInvTabl,[] +Bell,,InvRevTabl,[] Bell,,AccTabl,[1 1 3 2 5 10 5 12 22 37 15 35 62 99 151 52 119 206 320 471 674 203 ] Bell,,RevAccTabl,[1 3 1 10 5 2 37 22 12 5 151 99 62 35 15 674 471 320 206 119 52 ] Bell,,AccRevTabl,[1 2 3 5 8 10 15 25 32 37 52 89 116 136 151 203 354 468 555 622 674 ] Bell,,AntiDiagTabl,[1 1 2 2 5 3 15 7 5 52 20 10 203 67 27 15 877 255 87 37 4140 1080 ] -Bell,,DiffxTabl,[1 1 4 2 6 15 5 14 30 60 15 40 81 148 260 52 134 261 456 755 1218 ] Bell,,BinTabl,[1 1 3 2 5 13 5 12 29 71 15 35 82 193 457 52 119 273 628 1449 3355 ] Bell,,InvBinTabl,[1 1 1 2 1 1 5 2 1 1 15 5 2 1 1 52 15 5 2 1 1 203 52 15 5 2 1 1 877 ] +Bell,,DiffxTabl,[1 1 4 2 6 15 5 14 30 60 15 40 81 148 260 52 134 261 456 755 1218 ] Bell,A005493,RowSum,[1 3 10 37 151 674 3263 17007 94828 562595 3535027 23430840 ] Bell,,EvenSum,[1 1 7 15 94 290 1925 7541 54217 254189 1979704 10725568 90086877 ] Bell,,OddSum,[0 2 3 22 57 384 1338 9466 40611 308406 1555323 12705272 73168008 ] @@ -28,9 +28,9 @@ Bell,A000110,ColLeft,[1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 276 Bell,A000110,ColRight,[1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 ] Bell,A126390,BinConv,[1 3 13 71 457 3355 27509 248127 2434129 25741939 291397789 ] Bell,A000012,InvBinConv,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -Bell,,TransSqrs,[0 2 23 182 1293 8932 62014 439442 3202127 24081458 187213595 ] Bell,,TransNat0,[0 2 13 72 393 2202 12850 78488 502327 3366648 23597297 172691956 ] Bell,A278677,TransNat1,[1 5 23 109 544 2876 16113 95495 597155 3929243 27132324 196122796 ] +Bell,,TransSqrs,[0 2 23 182 1293 8932 62014 439442 3202127 24081458 187213595 ] Bell,A005493,DiagRow1,[1 3 10 37 151 674 3263 17007 94828 562595 3535027 23430840 ] Bell,A011965,DiagRow2,[2 7 27 114 523 2589 13744 77821 467767 2972432 19895813 139824045 ] Bell,A011966,DiagRow3,[5 20 87 409 2066 11155 64077 389946 2504665 16923381 119928232 ] diff --git a/data/csv/Bessel.csv b/data/csv/Bessel.csv index 8c2a9f6..7106fd4 100644 --- a/data/csv/Bessel.csv +++ b/data/csv/Bessel.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -Bessel,A122850,Tabl,[1 0 1 0 1 1 0 3 3 1 0 15 15 6 1 0 105 105 45 10 1 0 945 945 420 ] +Bessel,A132062,Tabl,[1 0 1 0 1 1 0 3 3 1 0 15 15 6 1 0 105 105 45 10 1 0 945 945 420 ] Bessel,A104548,RevTabl,[1 1 0 1 1 0 1 3 3 0 1 6 15 15 0 1 10 45 105 105 0 1 15 105 420 945 ] Bessel,A122848,InvTabl,[1 0 1 0 -1 1 0 0 -3 1 0 0 3 -6 1 0 0 0 15 -10 1 0 0 0 -15 45 -15 1 ] -Bessel,,InvRevTabl,[] Bessel,A144299,RevInvTabl,[1 1 0 1 -1 0 1 -3 0 0 1 -6 3 0 0 1 -10 15 0 0 0 1 -15 45 -15 0 0 0 ] +Bessel,,InvRevTabl,[] Bessel,,AccTabl,[1 0 1 0 1 2 0 3 6 7 0 15 30 36 37 0 105 210 255 265 266 0 945 1890 ] Bessel,,RevAccTabl,[1 1 0 2 1 0 7 6 3 0 37 36 30 15 0 266 265 255 210 105 0 2431 2430 ] Bessel,,AccRevTabl,[1 1 1 1 2 2 1 4 7 7 1 7 22 37 37 1 11 56 161 266 266 1 16 121 541 ] Bessel,,AntiDiagTabl,[1 0 0 1 0 1 0 3 1 0 15 3 0 105 15 1 0 945 105 6 0 10395 945 45 1 0 ] -Bessel,,DiffxTabl,[1 0 2 0 2 3 0 6 9 4 0 30 45 24 5 0 210 315 180 50 6 0 1890 2835 ] Bessel,,BinTabl,[1 0 1 0 1 3 0 3 9 19 0 15 45 96 175 0 105 315 675 1240 2076 0 945 ] Bessel,,InvBinTabl,[1 0 1 0 1 -1 0 3 -3 1 0 15 -15 6 7 0 105 -105 45 40 -124 0 945 ] +Bessel,,DiffxTabl,[1 0 2 0 2 3 0 6 9 4 0 30 45 24 5 0 210 315 180 50 6 0 1890 2835 ] Bessel,A144301,RowSum,[1 1 2 7 37 266 2431 27007 353522 5329837 90960751 1733584106 ] Bessel,A036244,EvenSum,[1 0 1 3 16 115 1051 11676 152839 2304261 39325276 749484505 ] Bessel,A025164,OddSum,[0 1 1 4 21 151 1380 15331 200683 3025576 51635475 984099601 ] @@ -28,9 +28,9 @@ Bessel,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Bessel,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Bessel,,BinConv,[1 1 3 19 175 2076 29911 505093 9757539 211883500 5102919316 ] Bessel,,InvBinConv,[1 1 -1 1 7 -124 1591 -19991 256691 -3335228 41759236 -430787224 ] -Bessel,,TransSqrs,[0 1 5 24 145 1115 10596 120715 1607999 24555060 423178255 ] Bessel,A107103,TransNat0,[0 1 3 12 67 495 4596 51583 680037 10306152 176591665 3376207461 ] Bessel,,TransNat1,[1 2 5 19 104 761 7027 78590 1033559 15635989 267552416 5109791567 ] +Bessel,,TransSqrs,[0 1 5 24 145 1115 10596 120715 1607999 24555060 423178255 ] Bessel,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] Bessel,A050534,DiagRow2,[0 3 15 45 105 210 378 630 990 1485 2145 3003 4095 5460 7140 9180 ] Bessel,A240440,DiagRow3,[0 15 105 420 1260 3150 6930 13860 25740 45045 75075 120120 185640 ] diff --git a/data/csv/Bessel2.csv b/data/csv/Bessel2.csv index e221380..e266d66 100644 --- a/data/csv/Bessel2.csv +++ b/data/csv/Bessel2.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -Bessel2,A073278,Tabl,[1 1 0 1 0 1 1 0 3 0 1 0 6 0 3 1 0 10 0 15 0 1 0 15 0 45 0 15 1 0 ] +Bessel2,A359760,Tabl,[1 1 0 1 0 1 1 0 3 0 1 0 6 0 3 1 0 10 0 15 0 1 0 15 0 45 0 15 1 0 ] Bessel2,A066325,RevTabl,[1 0 1 1 0 1 0 3 0 1 3 0 6 0 1 0 15 0 10 0 1 15 0 45 0 15 0 1 0 105 ] Bessel2,,InvTabl,[] -Bessel2,A066325,InvRevTabl,[1 0 1 -1 0 1 0 -3 0 1 3 0 -6 0 1 0 15 0 -10 0 1 -15 0 45 0 -15 0 1 ] Bessel2,,RevInvTabl,[] +Bessel2,A066325,InvRevTabl,[1 0 1 -1 0 1 0 -3 0 1 3 0 -6 0 1 0 15 0 -10 0 1 -15 0 45 0 -15 0 1 ] Bessel2,,AccTabl,[1 1 1 1 1 2 1 1 4 4 1 1 7 7 10 1 1 11 11 26 26 1 1 16 16 61 61 76 ] Bessel2,,RevAccTabl,[1 1 1 2 1 1 4 4 1 1 10 7 7 1 1 26 26 11 11 1 1 76 61 61 16 16 1 1 ] Bessel2,,AccRevTabl,[1 0 1 1 1 2 0 3 3 4 3 3 9 9 10 0 15 15 25 25 26 15 15 60 60 75 75 ] Bessel2,,AntiDiagTabl,[1 1 1 0 1 0 1 0 1 1 0 3 1 0 6 0 1 0 10 0 1 0 15 0 3 1 0 21 0 15 1 ] -Bessel2,,DiffxTabl,[1 1 0 1 0 3 1 0 9 0 1 0 18 0 15 1 0 30 0 75 0 1 0 45 0 225 0 105 1 ] Bessel2,,BinTabl,[1 1 1 1 1 2 1 1 4 10 1 1 7 19 40 1 1 11 31 76 176 1 1 16 46 136 ] Bessel2,,InvBinTabl,[1 1 -1 1 -1 2 1 -1 4 -10 1 -1 7 -19 40 1 -1 11 -31 76 -176 1 -1 16 ] +Bessel2,,DiffxTabl,[1 1 0 1 0 3 1 0 9 0 1 0 18 0 15 1 0 30 0 75 0 1 0 45 0 225 0 105 1 ] Bessel2,A000085,RowSum,[1 1 2 4 10 26 76 232 764 2620 9496 35696 140152 568504 2390480 ] Bessel2,A000085,EvenSum,[1 1 2 4 10 26 76 232 764 2620 9496 35696 140152 568504 2390480 ] Bessel2,A000004,OddSum,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] @@ -28,9 +28,9 @@ Bessel2,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Bessel2,A123022,ColRight,[1 0 1 0 3 0 15 0 105 0 945 0 10395 0 135135 0 2027025 0 34459425 0 ] Bessel2,A344501,BinConv,[1 1 2 10 40 176 916 4852 27350 163270 1009396 6504356 43400512 ] Bessel2,A344501,InvBinConv,[1 -1 2 -10 40 -176 916 -4852 27350 -163270 1009396 -6504356 ] -Bessel2,,TransSqrs,[0 0 4 12 72 280 1320 5544 25312 112032 520560 2413840 11583264 ] Bessel2,,TransNat0,[0 0 2 6 24 80 300 1092 4256 16704 68760 288200 1253472 5568576 ] Bessel2,,TransNat1,[1 1 4 10 34 106 376 1324 5020 19324 78256 323896 1393624 6137080 ] +Bessel2,,TransSqrs,[0 0 4 12 72 280 1320 5544 25312 112032 520560 2413840 11583264 ] Bessel2,A123022,DiagRow1,[1 0 3 0 15 0 105 0 945 0 10395 0 135135 0 2027025 0 34459425 0 ] Bessel2,A001879,DiagRow2,[1 0 6 0 45 0 420 0 4725 0 62370 0 945945 0 16216200 0 310134825 0 ] Bessel2,A000457,DiagRow3,[1 0 10 0 105 0 1260 0 17325 0 270270 0 4729725 0 91891800 0 ] diff --git a/data/csv/Binomial.csv b/data/csv/Binomial.csv index 1632d6a..52edc58 100644 --- a/data/csv/Binomial.csv +++ b/data/csv/Binomial.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Binomial,A007318,Tabl,[1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 ] Binomial,A007318,RevTabl,[1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 ] Binomial,A007318,InvTabl,[1 -1 1 1 -2 1 -1 3 -3 1 1 -4 6 -4 1 -1 5 -10 10 -5 1 1 -6 15 -20 ] -Binomial,A007318,InvRevTabl,[1 -1 1 1 -2 1 -1 3 -3 1 1 -4 6 -4 1 -1 5 -10 10 -5 1 1 -6 15 -20 ] Binomial,A007318,RevInvTabl,[1 1 -1 1 -2 1 1 -3 3 -1 1 -4 6 -4 1 1 -5 10 -10 5 -1 1 -6 15 -20 ] +Binomial,A007318,InvRevTabl,[1 -1 1 1 -2 1 -1 3 -3 1 1 -4 6 -4 1 -1 5 -10 10 -5 1 1 -6 15 -20 ] Binomial,A008949,AccTabl,[1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 ] Binomial,A055248,RevAccTabl,[1 2 1 4 3 1 8 7 4 1 16 15 11 5 1 32 31 26 16 6 1 64 63 57 42 22 7 ] Binomial,A008949,AccRevTabl,[1 1 2 1 3 4 1 4 7 8 1 5 11 15 16 1 6 16 26 31 32 1 7 22 42 57 63 ] Binomial,A011973,AntiDiagTabl,[1 1 1 1 1 2 1 3 1 1 4 3 1 5 6 1 1 6 10 4 1 7 15 10 1 1 8 21 20 5 1 ] -Binomial,A103406,DiffxTabl,[1 1 2 1 4 3 1 6 9 4 1 8 18 16 5 1 10 30 40 25 6 1 12 45 80 75 36 7 ] Binomial,A046899,BinTabl,[1 1 2 1 3 6 1 4 10 20 1 5 15 35 70 1 6 21 56 126 252 1 7 28 84 210 ] Binomial,,InvBinTabl,[1 1 0 1 1 -2 1 2 -2 0 1 3 -1 -3 6 1 4 1 -6 6 0 1 5 4 -8 2 10 -20 1 ] +Binomial,A103406,DiffxTabl,[1 1 2 1 4 3 1 6 9 4 1 8 18 16 5 1 10 30 40 25 6 1 12 45 80 75 36 7 ] Binomial,A000079,RowSum,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] Binomial,A011782,EvenSum,[1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] Binomial,A034008,OddSum,[0 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] @@ -28,9 +28,9 @@ Binomial,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Binomial,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Binomial,A000984,BinConv,[1 2 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 ] Binomial,A126869,InvBinConv,[1 0 -2 0 6 0 -20 0 70 0 -252 0 924 0 -3432 0 12870 0 -48620 0 ] -Binomial,A001788,TransSqrs,[0 1 6 24 80 240 672 1792 4608 11520 28160 67584 159744 372736 ] Binomial,A001787,TransNat0,[0 1 4 12 32 80 192 448 1024 2304 5120 11264 24576 53248 114688 ] Binomial,A001792,TransNat1,[1 3 8 20 48 112 256 576 1280 2816 6144 13312 28672 61440 131072 ] +Binomial,A001788,TransSqrs,[0 1 6 24 80 240 672 1792 4608 11520 28160 67584 159744 372736 ] Binomial,A000027,DiagRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] Binomial,A000217,DiagRow2,[1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 ] Binomial,A000292,DiagRow3,[1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 ] diff --git a/data/csv/Catalan.csv b/data/csv/Catalan.csv index 1258cbf..22b753b 100644 --- a/data/csv/Catalan.csv +++ b/data/csv/Catalan.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Catalan,A128899,Tabl,[1 0 1 0 2 1 0 5 4 1 0 14 14 6 1 0 42 48 27 8 1 0 132 165 110 44 10 ] Catalan,A050166,RevTabl,[1 1 0 1 2 0 1 4 5 0 1 6 14 14 0 1 8 27 48 42 0 1 10 44 110 165 132 ] Catalan,A128908,InvTabl,[1 0 1 0 -2 1 0 3 -4 1 0 -4 10 -6 1 0 5 -20 21 -8 1 0 -6 35 -56 36 ] -Catalan,,InvRevTabl,[] Catalan,A053123,RevInvTabl,[1 1 0 1 -2 0 1 -4 3 0 1 -6 10 -4 0 1 -8 21 -20 5 0 1 -10 36 -56 35 ] +Catalan,,InvRevTabl,[] Catalan,A050158,AccTabl,[1 0 1 0 2 3 0 5 9 10 0 14 28 34 35 0 42 90 117 125 126 0 132 297 ] Catalan,,RevAccTabl,[1 1 0 3 2 0 10 9 5 0 35 34 28 14 0 126 125 117 90 42 0 462 461 451 ] Catalan,,AccRevTabl,[1 1 1 1 3 3 1 5 10 10 1 7 21 35 35 1 9 36 84 126 126 1 11 55 165 ] Catalan,A114494,AntiDiagTabl,[1 0 0 1 0 2 0 5 1 0 14 4 0 42 14 1 0 132 48 6 0 429 165 27 1 0 ] -Catalan,,DiffxTabl,[1 0 2 0 4 3 0 10 12 4 0 28 42 24 5 0 84 144 108 40 6 0 264 495 440 ] Catalan,,BinTabl,[1 0 1 0 2 5 0 5 14 28 0 14 42 90 165 0 42 132 297 572 1001 0 132 ] Catalan,,InvBinTabl,[1 0 1 0 2 -3 0 5 -6 4 0 14 -14 6 5 0 42 -36 9 20 -39 0 132 -99 11 ] +Catalan,,DiffxTabl,[1 0 2 0 4 3 0 10 12 4 0 28 42 24 5 0 84 144 108 40 6 0 264 495 440 ] Catalan,A088218,RowSum,[1 1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 ] Catalan,A001791,EvenSum,[1 0 1 4 15 56 210 792 3003 11440 43758 167960 646646 2496144 ] Catalan,A000984,OddSum,[0 1 2 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 ] @@ -28,9 +28,9 @@ Catalan,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Catalan,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Catalan,A025174,BinConv,[1 1 5 28 165 1001 6188 38760 245157 1562275 10015005 64512240 ] Catalan,,InvBinConv,[1 1 -3 4 5 -39 84 8 -603 1795 -1243 -9096 37388 -50036 -119640 ] -Catalan,A002457,TransSqrs,[0 1 6 30 140 630 2772 12012 51480 218790 923780 3879876 16224936 ] Catalan,A000302,TransNat0,[0 1 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 ] Catalan,A114121,TransNat1,[1 2 7 26 99 382 1486 5812 22819 89846 354522 1401292 5546382 ] +Catalan,A002457,TransSqrs,[0 1 6 30 140 630 2772 12012 51480 218790 923780 3879876 16224936 ] Catalan,A005843,DiagRow1,[0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 ] Catalan,A014106,DiagRow2,[0 5 14 27 44 65 90 119 152 189 230 275 324 377 434 495 560 629 702 ] Catalan,,DiagRow3,[0 14 48 110 208 350 544 798 1120 1518 2000 2574 3248 4030 4928 ] diff --git a/data/csv/CatalanAer.csv b/data/csv/CatalanAer.csv index 8a8832d..c541178 100644 --- a/data/csv/CatalanAer.csv +++ b/data/csv/CatalanAer.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence CatalanAer,A053121,Tabl,[1 0 1 1 0 1 0 2 0 1 2 0 3 0 1 0 5 0 4 0 1 5 0 9 0 5 0 1 0 14 0 14 ] CatalanAer,A052173,RevTabl,[1 1 0 1 0 1 1 0 2 0 1 0 3 0 2 1 0 4 0 5 0 1 0 5 0 9 0 5 1 0 6 0 14 ] CatalanAer,A049310,InvTabl,[1 0 1 -1 0 1 0 -2 0 1 1 0 -3 0 1 0 3 0 -4 0 1 -1 0 6 0 -5 0 1 0 -4 ] -CatalanAer,,InvRevTabl,[] CatalanAer,A053119,RevInvTabl,[1 1 0 1 0 -1 1 0 -2 0 1 0 -3 0 1 1 0 -4 0 3 0 1 0 -5 0 6 0 -1 1 0 ] +CatalanAer,,InvRevTabl,[] CatalanAer,,AccTabl,[1 0 1 1 1 2 0 2 2 3 2 2 5 5 6 0 5 5 9 9 10 5 5 14 14 19 19 20 0 14 ] CatalanAer,,RevAccTabl,[1 1 0 2 1 1 3 2 2 0 6 5 5 2 2 10 9 9 5 5 0 20 19 19 14 14 5 5 35 ] CatalanAer,A107430,AccRevTabl,[1 1 1 1 1 2 1 1 3 3 1 1 4 4 6 1 1 5 5 10 10 1 1 6 6 15 15 20 1 1 7 ] CatalanAer,A033184,AntiDiagTabl,[1 0 1 1 0 0 2 2 1 0 0 0 5 5 3 1 0 0 0 0 14 14 9 4 1 0 0 0 0 0 42 ] -CatalanAer,,DiffxTabl,[1 0 2 1 0 3 0 4 0 4 2 0 9 0 5 0 10 0 16 0 6 5 0 27 0 25 0 7 0 28 0 ] CatalanAer,,BinTabl,[1 0 1 1 1 2 0 2 4 7 2 2 5 11 21 0 5 10 19 36 66 5 5 14 32 64 120 ] CatalanAer,,InvBinTabl,[1 0 1 1 -1 2 0 2 -4 7 2 -2 5 -11 21 0 5 -10 19 -36 66 5 -5 14 -32 ] +CatalanAer,,DiffxTabl,[1 0 2 1 0 3 0 4 0 4 2 0 9 0 5 0 10 0 16 0 6 5 0 27 0 25 0 7 0 28 0 ] CatalanAer,A001405,RowSum,[1 1 2 3 6 10 20 35 70 126 252 462 924 1716 3432 6435 12870 24310 ] CatalanAer,A126869,EvenSum,[1 0 2 0 6 0 20 0 70 0 252 0 924 0 3432 0 12870 0 48620 0 ] CatalanAer,A138364,OddSum,[0 1 0 3 0 10 0 35 0 126 0 462 0 1716 0 6435 0 24310 0 92378 ] @@ -28,9 +28,9 @@ CatalanAer,A090192,ColLeft,[1 0 1 0 2 0 5 0 14 0 42 0 132 0 429 0 1430 0 4862 0 CatalanAer,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] CatalanAer,A344500,BinConv,[1 1 2 7 21 66 216 715 2395 8101 27598 94568 325612 1125632 3904512 ] CatalanAer,A344500,InvBinConv,[1 1 2 7 21 66 216 715 2395 8101 27598 94568 325612 1125632 3904512 ] -CatalanAer,,TransSqrs,[0 1 4 11 28 66 152 339 748 1622 3496 7454 15832 33380 70192 146819 ] CatalanAer,A045621,TransNat0,[0 1 2 5 10 22 44 93 186 386 772 1586 3172 6476 12952 26333 52666 ] CatalanAer,A000079,TransNat1,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] +CatalanAer,,TransSqrs,[0 1 4 11 28 66 152 339 748 1622 3496 7454 15832 33380 70192 146819 ] CatalanAer,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] CatalanAer,A000027,DiagRow2,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] CatalanAer,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/CentralCyc.csv b/data/csv/CentralCyc.csv deleted file mode 100644 index 5a6ff99..0000000 --- a/data/csv/CentralCyc.csv +++ /dev/null @@ -1,52 +0,0 @@ -Triangle,Trait,ANumber,Sequence -OEIS Similars: A111999 A259456 A269940,,, -central_cycle,Triangle ,A269940,[1 0 1 0 2 3 0 6 20 15 0 24 130 210 105 0 120 924 2380 2520 945 0 ] -central_cycle,TriRev ,A000000,[1 1 0 3 2 0 15 20 6 0 105 210 130 24 0 945 2520 2380 924 120 0 ] -central_cycle,TriAcc ,A000000,[1 0 1 0 2 5 0 6 26 41 0 24 154 364 469 0 120 1044 3424 5944 6889 0 ] -central_cycle,TriRevAcc,A000000,[1 1 0 5 2 0 41 26 6 0 469 364 154 24 0 6889 5944 3424 1044 120 0 ] -central_cycle,TriAccRev,A000000,[1 1 1 3 5 5 15 35 41 41 105 315 445 469 469 945 3465 5845 6769 ] -central_cycle,TriDiag ,A106828,[1 0 0 1 0 2 0 6 3 0 24 20 0 120 130 15 0 720 924 210 0 5040 7308 ] -central_cycle,TriPoly ,A000000,[1 0 1 0 1 1 0 5 2 1 0 41 16 3 1 0 469 212 33 4 1 0 6889 3928 603 ] -central_cycle,TriConv ,A000000,[1 0 1 0 2 13 0 6 72 661 0 24 438 5894 72601 0 120 3012 54900 ] -central_cycle,TriBin ,A000000,[1 0 1 0 2 7 0 6 32 93 0 24 178 672 1821 0 120 1164 5512 18064 ] -central_cycle,TriInvBin,A000000,[1 0 1 0 2 -1 0 6 8 -27 0 24 82 -108 -51 0 120 684 -32 -1936 3505 0 ] -central_cycle,Sum ,A032188,[1 1 5 41 469 6889 123605 2620169 64074901 1775623081 54989743445 ] -central_cycle,EvenSum ,A000000,[1 0 3 20 235 3444 61803 1310084 32037451 887811540 27494871723 ] -central_cycle,OddSum ,A000000,[0 1 2 21 234 3445 61802 1310085 32037450 887811541 27494871722 ] -central_cycle,AltSum ,A000012,[1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 ] -central_cycle,AccSum ,A000000,[1 1 7 73 1011 17421 358583 8575169 233499227 7129778357 ] -central_cycle,AccRevSum,A000000,[1 1 7 73 1011 17421 358583 8575169 233499227 7129778357 ] -central_cycle,RevAccSum,A000000,[1 2 13 132 1803 30802 630257 15006352 407249783 12402075534 ] -central_cycle,DiagSum ,A000166,[1 0 1 2 9 44 265 1854 14833 133496 1334961 14684570 176214841 ] -central_cycle,RowLcm ,A000000,[1 1 6 60 10920 1413720 39840292800 6652505239380000 ] -central_cycle,RowGcd ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -central_cycle,RowMax ,A000000,[1 1 3 20 210 2520 44100 866250 18858840 520059540 14980405440 ] -central_cycle,Middle ,A000000,[1 0 2 6 130 924 26432 303660 11098780 177331440 7934927000 ] -central_cycle,Central ,A000000,[1 2 130 26432 11098780 7934927000 8637235647040 13306928148113600 ] -central_cycle,LeftSide ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -central_cycle,RightSide,A001147,[1 1 3 15 105 945 10395 135135 2027025 34459425 654729075 ] -central_cycle,PosHalf ,A000000,[1 1 7 79 1237 24817 607519 17560063 585330109 22104345409 ] -central_cycle,NegHalf ,A001662,[1 1 -1 -1 13 -47 -73 2447 -16811 -15551 1726511 -18994849 10979677 ] -central_cycle,Bin ,A000000,[1 1 7 93 1821 47185 1522375 58808449 2646575953 135989069049 ] -central_cycle,InvBin ,A000000,[1 1 -1 -27 -51 3505 40655 -828919 -29640527 145577529 27107045415 ] -central_cycle,TransSqrs,A000000,[0 1 14 221 4114 89181 2213910 62017301 1936315234 66696441109 ] -central_cycle,TransNat0,A000000,[0 1 8 91 1334 23913 506652 12386183 343174882 10626452453 ] -central_cycle,TransNat1,A000000,[1 2 13 132 1803 30802 630257 15006352 407249783 12402075534 ] -central_cycle,DiagRow0 ,A001147,[1 1 3 15 105 945 10395 135135 2027025 34459425 654729075 ] -central_cycle,DiagRow1 ,A000906,[0 2 20 210 2520 34650 540540 9459450 183783600 3928374450 ] -central_cycle,DiagRow2 ,A000907,[0 6 130 2380 44100 866250 18288270 416215800 10199989800 ] -central_cycle,DiagRow3 ,A001784,[0 24 924 26432 705320 18858840 520059540 14980405440 453247114320 ] -central_cycle,DiagCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -central_cycle,DiagCol1 ,A000142,[1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] -central_cycle,DiagCol2 ,A000276,[3 20 130 924 7308 64224 623376 6636960 76998240 967524480 ] -central_cycle,DiagCol3 ,A000483,[15 210 2380 26432 303660 3678840 47324376 647536032 9418945536 ] -central_cycle,PolyRow0 ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -central_cycle,PolyRow1 ,A001477,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ] -central_cycle,PolyRow2 ,A045944,[0 5 16 33 56 85 120 161 208 261 320 385 456 533 616 705 800 901 ] -central_cycle,PolyRow3 ,A000000,[0 41 212 603 1304 2405 3996 6167 9008 12609 17060 22451 28872 ] -central_cycle,PolyCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -central_cycle,PolyCol1 ,A032188,[1 1 5 41 469 6889 123605 2620169 64074901 1775623081 54989743445 ] -central_cycle,PolyCol2 ,A000000,[1 2 16 212 3928 93536 2721808 93593216 3713245504 166956801728 ] -central_cycle,PolyCol3 ,A000000,[1 3 33 603 15417 506691 20351601 966015531 52906036329 ] -central_cycle,PolyDiag ,A000000,[1 1 16 603 42496 4849325 817557840 191243348471 59305251856384 ] -# central_cycle, 49 traits, 22 A-numbers,44% \ No newline at end of file diff --git a/data/csv/CentralSet.csv b/data/csv/CentralSet.csv index d6cc25a..bee64d1 100644 --- a/data/csv/CentralSet.csv +++ b/data/csv/CentralSet.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence CentralSet,A269945,Tabl,[1 0 1 0 1 1 0 1 5 1 0 1 21 14 1 0 1 85 147 30 1 0 1 341 1408 627 ] CentralSet,A008957,RevTabl,[1 1 0 1 1 0 1 5 1 0 1 14 21 1 0 1 30 147 85 1 0 1 55 627 1408 341 ] CentralSet,A269944,InvTabl,[1 0 1 0 -1 1 0 4 -5 1 0 -36 49 -14 1 0 576 -820 273 -30 1 0 -14400 ] -CentralSet,,InvRevTabl,[] CentralSet,A008955,RevInvTabl,[1 1 0 1 -1 0 1 -5 4 0 1 -14 49 -36 0 1 -30 273 -820 576 0 1 -55 ] +CentralSet,,InvRevTabl,[] CentralSet,,AccTabl,[1 0 1 0 1 2 0 1 6 7 0 1 22 36 37 0 1 86 233 263 264 0 1 342 1750 ] CentralSet,,RevAccTabl,[1 1 0 2 1 0 7 6 1 0 37 36 22 1 0 264 263 233 86 1 0 2433 2432 2377 ] CentralSet,,AccRevTabl,[1 1 1 1 2 2 1 6 7 7 1 15 36 37 37 1 31 178 263 264 264 1 56 683 ] CentralSet,,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 1 5 0 1 21 1 0 1 85 14 0 1 341 147 1 0 1 1365 ] -CentralSet,,DiffxTabl,[1 0 2 0 2 3 0 2 15 4 0 2 63 56 5 0 2 255 588 150 6 0 2 1023 5632 ] CentralSet,,BinTabl,[1 0 1 0 1 3 0 1 7 19 0 1 23 80 187 0 1 87 405 1132 2476 0 1 343 ] CentralSet,,InvBinTabl,[1 0 1 0 1 -1 0 1 3 -11 0 1 19 -46 67 0 1 83 -105 -52 476 0 1 339 ] +CentralSet,,DiffxTabl,[1 0 2 0 2 3 0 2 15 4 0 2 63 56 5 0 2 255 588 150 6 0 2 1023 5632 ] CentralSet,A135920,RowSum,[1 1 2 7 37 264 2433 27913 386906 6346119 121159373 2655174768 ] CentralSet,,EvenSum,[1 0 1 5 22 115 969 12896 206793 3528877 64056886 1297918107 ] CentralSet,,OddSum,[0 1 1 2 15 149 1464 15017 180113 2817242 57102487 1357256661 ] @@ -28,9 +28,9 @@ CentralSet,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] CentralSet,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] CentralSet,,BinConv,[1 1 3 19 187 2476 43017 927207 24103963 740949940 26435241416 ] CentralSet,,InvBinConv,[1 1 -1 -11 67 476 -13975 67803 3944843 -123221396 431500536 ] -CentralSet,,TransSqrs,[0 1 5 30 227 2169 25480 358993 5959213 114813254 2534015395 ] CentralSet,,TransNat0,[0 1 3 14 89 737 7696 98093 1490687 26542518 545598045 12787451789 ] CentralSet,,TransNat1,[1 2 5 21 126 1001 10129 126006 1877593 32888637 666757418 ] +CentralSet,,TransSqrs,[0 1 5 30 227 2169 25480 358993 5959213 114813254 2534015395 ] CentralSet,A000330,DiagRow1,[0 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 ] CentralSet,A060493,DiagRow2,[0 1 21 147 627 2002 5278 12138 25194 48279 86779 148005 241605 ] CentralSet,A351105,DiagRow3,[0 1 85 1408 11440 61490 251498 846260 2458676 6369275 15047175 ] diff --git a/data/csv/ChebyshevS.csv b/data/csv/ChebyshevS.csv index 6bee692..3163426 100644 --- a/data/csv/ChebyshevS.csv +++ b/data/csv/ChebyshevS.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence ChebyshevS,A049310,Tabl,[1 0 1 -1 0 1 0 -2 0 1 1 0 -3 0 1 0 3 0 -4 0 1 -1 0 6 0 -5 0 1 0 -4 ] ChebyshevS,A053119,RevTabl,[1 1 0 1 0 -1 1 0 -2 0 1 0 -3 0 1 1 0 -4 0 3 0 1 0 -5 0 6 0 -1 1 0 ] ChebyshevS,A053121,InvTabl,[1 0 1 1 0 1 0 2 0 1 2 0 3 0 1 0 5 0 4 0 1 5 0 9 0 5 0 1 0 14 0 14 ] -ChebyshevS,,InvRevTabl,[] ChebyshevS,A052173,RevInvTabl,[1 1 0 1 0 1 1 0 2 0 1 0 3 0 2 1 0 4 0 5 0 1 0 5 0 9 0 5 1 0 6 0 14 ] +ChebyshevS,,InvRevTabl,[] ChebyshevS,,AccTabl,[1 0 1 -1 -1 0 0 -2 -2 -1 1 1 -2 -2 -1 0 3 3 -1 -1 0 -1 -1 5 5 0 0 ] ChebyshevS,,RevAccTabl,[1 1 0 0 -1 -1 -1 -2 -2 0 -1 -2 -2 1 1 0 -1 -1 3 3 0 1 0 0 5 5 -1 ] ChebyshevS,,AccRevTabl,[1 1 1 1 1 0 1 1 -1 -1 1 1 -2 -2 -1 1 1 -3 -3 0 0 1 1 -4 -4 2 2 1 1 ] ChebyshevS,A007318,AntiDiagTabl,[1 0 -1 1 0 0 1 -2 1 0 0 0 -1 3 -3 1 0 0 0 0 1 -4 6 -4 1 0 0 0 0 0 ] -ChebyshevS,,DiffxTabl,[1 0 2 -1 0 3 0 -4 0 4 1 0 -9 0 5 0 6 0 -16 0 6 -1 0 18 0 -25 0 7 0 ] ChebyshevS,,BinTabl,[1 0 1 -1 -1 0 0 -2 -4 -5 1 1 -2 -8 -16 0 3 6 5 -4 -24 -1 -1 5 17 ] ChebyshevS,,InvBinTabl,[1 0 1 -1 1 0 0 -2 4 -5 1 -1 -2 8 -16 0 3 -6 5 4 -24 -1 1 5 -17 30 ] +ChebyshevS,,DiffxTabl,[1 0 2 -1 0 3 0 -4 0 4 1 0 -9 0 5 0 6 0 -16 0 6 -1 0 18 0 -25 0 7 0 ] ChebyshevS,A000484,RowSum,[1 1 0 -1 -1 0 1 1 0 -1 -1 0 1 1 0 -1 -1 0 1 1 ] ChebyshevS,A014021,EvenSum,[1 0 0 0 -1 0 1 0 0 0 -1 0 1 0 0 0 -1 0 1 0 ] ChebyshevS,A010051,OddSum,[0 1 0 -1 0 0 0 1 0 -1 0 0 0 1 0 -1 0 0 0 1 ] @@ -18,8 +18,8 @@ ChebyshevS,A000484,AltSum,[1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 ] ChebyshevS,A000007,AntiDiagSum,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ChebyshevS,,AccSum,[1 1 -2 -5 -3 4 9 5 -6 -13 -7 8 17 9 -10 -21 -11 12 25 13 ] ChebyshevS,,AccRevSum,[1 2 2 0 -3 -4 -1 4 6 2 -5 -8 -3 6 10 4 -7 -12 -5 8 ] -ChebyshevS,A073618,RowLcm,[1 1 1 -2 -3 12 30 60 210 840 1260 2520 13860 27720 180180 360360 ] -ChebyshevS,,RowGcd,[1 1 1 -2 -3 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 ] +ChebyshevS,A073618,RowLcm,[1 1 1 2 3 12 30 60 210 840 1260 2520 13860 27720 180180 360360 ] +ChebyshevS,,RowGcd,[1 1 1 2 3 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 ] ChebyshevS,A073028,RowMax,[1 1 1 2 3 4 6 10 15 21 35 56 84 126 210 330 495 792 1287 2002 ] ChebyshevS,A081204,ColMiddle,[1 0 0 -2 -3 0 0 10 15 0 0 -56 -84 0 0 330 495 0 0 -2002 ] ChebyshevS,A005809,ColECentral,[1 0 -3 0 15 0 -84 0 495 0 ] @@ -28,9 +28,9 @@ ChebyshevS,A000035,ColLeft,[1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 ] ChebyshevS,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ChebyshevS,A278415,BinConv,[1 1 0 -5 -16 -24 15 197 576 724 -1200 -8832 -22801 -21293 76440 ] ChebyshevS,A278415,InvBinConv,[1 1 0 -5 -16 -24 15 197 576 724 -1200 -8832 -22801 -21293 76440 ] -ChebyshevS,,TransSqrs,[0 1 4 7 4 -8 -20 -15 12 39 32 -16 -64 -55 20 95 84 -24 -132 -119 ] ChebyshevS,A186731,TransNat0,[0 1 2 1 -2 -4 -2 3 6 3 -4 -8 -4 5 10 5 -6 -12 -6 7 ] ChebyshevS,,TransNat1,[1 2 2 0 -3 -4 -1 4 6 2 -5 -8 -3 6 10 4 -7 -12 -5 8 ] +ChebyshevS,,TransSqrs,[0 1 4 7 4 -8 -20 -15 12 39 32 -16 -64 -55 20 95 84 -24 -132 -119 ] ChebyshevS,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ChebyshevS,A000027,DiagRow2,[-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 ] ChebyshevS,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/ChebyshevT.csv b/data/csv/ChebyshevT.csv index cc0c88b..3670269 100644 --- a/data/csv/ChebyshevT.csv +++ b/data/csv/ChebyshevT.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence ChebyshevT,A053120,Tabl,[1 0 1 -1 0 2 0 -3 0 4 1 0 -8 0 8 0 5 0 -20 0 16 -1 0 18 0 -48 0 32 ] ChebyshevT,A039991,RevTabl,[1 1 0 2 0 -1 4 0 -3 0 8 0 -8 0 1 16 0 -20 0 5 0 32 0 -48 0 18 0 -1 ] ChebyshevT,,InvTabl,[] -ChebyshevT,,InvRevTabl,[] ChebyshevT,,RevInvTabl,[] +ChebyshevT,,InvRevTabl,[] ChebyshevT,,AccTabl,[1 0 1 -1 -1 1 0 -3 -3 1 1 1 -7 -7 1 0 5 5 -15 -15 1 -1 -1 17 17 ] ChebyshevT,,RevAccTabl,[1 1 0 1 -1 -1 1 -3 -3 0 1 -7 -7 1 1 1 -15 -15 5 5 0 1 -31 -31 17 ] ChebyshevT,,AccRevTabl,[1 1 1 2 2 1 4 4 1 1 8 8 0 0 1 16 16 -4 -4 1 1 32 32 -16 -16 2 2 1 ] ChebyshevT,A079628,AntiDiagTabl,[1 0 -1 1 0 0 1 -3 2 0 0 0 -1 5 -8 4 0 0 0 0 1 -7 18 -20 8 0 0 0 0 ] -ChebyshevT,,DiffxTabl,[1 0 2 -1 0 6 0 -6 0 16 1 0 -24 0 40 0 10 0 -80 0 96 -1 0 54 0 -240 ] ChebyshevT,,BinTabl,[1 0 1 -1 -1 1 0 -3 -6 -5 1 1 -7 -23 -39 0 5 10 -5 -60 -159 -1 -1 ] ChebyshevT,,InvBinTabl,[1 0 1 -1 1 1 0 -3 6 -5 1 -1 -7 23 -39 0 5 -10 -5 60 -159 -1 1 17 ] +ChebyshevT,,DiffxTabl,[1 0 2 -1 0 6 0 -6 0 16 1 0 -24 0 40 0 10 0 -80 0 96 -1 0 54 0 -240 ] ChebyshevT,A000012,RowSum,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ChebyshevT,A000035,EvenSum,[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ] ChebyshevT,A000035,OddSum,[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ] @@ -28,9 +28,9 @@ ChebyshevT,A000035,ColLeft,[1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 ] ChebyshevT,A011782,ColRight,[1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] ChebyshevT,,BinConv,[1 1 1 -5 -39 -159 -419 -377 3265 23953 96361 242419 117417 ] ChebyshevT,,InvBinConv,[1 1 1 -5 -39 -159 -419 -377 3265 23953 96361 242419 117417 ] -ChebyshevT,A014820,TransSqrs,[0 1 8 33 96 225 456 833 1408 2241 3400 4961 7008 9633 12936 17025 ] ChebyshevT,A000290,TransNat0,[0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 ] ChebyshevT,A002522,TransNat1,[1 2 5 10 17 26 37 50 65 82 101 122 145 170 197 226 257 290 325 362 ] +ChebyshevT,A014820,TransSqrs,[0 1 8 33 96 225 456 833 1408 2241 3400 4961 7008 9633 12936 17025 ] ChebyshevT,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ChebyshevT,A001792,DiagRow2,[-1 -3 -8 -20 -48 -112 -256 -576 -1280 -2816 -6144 -13312 -28672 ] ChebyshevT,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/ChebyshevU.csv b/data/csv/ChebyshevU.csv index d201055..08cb318 100644 --- a/data/csv/ChebyshevU.csv +++ b/data/csv/ChebyshevU.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence ChebyshevU,A053117,Tabl,[1 0 2 -1 0 4 0 -4 0 8 1 0 -12 0 16 0 6 0 -32 0 32 -1 0 24 0 -80 0 ] ChebyshevU,A053118,RevTabl,[1 2 0 4 0 -1 8 0 -4 0 16 0 -12 0 1 32 0 -32 0 6 0 64 0 -80 0 24 0 ] ChebyshevU,,InvTabl,[] -ChebyshevU,,InvRevTabl,[] ChebyshevU,,RevInvTabl,[] +ChebyshevU,,InvRevTabl,[] ChebyshevU,,AccTabl,[1 0 2 -1 -1 3 0 -4 -4 4 1 1 -11 -11 5 0 6 6 -26 -26 6 -1 -1 23 23 ] ChebyshevU,,RevAccTabl,[1 2 0 3 -1 -1 4 -4 -4 0 5 -11 -11 1 1 6 -26 -26 6 6 0 7 -57 -57 23 ] ChebyshevU,,AccRevTabl,[1 2 2 4 4 3 8 8 4 4 16 16 4 4 5 32 32 0 0 6 6 64 64 -16 -16 8 8 7 ] ChebyshevU,A013609,AntiDiagTabl,[1 0 -1 2 0 0 1 -4 4 0 0 0 -1 6 -12 8 0 0 0 0 1 -8 24 -32 16 0 0 0 ] -ChebyshevU,,DiffxTabl,[1 0 4 -1 0 12 0 -8 0 32 1 0 -36 0 80 0 12 0 -128 0 192 -1 0 72 0 ] ChebyshevU,,BinTabl,[1 0 2 -1 -1 3 0 -4 -8 -4 1 1 -11 -35 -55 0 6 12 -14 -104 -258 -1 ] ChebyshevU,,InvBinTabl,[1 0 2 -1 1 3 0 -4 8 -4 1 -1 -11 35 -55 0 6 -12 -14 104 -258 -1 1 ] +ChebyshevU,,DiffxTabl,[1 0 4 -1 0 12 0 -8 0 32 1 0 -36 0 80 0 12 0 -128 0 192 -1 0 72 0 ] ChebyshevU,A000027,RowSum,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] ChebyshevU,A193356,EvenSum,[1 0 3 0 5 0 7 0 9 0 11 0 13 0 15 0 17 0 19 0 ] ChebyshevU,A237420,OddSum,[0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0 18 0 20 ] @@ -28,9 +28,9 @@ ChebyshevU,A000035,ColLeft,[1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 ] ChebyshevU,A000079,ColRight,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] ChebyshevU,,BinConv,[1 2 3 -4 -55 -258 -777 -1160 3393 34970 158763 459252 544505 ] ChebyshevU,,InvBinConv,[1 2 3 -4 -55 -258 -777 -1160 3393 34970 158763 459252 544505 ] -ChebyshevU,,TransSqrs,[0 2 16 68 208 518 1120 2184 3936 6666 10736 16588 24752 35854 ] ChebyshevU,A007290,TransNat0,[0 2 8 20 40 70 112 168 240 330 440 572 728 910 1120 1360 1632 1938 ] ChebyshevU,A006527,TransNat1,[1 4 11 24 45 76 119 176 249 340 451 584 741 924 1135 1376 1649 ] +ChebyshevU,,TransSqrs,[0 2 16 68 208 518 1120 2184 3936 6666 10736 16588 24752 35854 ] ChebyshevU,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ChebyshevU,A001787,DiagRow2,[-1 -4 -12 -32 -80 -192 -448 -1024 -2304 -5120 -11264 -24576 -53248 ] ChebyshevU,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/ChristTree.csv b/data/csv/ChristTree.csv index 1b2e883..b16022d 100644 --- a/data/csv/ChristTree.csv +++ b/data/csv/ChristTree.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence ChristTree,A106465,Tabl,[1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 ] ChristTree,A106465,RevTabl,[1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 ] ChristTree,A106467,InvTabl,[1 -1 1 -1 0 1 1 -1 -1 1 0 0 -1 0 1 0 0 1 -1 -1 1 0 0 0 0 -1 0 1 0 ] -ChristTree,A106467,InvRevTabl,[1 -1 1 -1 0 1 1 -1 -1 1 0 0 -1 0 1 0 0 1 -1 -1 1 0 0 0 0 -1 0 1 0 ] ChristTree,A000012,RevInvTabl,[1 1 -1 1 0 -1 1 -1 -1 1 1 0 -1 0 0 1 -1 -1 1 0 0 1 0 -1 0 0 0 0 1 ] +ChristTree,A106467,InvRevTabl,[1 -1 1 -1 0 1 1 -1 -1 1 0 0 -1 0 1 0 0 1 -1 -1 1 0 0 0 0 -1 0 1 0 ] ChristTree,,AccTabl,[1 1 2 1 1 2 1 2 3 4 1 1 2 2 3 1 2 3 4 5 6 1 1 2 2 3 3 4 1 2 3 4 5 ] ChristTree,,RevAccTabl,[1 2 1 2 1 1 4 3 2 1 3 2 2 1 1 6 5 4 3 2 1 4 3 3 2 2 1 1 8 7 6 5 4 ] ChristTree,,AccRevTabl,[1 1 2 1 1 2 1 2 3 4 1 1 2 2 3 1 2 3 4 5 6 1 1 2 2 3 3 4 1 2 3 4 5 ] ChristTree,A000012,AntiDiagTabl,[1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 ] -ChristTree,,DiffxTabl,[1 1 2 1 0 3 1 2 3 4 1 0 3 0 5 1 2 3 4 5 6 1 0 3 0 5 0 7 1 2 3 4 5 ] ChristTree,,BinTabl,[1 1 2 1 1 2 1 2 4 8 1 1 2 4 8 1 2 4 8 16 32 1 1 2 4 8 16 32 1 2 4 ] ChristTree,,InvBinTabl,[1 1 0 1 -1 2 1 0 0 0 1 -1 2 -4 8 1 0 0 0 0 0 1 -1 2 -4 8 -16 32 1 ] +ChristTree,,DiffxTabl,[1 1 2 1 0 3 1 2 3 4 1 0 3 0 5 1 2 3 4 5 6 1 0 3 0 5 0 7 1 2 3 4 5 ] ChristTree,A029578,RowSum,[1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 ] ChristTree,A001057,EvenSum,[1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 ] ChristTree,A142150,OddSum,[0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 ] @@ -28,9 +28,9 @@ ChristTree,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ChristTree,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ChristTree,A158302,BinConv,[1 2 2 8 8 32 32 128 128 512 512 2048 2048 8192 8192 32768 32768 ] ChristTree,A199573,InvBinConv,[1 0 2 0 8 0 32 0 128 0 512 0 2048 0 8192 0 32768 0 131072 0 ] -ChristTree,,TransSqrs,[0 1 4 14 20 55 56 140 120 285 220 506 364 819 560 1240 816 1785 ] ChristTree,A056136,TransNat0,[0 1 2 6 6 15 12 28 20 45 30 66 42 91 56 120 72 153 90 190 ] ChristTree,A359366,TransNat1,[1 3 4 10 9 21 16 36 25 55 36 78 49 105 64 136 81 171 100 210 ] +ChristTree,,TransSqrs,[0 1 4 14 20 55 56 140 120 285 220 506 364 819 560 1240 816 1785 ] ChristTree,A000035,DiagRow1,[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ] ChristTree,A000012,DiagRow2,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ChristTree,A000035,DiagRow3,[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ] diff --git a/data/csv/CompositionMax.csv b/data/csv/CompositionMax.csv index 9181385..6ee3755 100644 --- a/data/csv/CompositionMax.csv +++ b/data/csv/CompositionMax.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence CompositionMax,A126198,Tabl,[1 0 1 0 1 2 0 1 3 4 0 1 5 7 8 0 1 8 13 15 16 0 1 13 24 29 31 32 0 ] CompositionMax,,RevTabl,[1 1 0 2 1 0 4 3 1 0 8 7 5 1 0 16 15 13 8 1 0 32 31 29 24 13 1 0 64 ] CompositionMax,,InvTabl,[] -CompositionMax,,InvRevTabl,[] CompositionMax,,RevInvTabl,[] +CompositionMax,,InvRevTabl,[] CompositionMax,,AccTabl,[1 0 1 0 1 3 0 1 4 8 0 1 6 13 21 0 1 9 22 37 53 0 1 14 38 67 98 130 ] CompositionMax,,RevAccTabl,[1 1 0 3 1 0 8 4 1 0 21 13 6 1 0 53 37 22 9 1 0 130 98 67 38 14 1 0 ] CompositionMax,,AccRevTabl,[1 1 1 2 3 3 4 7 8 8 8 15 20 21 21 16 31 44 52 53 53 32 63 92 116 ] CompositionMax,,AntiDiagTabl,[1 0 0 1 0 1 0 1 2 0 1 3 0 1 5 4 0 1 8 7 0 1 13 13 8 0 1 21 24 15 0 ] -CompositionMax,,DiffxTabl,[1 0 2 0 2 6 0 2 9 16 0 2 15 28 40 0 2 24 52 75 96 0 2 39 96 145 ] CompositionMax,,BinTabl,[1 0 1 0 1 4 0 1 5 16 0 1 7 25 70 0 1 10 40 119 306 0 1 15 66 207 ] CompositionMax,,InvBinTabl,[1 0 1 0 1 0 0 1 1 -2 0 1 3 -5 6 0 1 6 -8 7 -4 0 1 11 -12 7 1 -10 0 ] +CompositionMax,,DiffxTabl,[1 0 2 0 2 6 0 2 9 16 0 2 15 28 40 0 2 24 52 75 96 0 2 39 96 145 ] CompositionMax,A039671,RowSum,[1 1 3 8 21 53 130 310 724 1661 3757 8398 18588 40800 88918 192592 ] CompositionMax,,EvenSum,[1 0 2 3 13 23 74 140 395 766 2003 3932 9780 19320 46396 91990 ] CompositionMax,,OddSum,[0 1 1 5 8 30 56 170 329 895 1754 4466 8808 21480 42522 100602 ] @@ -28,9 +28,9 @@ CompositionMax,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] CompositionMax,A011782,ColRight,[1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] CompositionMax,,BinConv,[1 1 4 16 70 306 1334 5734 24420 102940 430360 1785829 7364042 ] CompositionMax,,InvBinConv,[1 1 0 -2 6 -4 -10 50 -140 310 -456 199 1298 -5953 19032 -51192 ] -CompositionMax,,TransSqrs,[0 1 9 49 212 790 2660 8306 24509 69171 188414 498542 1287768 ] CompositionMax,,TransNat0,[0 1 5 19 64 196 562 1530 4007 10173 25196 61148 145916 343270 ] CompositionMax,,TransNat1,[1 2 8 27 85 249 692 1840 4731 11834 28953 69546 164504 384070 ] +CompositionMax,,TransSqrs,[0 1 9 49 212 790 2660 8306 24509 69171 188414 498542 1287768 ] CompositionMax,A000225,DiagRow1,[0 1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 16383 32767 65535 ] CompositionMax,A344920,DiagRow2,[0 1 5 13 29 61 125 253 509 1021 2045 4093 8189 16381 32765 65533 ] CompositionMax,A159741,DiagRow3,[0 1 8 24 56 120 248 504 1016 2040 4088 8184 16376 32760 65528 ] diff --git a/data/csv/Compositions.csv b/data/csv/Compositions.csv deleted file mode 100644 index 141fe8a..0000000 --- a/data/csv/Compositions.csv +++ /dev/null @@ -1,54 +0,0 @@ -Triangle,Trait,ANumber,Sequence -OEIS Similars: A048004,,, -compo,Triangle ,A000000,[1 0 1 0 1 1 0 1 2 1 0 1 4 2 1 0 1 7 5 2 1 0 1 12 11 5 2 1 0 1 20 ] -compo,TriRev ,A000000,[1 1 0 1 1 0 1 2 1 0 1 2 4 1 0 1 2 5 7 1 0 1 2 5 11 12 1 0 1 2 5 12 ] -compo,TriInv ,A000000,[1 0 1 0 -1 1 0 1 -2 1 0 1 0 -2 1 0 -1 3 -1 -2 1 0 -3 4 1 -1 -2 1 0 ] -compo,TriRevInv,A000000,[1 1 0 1 -1 0 1 -2 1 0 1 -2 0 1 0 1 -2 -1 3 -1 0 1 -2 -1 1 4 -3 0 1 ] -compo,TriAcc ,A000000,[1 0 1 0 1 2 0 1 3 4 0 1 5 7 8 0 1 8 13 15 16 0 1 13 24 29 31 32 0 ] -compo,TriRevAcc,A000000,[1 1 0 2 1 0 4 3 1 0 8 7 5 1 0 16 15 13 8 1 0 32 31 29 24 13 1 0 64 ] -compo,TriAccRev,A000000,[1 1 1 1 2 2 1 3 4 4 1 3 7 8 8 1 3 8 15 16 16 1 3 8 19 31 32 32 1 3 ] -compo,TriDiag ,A000000,[1 0 0 1 0 1 0 1 1 0 1 2 0 1 4 1 0 1 7 2 0 1 12 5 1 0 1 20 11 2 0 1 ] -compo,TriPoly ,A000000,[1 0 1 0 1 1 0 2 2 1 0 4 6 3 1 0 8 18 12 4 1 0 16 50 48 20 5 1 0 32 ] -compo,TriConv ,A000000,[1 0 1 0 1 2 0 1 3 6 0 1 5 11 22 0 1 8 20 41 80 0 1 13 36 76 152 ] -compo,TriBin ,A000000,[1 0 1 0 1 3 0 1 4 10 0 1 6 17 37 0 1 9 29 68 136 0 1 14 50 125 262 ] -compo,TriInvBin,A000000,[1 0 1 0 1 -1 0 1 0 -2 0 1 2 -7 13 0 1 5 -13 20 -24 0 1 10 -22 29 ] -compo,Sum ,A011782,[1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] -compo,EvenSum ,A103422,[1 0 1 2 5 9 18 34 66 127 249 490 972 1936 3874 7772 15623 31439 ] -compo,OddSum ,A103421,[0 1 1 2 3 7 14 30 62 129 263 534 1076 2160 4318 8612 17145 34097 ] -compo,AltSum ,A000000,[1 -1 0 0 2 2 4 4 4 -2 -14 -44 -104 -224 -444 -840 -1522 -2658 ] -compo,AccSum ,A039671,[1 1 3 8 21 53 130 310 724 1661 3757 8398 18588 40800 88918 192592 ] -compo,AccRevSum,A039671,[1 1 3 8 21 53 130 310 724 1661 3757 8398 18588 40800 88918 192592 ] -compo,RevAccSum,A000000,[1 2 5 12 27 59 126 266 556 1155 2387 4914 10084 20640 42154 85936 ] -compo,DiagSum ,A000000,[1 0 1 1 2 3 6 10 19 34 63 116 216 402 754 1417 2674 5061 9608 ] -compo,RowLcm ,A000000,[1 1 1 2 4 70 660 1380 279180 10481940 130256280 107793766080 ] -compo,RowGcd ,A030409,[1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -compo,RowMax ,A000000,[1 1 1 2 4 7 12 23 47 94 185 360 694 1328 2526 4903 9960 20135 ] -compo,Middle ,A000000,[1 0 1 1 4 7 11 23 27 59 63 139 143 315 319 699 703 1531 1535 3323 ] -compo,Central ,A047859,[1 1 4 11 27 63 143 319 703 1535 3327 7167 ] -compo,LeftSide ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -compo,RightSide,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -compo,PosHalf ,A000000,[1 1 3 9 29 97 337 1193 4297 15641 57433 212345 789625 2950841 ] -compo,NegHalf ,A000000,[1 1 -1 1 5 -23 89 -287 833 -2159 4817 -7247 -6991 130801 -800527 ] -compo,Bin ,A000000,[1 1 3 10 37 136 494 1772 6283 22018 76282 261350 886340 2977846 ] -compo,InvBin ,A000000,[1 1 -1 -2 13 -24 18 64 -373 1210 -2810 4940 -5700 -2352 44198 ] -compo,TransSqrs,A000000,[0 1 5 18 51 131 314 726 1630 3593 7803 16762 35692 75476 158690 ] -compo,TransNat0,A102712,[0 1 3 8 19 43 94 202 428 899 1875 3890 8036 16544 33962 69552 ] -compo,TransNat1,A000000,[1 2 5 12 27 59 126 266 556 1155 2387 4914 10084 20640 42154 85936 ] -compo,DiagRow0 ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -compo,DiagRow1 ,A065685,[0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ] -compo,DiagRow2 ,A064099,[0 1 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 ] -compo,DiagRow3 ,A000000,[0 1 7 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 ] -compo,DiagCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -compo,DiagCol1 ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -compo,DiagCol2 ,A000071,[1 2 4 7 12 20 33 54 88 143 232 376 609 986 1596 2583 4180 6764 ] -compo,DiagCol3 ,A000100,[1 2 5 11 23 47 94 185 360 694 1328 2526 4781 9012 16929 31709 ] -compo,PolyRow0 ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -compo,PolyRow1 ,A001477,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ] -compo,PolyRow2 ,A002378,[0 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] -compo,PolyRow3 ,A045991,[0 4 18 48 100 180 294 448 648 900 1210 1584 2028 2548 3150 3840 ] -compo,PolyCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -compo,PolyCol1 ,A011782,[1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 ] -compo,PolyCol2 ,A000000,[1 2 6 18 50 134 346 874 2158 5242 12538 29614 69154 159922 366582 ] -compo,PolyCol3 ,A000000,[1 3 12 48 174 606 2028 6636 21252 67098 209262 646860 1985160 ] -compo,PolyDiag ,A000000,[1 1 6 48 452 5180 71502 1180564 22812232 505919448 12655763810 ] -# compo, 51 traits, 23 A-numbers,45% \ No newline at end of file diff --git a/data/csv/Delannoy.csv b/data/csv/Delannoy.csv index e0c2776..5be7b46 100644 --- a/data/csv/Delannoy.csv +++ b/data/csv/Delannoy.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Delannoy,A008288,Tabl,[1 1 1 1 3 1 1 5 5 1 1 7 13 7 1 1 9 25 25 9 1 1 11 41 63 41 11 1 1 ] Delannoy,A008288,RevTabl,[1 1 1 1 3 1 1 5 5 1 1 7 13 7 1 1 9 25 25 9 1 1 11 41 63 41 11 1 1 ] Delannoy,A103136,InvTabl,[1 -1 1 2 -3 1 -6 10 -5 1 22 -38 22 -7 1 -90 158 -98 38 -9 1 394 ] -Delannoy,A103136,InvRevTabl,[1 -1 1 2 -3 1 -6 10 -5 1 22 -38 22 -7 1 -90 158 -98 38 -9 1 394 ] Delannoy,A033878,RevInvTabl,[1 1 -1 1 -3 2 1 -5 10 -6 1 -7 22 -38 22 1 -9 38 -98 158 -90 1 -11 ] +Delannoy,A103136,InvRevTabl,[1 -1 1 2 -3 1 -6 10 -5 1 22 -38 22 -7 1 -90 158 -98 38 -9 1 394 ] Delannoy,,AccTabl,[1 1 2 1 4 5 1 6 11 12 1 8 21 28 29 1 10 35 60 69 70 1 12 53 116 ] Delannoy,A103415,RevAccTabl,[1 2 1 5 4 1 12 11 6 1 29 28 21 8 1 70 69 60 35 10 1 169 168 157 ] Delannoy,,AccRevTabl,[1 1 2 1 4 5 1 6 11 12 1 8 21 28 29 1 10 35 60 69 70 1 12 53 116 ] Delannoy,,AntiDiagTabl,[1 1 1 1 1 3 1 5 1 1 7 5 1 9 13 1 1 11 25 7 1 13 41 25 1 1 15 61 63 ] -Delannoy,,DiffxTabl,[1 1 2 1 6 3 1 10 15 4 1 14 39 28 5 1 18 75 100 45 6 1 22 123 252 ] Delannoy,,BinTabl,[1 1 2 1 4 8 1 6 16 32 1 8 28 68 136 1 10 44 128 296 592 1 12 64 ] Delannoy,,InvBinTabl,[1 1 0 1 2 -4 1 4 -4 0 1 6 0 -12 24 1 8 8 -24 24 0 1 10 20 -28 -8 ] +Delannoy,,DiffxTabl,[1 1 2 1 6 3 1 10 15 4 1 14 39 28 5 1 18 75 100 45 6 1 22 123 252 ] Delannoy,A000129,RowSum,[1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 ] Delannoy,A116404,EvenSum,[1 1 2 6 15 35 84 204 493 1189 2870 6930 16731 40391 97512 235416 ] Delannoy,,OddSum,[0 1 3 6 14 35 85 204 492 1189 2871 6930 16730 40391 97513 235416 ] @@ -28,9 +28,9 @@ Delannoy,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Delannoy,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Delannoy,A006139,BinConv,[1 2 8 32 136 592 2624 11776 53344 243392 1116928 5149696 23835904 ] Delannoy,A059304,InvBinConv,[1 0 -4 0 24 0 -160 0 1120 0 -8064 0 59136 0 -439296 0 3294720 0 ] -Delannoy,,TransSqrs,[0 1 7 34 138 503 1709 5524 17204 52061 153971 446934 1277310 ] Delannoy,,TransNat0,[0 1 5 18 58 175 507 1428 3940 10701 28705 76230 200766 525083 ] Delannoy,A026937,TransNat1,[1 3 10 30 87 245 676 1836 4925 13079 34446 90090 234227 605865 ] +Delannoy,,TransSqrs,[0 1 7 34 138 503 1709 5524 17204 52061 153971 446934 1277310 ] Delannoy,A004273,DiagRow1,[1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 ] Delannoy,A001844,DiagRow2,[1 5 13 25 41 61 85 113 145 181 221 265 313 365 421 481 545 613 685 ] Delannoy,A001845,DiagRow3,[1 7 25 63 129 231 377 575 833 1159 1561 2047 2625 3303 4089 4991 ] diff --git a/data/csv/Euler.csv b/data/csv/Euler.csv index 78f9a7f..443408c 100644 --- a/data/csv/Euler.csv +++ b/data/csv/Euler.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -Euler,A109449,Tabl,[1 -1 1 1 -2 1 -2 3 -3 1 5 -8 6 -4 1 -16 25 -20 10 -5 1 61 -96 75 ] +Euler,A247453,Tabl,[1 -1 1 1 -2 1 -2 3 -3 1 5 -8 6 -4 1 -16 25 -20 10 -5 1 61 -96 75 ] Euler,,RevTabl,[1 1 -1 1 -2 1 1 -3 3 -2 1 -4 6 -8 5 1 -5 10 -20 25 -16 1 -6 15 -40 ] Euler,A109449,InvTabl,[1 1 1 1 2 1 2 3 3 1 5 8 6 4 1 16 25 20 10 5 1 61 96 75 40 15 6 1 ] -Euler,,InvRevTabl,[] Euler,,RevInvTabl,[1 1 1 1 2 1 1 3 3 2 1 4 6 8 5 1 5 10 20 25 16 1 6 15 40 75 96 61 1 ] +Euler,,InvRevTabl,[] Euler,,AccTabl,[1 -1 0 1 -1 0 -2 1 -2 -1 5 -3 3 -1 0 -16 9 -11 -1 -6 -5 61 -35 40 ] Euler,,RevAccTabl,[1 0 -1 0 -1 1 -1 -2 1 -2 0 -1 3 -3 5 -5 -6 -1 -11 9 -16 10 9 15 0 ] Euler,,AccRevTabl,[1 1 0 1 -1 0 1 -2 1 -1 1 -3 3 -5 0 1 -4 6 -14 11 -5 1 -5 10 -30 45 ] Euler,,AntiDiagTabl,[1 -1 1 1 -2 -2 5 3 1 -16 -8 -3 61 25 6 1 -272 -96 -20 -4 1385 427 ] -Euler,,DiffxTabl,[1 -1 2 1 -4 3 -2 6 -9 4 5 -16 18 -16 5 -16 50 -60 40 -25 6 61 -192 ] Euler,,BinTabl,[1 -1 0 1 -1 -2 -2 1 1 -1 5 -3 -5 -5 -6 -16 9 14 9 -1 -15 61 -35 ] Euler,,InvBinTabl,[1 -1 2 1 -3 6 -2 5 -11 21 5 -13 27 -51 90 -16 41 -86 161 -281 467 ] +Euler,,DiffxTabl,[1 -1 2 1 -4 3 -2 6 -9 4 5 -16 18 -16 5 -16 50 -60 40 -25 6 61 -192 ] Euler,A062162,RowSum,[1 0 0 -1 0 -5 10 -61 280 -1665 10470 -73621 561660 -4650425 ] Euler,A062272,EvenSum,[1 -1 2 -5 12 -41 152 -685 3472 -19921 126752 -887765 6781632 ] Euler,A062161,OddSum,[0 1 -2 4 -12 36 -142 624 -3192 18256 -116282 814144 -6219972 ] @@ -18,8 +18,8 @@ Euler,A000667,AltSum,[1 -2 4 -9 24 -77 294 -1309 6664 -38177 243034 -1701909 130 Euler,A343845,AntiDiagSum,[1 -1 2 -4 9 -27 93 -392 1898 -10493 64885 -443916 3326317 ] Euler,,AccSum,[1 -1 0 -4 4 -30 100 -558 3008 -19170 131820 -998622 8185032 ] Euler,A337443,AccRevSum,[1 1 0 -1 -4 -5 -20 9 -208 855 -6180 41549 -321792 2651155 ] -Euler,,RowLcm,[1 1 -2 6 120 400 146400 8710800 6434377600 718076540160 ] -Euler,A062557,RowGcd,[1 1 -2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] +Euler,,RowLcm,[1 1 2 6 120 400 146400 8710800 6434377600 718076540160 ] +Euler,A062557,RowGcd,[1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Euler,A065619,RowMax,[1 1 2 3 8 25 96 427 2176 12465 79360 555731 4245504 35135945 ] Euler,,ColMiddle,[1 -1 -2 3 6 -20 -40 175 350 -2016 -4032 28182 56364 -466752 ] Euler,A343846,ColECentral,[1 -2 6 -40 350 -4032 56364 -933504 17824950 -385848320 ] @@ -28,9 +28,9 @@ Euler,A000111,ColLeft,[1 -1 1 -2 5 -16 61 -272 1385 -7936 50521 -353792 2702765 Euler,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Euler,,BinConv,[1 0 -2 -1 -6 -15 0 -271 574 -5379 12408 -63391 -355740 5907811 ] Euler,,InvBinConv,[1 2 6 21 90 467 2824 19383 148414 1251935 11523136 114837943 ] -Euler,,TransSqrs,[0 1 2 0 -4 -20 -30 -140 72 -1872 8550 -67980 498588 -4183296 ] Euler,,TransNat0,[0 1 0 0 -4 0 -30 70 -488 2520 -16650 115170 -883452 7301580 ] Euler,A337443,TransNat1,[1 1 0 -1 -4 -5 -20 9 -208 855 -6180 41549 -321792 2651155 ] +Euler,,TransSqrs,[0 1 2 0 -4 -20 -30 -140 72 -1872 8550 -67980 498588 -4183296 ] Euler,A000027,DiagRow1,[-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 ] Euler,A000217,DiagRow2,[1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 ] Euler,A007290,DiagRow3,[-2 -8 -20 -40 -70 -112 -168 -240 -330 -440 -572 -728 -910 -1120 ] diff --git a/data/csv/EulerSec.csv b/data/csv/EulerSec.csv index 2945938..ca4c1eb 100644 --- a/data/csv/EulerSec.csv +++ b/data/csv/EulerSec.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence EulerSec,A119879,Tabl,[1 0 1 -1 0 1 0 -3 0 1 5 0 -6 0 1 0 25 0 -10 0 1 -61 0 75 0 -15 0 1 ] EulerSec,A153641,RevTabl,[1 1 0 1 0 -1 1 0 -3 0 1 0 -6 0 5 1 0 -10 0 25 0 1 0 -15 0 75 0 -61 ] EulerSec,A119467,InvTabl,[1 0 1 1 0 1 0 3 0 1 1 0 6 0 1 0 5 0 10 0 1 1 0 15 0 15 0 1 0 7 0 ] -EulerSec,,InvRevTabl,[] EulerSec,A141665,RevInvTabl,[1 1 0 1 0 1 1 0 3 0 1 0 6 0 1 1 0 10 0 5 0 1 0 15 0 15 0 1 1 0 21 ] +EulerSec,,InvRevTabl,[] EulerSec,,AccTabl,[1 0 1 -1 -1 0 0 -3 -3 -2 5 5 -1 -1 0 0 25 25 15 15 16 -61 -61 14 ] EulerSec,,RevAccTabl,[1 1 0 0 -1 -1 -2 -3 -3 0 0 -1 -1 5 5 16 15 15 25 25 0 0 -1 -1 14 ] EulerSec,,AccRevTabl,[1 1 1 1 1 0 1 1 -2 -2 1 1 -5 -5 0 1 1 -9 -9 16 16 1 1 -14 -14 61 ] EulerSec,,AntiDiagTabl,[1 0 -1 1 0 0 5 -3 1 0 0 0 -61 25 -6 1 0 0 0 0 1385 -427 75 -10 1 0 ] -EulerSec,,DiffxTabl,[1 0 2 -1 0 3 0 -6 0 4 5 0 -18 0 5 0 50 0 -40 0 6 -61 0 225 0 -75 0 ] EulerSec,,BinTabl,[1 0 1 -1 -1 0 0 -3 -6 -8 5 5 -1 -13 -30 0 25 50 65 60 26 -61 -61 ] EulerSec,,InvBinTabl,[1 0 1 -1 1 0 0 -3 6 -8 5 -5 -1 13 -30 0 25 -50 65 -60 26 -61 61 14 ] +EulerSec,,DiffxTabl,[1 0 2 -1 0 3 0 -6 0 4 5 0 -18 0 5 0 50 0 -40 0 6 -61 0 225 0 -75 0 ] EulerSec,A009006,RowSum,[1 1 0 -2 0 16 0 -272 0 7936 0 -353792 0 22368256 0 -1903757312 0 ] EulerSec,A000007,EvenSum,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] EulerSec,A350972,OddSum,[0 1 0 -2 0 16 0 -272 0 7936 0 -353792 0 22368256 0 -1903757312 0 ] @@ -18,8 +18,8 @@ EulerSec,A009006,AltSum,[1 -1 0 2 0 -16 0 272 0 -7936 0 353792 0 -22368256 0 190 EulerSec,,AntiDiagSum,[1 0 0 0 3 0 -41 0 1024 0 -39603 0 2204565 0 -167343104 0 ] EulerSec,,AccSum,[1 1 -2 -8 8 96 -96 -2176 2176 79360 -79360 -4245504 4245504 ] EulerSec,A009725,AccRevSum,[1 2 2 -2 -8 16 96 -272 -2176 7936 79360 -353792 -4245504 22368256 ] -EulerSec,,RowLcm,[1 1 1 -3 30 50 4575 32025 11827900 21290220 2689008011550 ] -EulerSec,A155457,RowGcd,[1 1 1 -3 1 5 1 7 1 3 1 11 1 13 1 1 1 17 1 19 ] +EulerSec,,RowLcm,[1 1 1 3 30 50 4575 32025 11827900 21290220 2689008011550 ] +EulerSec,A155457,RowGcd,[1 1 1 3 1 5 1 7 1 3 1 11 1 13 1 1 1 17 1 19 ] EulerSec,,RowMax,[1 1 1 3 6 25 75 427 1708 12465 62325 555731 3334386 35135945 ] EulerSec,,ColMiddle,[1 0 0 -3 -6 0 0 175 350 0 0 -28182 -56364 0 0 8912475 17824950 0 0 ] EulerSec,A214445,ColECentral,[1 0 -6 0 350 0 -56364 0 17824950 0 ] @@ -28,9 +28,9 @@ EulerSec,A122045,ColLeft,[1 0 -1 0 5 0 -61 0 1385 0 -50521 0 2702765 0 -19936098 EulerSec,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] EulerSec,,BinConv,[1 1 0 -8 -30 26 840 2696 -22722 -240146 282480 19114976 71133348 ] EulerSec,,InvBinConv,[1 1 0 -8 -30 26 840 2696 -22722 -240146 282480 19114976 71133348 ] -EulerSec,,TransSqrs,[0 1 4 6 -8 -40 96 672 -2176 -19584 79360 872960 -4245504 -55191552 ] EulerSec,A109573,TransNat0,[0 1 2 0 -8 0 96 0 -2176 0 79360 0 -4245504 0 313155584 0 ] EulerSec,A009725,TransNat1,[1 2 2 -2 -8 16 96 -272 -2176 7936 79360 -353792 -4245504 22368256 ] +EulerSec,,TransSqrs,[0 1 4 6 -8 -40 96 672 -2176 -19584 79360 872960 -4245504 -55191552 ] EulerSec,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] EulerSec,A000217,DiagRow2,[-1 -3 -6 -10 -15 -21 -28 -36 -45 -55 -66 -78 -91 -105 -120 -136 ] EulerSec,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/EulerTan.csv b/data/csv/EulerTan.csv index b024590..ae76f7e 100644 --- a/data/csv/EulerTan.csv +++ b/data/csv/EulerTan.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence EulerTan,A162660,Tabl,[0 1 0 0 2 0 -2 0 3 0 0 -8 0 4 0 16 0 -20 0 5 0 0 96 0 -40 0 6 0 ] EulerTan,,RevTabl,[0 0 1 0 2 0 0 3 0 -2 0 4 0 -8 0 0 5 0 -20 0 16 0 6 0 -40 0 96 0 0 ] EulerTan,,InvTabl,[] -EulerTan,,InvRevTabl,[] EulerTan,,RevInvTabl,[] +EulerTan,,InvRevTabl,[] EulerTan,,AccTabl,[0 1 1 0 2 2 -2 -2 1 1 0 -8 -8 -4 -4 16 16 -4 -4 1 1 0 96 96 56 56 ] EulerTan,,RevAccTabl,[0 1 1 2 2 0 1 1 -2 -2 -4 -4 -8 -8 0 1 1 -4 -4 16 16 62 62 56 56 96 ] EulerTan,,AccRevTabl,[0 0 1 0 2 2 0 3 3 1 0 4 4 -4 -4 0 5 5 -15 -15 1 0 6 6 -34 -34 62 ] EulerTan,,AntiDiagTabl,[0 1 0 0 -2 2 0 0 0 16 -8 3 0 0 0 0 -272 96 -20 4 0 0 0 0 0 7936 ] -EulerTan,,DiffxTabl,[0 1 0 0 4 0 -2 0 9 0 0 -16 0 16 0 16 0 -60 0 25 0 0 192 0 -160 0 ] EulerTan,,BinTabl,[0 1 1 0 2 4 -2 -2 1 7 0 -8 -16 -20 -16 16 16 -4 -44 -99 -159 0 96 ] EulerTan,,InvBinTabl,[0 1 -1 0 2 -4 -2 2 1 -7 0 -8 16 -20 16 16 -16 -4 44 -99 159 0 96 ] +EulerTan,,DiffxTabl,[0 1 0 0 4 0 -2 0 9 0 0 -16 0 16 0 16 0 -60 0 25 0 0 192 0 -160 0 ] EulerTan,A009832,RowSum,[0 1 2 1 -4 1 62 1 -1384 1 50522 1 -2702764 1 199360982 1 ] EulerTan,A000035,EvenSum,[0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ] EulerTan,A009744,OddSum,[0 0 2 0 -4 0 62 0 -1384 0 50522 0 -2702764 0 199360982 0 ] @@ -28,9 +28,9 @@ EulerTan,A350972,ColLeft,[0 1 0 -2 0 16 0 -272 0 7936 0 -353792 0 22368256 0 -19 EulerTan,A000004,ColRight,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] EulerTan,,BinConv,[0 1 4 7 -16 -159 -188 4383 26560 -104591 -2135836 -2607417 ] EulerTan,,InvBinConv,[0 -1 -4 -7 16 159 188 -4383 -26560 104591 2135836 2607417 ] -EulerTan,,TransSqrs,[0 0 2 12 28 0 -114 476 3480 -12384 -124550 555852 6668916 ] EulerTan,,TransNat0,[0 0 2 6 4 -20 6 434 8 -12456 10 555742 12 -35135932 14 2990414730 ] EulerTan,,TransNat1,[0 1 4 7 0 -19 68 435 -1376 -12455 50532 555743 -2702752 -35135931 ] +EulerTan,,TransSqrs,[0 0 2 12 28 0 -114 476 3480 -12384 -124550 555852 6668916 ] EulerTan,A000027,DiagRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] EulerTan,A000004,DiagRow2,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] EulerTan,A007290,DiagRow3,[-2 -8 -20 -40 -70 -112 -168 -240 -330 -440 -572 -728 -910 -1120 ] diff --git a/data/csv/Eulerian.csv b/data/csv/Eulerian.csv index 0a34560..0e75bb6 100644 --- a/data/csv/Eulerian.csv +++ b/data/csv/Eulerian.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Eulerian,A173018,Tabl,[1 1 0 1 1 0 1 4 1 0 1 11 11 1 0 1 26 66 26 1 0 1 57 302 302 57 1 0 ] Eulerian,A123125,RevTabl,[1 0 1 0 1 1 0 1 4 1 0 1 11 11 1 0 1 26 66 26 1 0 1 57 302 302 57 1 ] Eulerian,,InvTabl,[] -Eulerian,A055325,InvRevTabl,[1 0 1 0 -1 1 0 3 -4 1 0 -23 33 -11 1 0 425 -620 220 -26 1 0 -18129 ] Eulerian,,RevInvTabl,[] +Eulerian,A055325,InvRevTabl,[1 0 1 0 -1 1 0 3 -4 1 0 -23 33 -11 1 0 425 -620 220 -26 1 0 -18129 ] Eulerian,,AccTabl,[1 1 1 1 2 2 1 5 6 6 1 12 23 24 24 1 27 93 119 120 120 1 58 360 662 ] Eulerian,,RevAccTabl,[1 1 1 2 2 1 6 6 5 1 24 24 23 12 1 120 120 119 93 27 1 720 720 719 ] Eulerian,A179457,AccRevTabl,[1 0 1 0 1 2 0 1 5 6 0 1 12 23 24 0 1 27 93 119 120 0 1 58 360 662 ] Eulerian,A344393,AntiDiagTabl,[1 1 1 0 1 1 1 4 0 1 11 1 1 26 11 0 1 57 66 1 1 120 302 26 0 1 247 ] -Eulerian,A065826,DiffxTabl,[1 1 0 1 2 0 1 8 3 0 1 22 33 4 0 1 52 198 104 5 0 1 114 906 1208 ] Eulerian,,BinTabl,[1 1 1 1 2 3 1 5 10 16 1 12 34 68 115 1 27 119 303 606 1056 1 58 ] Eulerian,,InvBinTabl,[1 1 -1 1 0 -1 1 3 -6 8 1 10 -10 0 19 1 25 15 -95 190 -276 1 56 189 ] +Eulerian,A065826,DiffxTabl,[1 1 0 1 2 0 1 8 3 0 1 22 33 4 0 1 52 198 104 5 0 1 114 906 1208 ] Eulerian,A000142,RowSum,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] Eulerian,A128103,EvenSum,[1 1 1 2 12 68 360 2384 20160 185408 1814400 19781504 239500800 ] Eulerian,A262745,OddSum,[0 0 1 4 12 52 360 2656 20160 177472 1814400 20135296 239500800 ] @@ -28,9 +28,9 @@ Eulerian,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Eulerian,A000007,ColRight,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Eulerian,A011818,BinConv,[1 1 3 16 115 1056 11774 154624 2337507 39984640 763546234 ] Eulerian,A344052,InvBinConv,[1 -1 -1 8 19 -276 -1002 21216 103395 -2881180 -17620142 609297072 ] -Eulerian,A344054,TransSqrs,[0 0 1 8 64 540 4920 48720 524160 6108480 76809600 1037836800 ] Eulerian,A180119,TransNat0,[0 0 1 6 36 240 1800 15120 141120 1451520 16329600 199584000 ] Eulerian,A001710,TransNat1,[1 1 3 12 60 360 2520 20160 181440 1814400 19958400 239500800 ] +Eulerian,A344054,TransSqrs,[0 0 1 8 64 540 4920 48720 524160 6108480 76809600 1037836800 ] Eulerian,A000012,DiagRow1,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Eulerian,A000295,DiagRow2,[1 4 11 26 57 120 247 502 1013 2036 4083 8178 16369 32752 65519 ] Eulerian,A000460,DiagRow3,[1 11 66 302 1191 4293 14608 47840 152637 478271 1479726 4537314 ] diff --git a/data/csv/Eulerian2.csv b/data/csv/Eulerian2.csv index fcc9eef..c16c7a7 100644 --- a/data/csv/Eulerian2.csv +++ b/data/csv/Eulerian2.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Eulerian2,A340556,Tabl,[1 0 1 0 1 2 0 1 8 6 0 1 22 58 24 0 1 52 328 444 120 0 1 114 1452 ] Eulerian2,A163936,RevTabl,[1 1 0 2 1 0 6 8 1 0 24 58 22 1 0 120 444 328 52 1 0 720 3708 4400 ] Eulerian2,,InvTabl,[] -Eulerian2,,InvRevTabl,[] Eulerian2,,RevInvTabl,[] +Eulerian2,,InvRevTabl,[] Eulerian2,,AccTabl,[1 0 1 0 1 3 0 1 9 15 0 1 23 81 105 0 1 53 381 825 945 0 1 115 1567 ] Eulerian2,,RevAccTabl,[1 1 0 3 1 0 15 9 1 0 105 81 23 1 0 945 825 381 53 1 0 10395 9675 ] Eulerian2,,AccRevTabl,[1 1 1 2 3 3 6 14 15 15 24 82 104 105 105 120 564 892 944 945 945 ] Eulerian2,,AntiDiagTabl,[1 0 0 1 0 1 0 1 2 0 1 8 0 1 22 6 0 1 52 58 0 1 114 328 24 0 1 240 ] -Eulerian2,,DiffxTabl,[1 0 2 0 2 6 0 2 24 24 0 2 66 232 120 0 2 156 1312 2220 720 0 2 342 ] Eulerian2,,BinTabl,[1 0 1 0 1 4 0 1 10 33 0 1 24 127 392 0 1 54 487 2072 6145 0 1 116 ] Eulerian2,,InvBinTabl,[1 0 1 0 1 0 0 1 6 -15 0 1 20 -5 -80 0 1 50 175 -560 665 0 1 112 ] +Eulerian2,,DiffxTabl,[1 0 2 0 2 6 0 2 24 24 0 2 66 232 120 0 2 156 1312 2220 720 0 2 342 ] Eulerian2,A001147,RowSum,[1 1 3 15 105 945 10395 135135 2027025 34459425 654729075 ] Eulerian2,,EvenSum,[1 0 2 8 46 496 5234 66344 1021918 17237488 326501282 6884152712 ] Eulerian2,,OddSum,[0 1 1 7 59 449 5161 68791 1005107 17221937 328227793 6865157863 ] @@ -28,9 +28,9 @@ Eulerian2,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Eulerian2,A000142,ColRight,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] Eulerian2,,BinConv,[1 1 4 33 392 6145 119724 2789465 75660080 2341894185 81464695900 ] Eulerian2,,InvBinConv,[1 1 0 -15 -80 665 17136 55209 -4162752 -83069415 626654600 ] -Eulerian2,,TransSqrs,[0 1 9 87 995 13265 202545 3489255 66981915 1418241825 32839832025 ] Eulerian2,A051577,TransNat0,[0 1 5 35 315 3465 45045 675675 11486475 218243025 4583103525 ] Eulerian2,,TransNat1,[1 2 8 50 420 4410 55440 810810 13513500 252702450 5237832600 ] +Eulerian2,,TransSqrs,[0 1 9 87 995 13265 202545 3489255 66981915 1418241825 32839832025 ] Eulerian2,A002538,DiagRow1,[0 1 8 58 444 3708 33984 341136 3733920 44339040 568356480 ] Eulerian2,A002539,DiagRow2,[0 1 22 328 4400 58140 785304 11026296 162186912 2507481216 ] Eulerian2,A112008,DiagRow3,[0 1 52 1452 32120 644020 12440064 238904904 4642163952 92199790224 ] diff --git a/data/csv/EulerianB.csv b/data/csv/EulerianB.csv index 8ecd1aa..c5d2880 100644 --- a/data/csv/EulerianB.csv +++ b/data/csv/EulerianB.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence EulerianB,A060187,Tabl,[1 1 1 1 6 1 1 23 23 1 1 76 230 76 1 1 237 1682 1682 237 1 1 722 ] EulerianB,A060187,RevTabl,[1 1 1 1 6 1 1 23 23 1 1 76 230 76 1 1 237 1682 1682 237 1 1 722 ] EulerianB,A171273,InvTabl,[1 -1 1 5 -6 1 -93 115 -23 1 5993 -7436 1518 -76 1 -1272089 1578757 ] -EulerianB,A171273,InvRevTabl,[1 -1 1 5 -6 1 -93 115 -23 1 5993 -7436 1518 -76 1 -1272089 1578757 ] EulerianB,,RevInvTabl,[1 1 -1 1 -6 5 1 -23 115 -93 1 -76 1518 -7436 5993 1 -237 16330 ] +EulerianB,A171273,InvRevTabl,[1 -1 1 5 -6 1 -93 115 -23 1 5993 -7436 1518 -76 1 -1272089 1578757 ] EulerianB,,AccTabl,[1 1 2 1 7 8 1 24 47 48 1 77 307 383 384 1 238 1920 3602 3839 3840 ] EulerianB,,RevAccTabl,[1 2 1 8 7 1 48 47 24 1 384 383 307 77 1 3840 3839 3602 1920 238 1 ] EulerianB,,AccRevTabl,[1 1 2 1 7 8 1 24 47 48 1 77 307 383 384 1 238 1920 3602 3839 3840 ] EulerianB,,AntiDiagTabl,[1 1 1 1 1 6 1 23 1 1 76 23 1 237 230 1 1 722 1682 76 1 2179 10543 ] -EulerianB,,DiffxTabl,[1 1 2 1 12 3 1 46 69 4 1 152 690 304 5 1 474 5046 6728 1185 6 1 ] EulerianB,,BinTabl,[1 1 2 1 7 14 1 24 70 140 1 77 383 995 1990 1 238 2157 7440 18006 ] EulerianB,,InvBinTabl,[1 1 0 1 5 -10 1 22 -22 0 1 75 79 -387 774 1 236 1209 -2654 2654 0 ] +EulerianB,,DiffxTabl,[1 1 2 1 12 3 1 46 69 4 1 152 690 304 5 1 474 5046 6728 1185 6 1 ] EulerianB,A000165,RowSum,[1 2 8 48 384 3840 46080 645120 10321920 185794560 3715891200 ] EulerianB,,EvenSum,[1 1 2 24 232 1920 21088 322560 5338240 92897280 1832078848 ] EulerianB,,OddSum,[0 1 6 24 152 1920 24992 322560 4983680 92897280 1883812352 ] @@ -28,9 +28,9 @@ EulerianB,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] EulerianB,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] EulerianB,,BinConv,[1 2 14 140 1990 36012 795916 20758712 624278342 21265899980 ] EulerianB,,InvBinConv,[1 0 -10 0 774 0 -163332 0 67364166 0 -45892468524 0 46687337311900 ] -EulerianB,,TransSqrs,[0 1 10 124 1696 25920 441600 8332800 172892160 3917168640 ] EulerianB,A014479,TransNat0,[0 1 8 72 768 9600 138240 2257920 41287680 836075520 18579456000 ] EulerianB,A187735,TransNat1,[1 3 16 120 1152 13440 184320 2903040 51609600 1021870080 ] +EulerianB,,TransSqrs,[0 1 10 124 1696 25920 441600 8332800 172892160 3917168640 ] EulerianB,A060188,DiagRow1,[1 6 23 76 237 722 2179 6552 19673 59038 177135 531428 1594309 ] EulerianB,A060189,DiagRow2,[1 23 230 1682 10543 60657 331612 1756340 9116141 46702427 ] EulerianB,A060190,DiagRow3,[1 76 1682 23548 259723 2485288 21707972 178300904 1403080725 ] diff --git a/data/csv/FallingFact.csv b/data/csv/FallingFact.csv index 73be4d1..0822282 100644 --- a/data/csv/FallingFact.csv +++ b/data/csv/FallingFact.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence FallingFact,A008279,Tabl,[1 1 1 1 2 2 1 3 6 6 1 4 12 24 24 1 5 20 60 120 120 1 6 30 120 360 ] FallingFact,A094587,RevTabl,[1 1 1 2 2 1 6 6 3 1 24 24 12 4 1 120 120 60 20 5 1 720 720 360 120 ] FallingFact,,InvTabl,[] -FallingFact,A128229,InvRevTabl,[1 -1 1 0 -2 1 0 0 -3 1 0 0 0 -4 1 0 0 0 0 -5 1 0 0 0 0 0 -6 1 0 0 ] FallingFact,,RevInvTabl,[] +FallingFact,A128229,InvRevTabl,[1 -1 1 0 -2 1 0 0 -3 1 0 0 0 -4 1 0 0 0 0 -5 1 0 0 0 0 0 -6 1 0 0 ] FallingFact,A347667,AccTabl,[1 1 2 1 3 5 1 4 10 16 1 5 17 41 65 1 6 26 86 206 326 1 7 37 157 ] FallingFact,,RevAccTabl,[1 2 1 5 3 1 16 10 4 1 65 41 17 5 1 326 206 86 26 6 1 1957 1237 517 ] FallingFact,,AccRevTabl,[1 1 2 2 4 5 6 12 15 16 24 48 60 64 65 120 240 300 320 325 326 720 ] FallingFact,A344391,AntiDiagTabl,[1 1 1 1 1 2 1 3 2 1 4 6 1 5 12 6 1 6 20 24 1 7 30 60 24 1 8 42 120 ] -FallingFact,A121757,DiffxTabl,[1 1 2 1 4 6 1 6 18 24 1 8 36 96 120 1 10 60 240 600 720 1 12 90 ] FallingFact,A176120,BinTabl,[1 1 2 1 3 7 1 4 13 34 1 5 21 73 209 1 6 31 136 501 1546 1 7 43 229 ] FallingFact,,InvBinTabl,[1 1 0 1 1 -1 1 2 1 -4 1 3 5 -1 -15 1 4 11 14 -19 -56 1 5 19 47 37 ] +FallingFact,A121757,DiffxTabl,[1 1 2 1 4 6 1 6 18 24 1 8 36 96 120 1 10 60 240 600 720 1 12 90 ] FallingFact,A000522,RowSum,[1 2 5 16 65 326 1957 13700 109601 986410 9864101 108505112 ] FallingFact,A087208,EvenSum,[1 1 3 7 37 141 1111 5923 62217 426457 5599531 46910271 739138093 ] FallingFact,A002747,OddSum,[0 1 2 9 28 185 846 7777 47384 559953 4264570 61594841 562923252 ] @@ -28,9 +28,9 @@ FallingFact,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] FallingFact,A000142,ColRight,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] FallingFact,A002720,BinConv,[1 2 7 34 209 1546 13327 130922 1441729 17572114 234662231 ] FallingFact,A009940,InvBinConv,[1 0 -1 -4 -15 -56 -185 -204 6209 112400 1520271 19165420 237686449 ] -FallingFact,A343276,TransSqrs,[0 1 10 81 652 5545 50886 506905 5480056 64116657 808856290 ] FallingFact,A093964,TransNat0,[0 1 6 33 196 1305 9786 82201 767208 7891281 88776910 1085051121 ] FallingFact,A001339,TransNat1,[1 3 11 49 261 1631 11743 95901 876809 8877691 98641011 1193556233 ] +FallingFact,A343276,TransSqrs,[0 1 10 81 652 5545 50886 506905 5480056 64116657 808856290 ] FallingFact,A000142,DiagRow1,[1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] FallingFact,A001710,DiagRow2,[1 3 12 60 360 2520 20160 181440 1814400 19958400 239500800 ] FallingFact,A001715,DiagRow3,[1 4 20 120 840 6720 60480 604800 6652800 79833600 1037836800 ] diff --git a/data/csv/Fibonacci.csv b/data/csv/Fibonacci.csv index 281e8ef..35c9321 100644 --- a/data/csv/Fibonacci.csv +++ b/data/csv/Fibonacci.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Fibonacci,A354267,Tabl,[1 0 1 1 1 1 1 2 2 1 2 3 4 3 1 3 5 7 7 4 1 5 8 12 14 11 5 1 8 13 20 ] Fibonacci,,RevTabl,[1 1 0 1 1 1 1 2 2 1 1 3 4 3 2 1 4 7 7 5 3 1 5 11 14 12 8 5 1 6 16 ] Fibonacci,,InvTabl,[1 0 1 -1 -1 1 1 0 -2 1 -1 1 2 -3 1 1 -2 -1 5 -4 1 -1 3 -1 -6 9 -5 ] -Fibonacci,,InvRevTabl,[] Fibonacci,,RevInvTabl,[1 1 0 1 -1 -1 1 -2 0 1 1 -3 2 1 -1 1 -4 5 -1 -2 1 1 -5 9 -6 -1 3 ] +Fibonacci,,InvRevTabl,[] Fibonacci,,AccTabl,[1 0 1 1 2 3 1 3 5 6 2 5 9 12 13 3 8 15 22 26 27 5 13 25 39 50 55 ] Fibonacci,,RevAccTabl,[1 1 0 3 2 1 6 5 3 1 13 12 9 5 2 27 26 22 15 8 3 56 55 50 39 25 13 ] Fibonacci,,AccRevTabl,[1 1 1 1 2 3 1 3 5 6 1 4 8 11 13 1 5 12 19 24 27 1 6 17 31 43 51 56 ] Fibonacci,,AntiDiagTabl,[1 0 1 1 1 1 2 2 1 3 3 2 5 5 4 1 8 8 7 3 13 13 12 7 1 21 21 20 14 4 ] -Fibonacci,,DiffxTabl,[1 0 2 1 2 3 1 4 6 4 2 6 12 12 5 3 10 21 28 20 6 5 16 36 56 55 30 7 ] Fibonacci,,BinTabl,[1 0 1 1 2 4 1 3 7 14 2 5 12 26 51 3 8 20 46 97 189 5 13 33 79 176 ] Fibonacci,,InvBinTabl,[1 0 1 1 0 0 1 1 -1 0 2 1 0 -2 3 3 2 0 -2 1 3 5 3 1 -3 0 5 -7 8 5 2 ] +Fibonacci,,DiffxTabl,[1 0 2 1 2 3 1 4 6 4 2 6 12 12 5 3 10 21 28 20 6 5 16 36 56 55 30 7 ] Fibonacci,A099036,RowSum,[1 1 3 6 13 27 56 115 235 478 969 1959 3952 7959 16007 32158 64549 ] Fibonacci,,EvenSum,[1 0 2 3 7 14 29 59 120 243 491 990 1993 4007 8048 16151 32391 ] Fibonacci,A099036,OddSum,[0 1 1 3 6 13 27 56 115 235 478 969 1959 3952 7959 16007 32158 ] @@ -28,9 +28,9 @@ Fibonacci,A039834,ColLeft,[1 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 15 Fibonacci,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Fibonacci,,BinConv,[1 1 4 14 51 189 709 2683 10220 39130 150438 580328 2245004 8705686 ] Fibonacci,,InvBinConv,[1 1 0 0 3 3 -7 -7 28 28 -98 -98 364 364 -1352 -1352 5083 5083 ] -Fibonacci,,TransSqrs,[0 1 5 19 62 185 519 1392 3607 9095 22430 54309 129475 304616 ] Fibonacci,A079282,TransNat0,[0 1 3 9 24 61 149 354 823 1881 4240 9449 20857 45666 99291 214589 ] Fibonacci,,TransNat1,[1 2 6 15 37 88 205 469 1058 2359 5209 11408 24809 53625 115298 ] +Fibonacci,,TransSqrs,[0 1 5 19 62 185 519 1392 3607 9095 22430 54309 129475 304616 ] Fibonacci,A001477,DiagRow1,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] Fibonacci,A000124,DiagRow2,[1 2 4 7 11 16 22 29 37 46 56 67 79 92 106 121 137 154 172 191 ] Fibonacci,A004006,DiagRow3,[1 3 7 14 25 41 63 92 129 175 231 298 377 469 575 696 833 987 1159 ] diff --git a/data/csv/Fubini.csv b/data/csv/Fubini.csv index afd3b9a..d5cea10 100644 --- a/data/csv/Fubini.csv +++ b/data/csv/Fubini.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Fubini,A131689,Tabl,[1 0 1 0 1 2 0 1 6 6 0 1 14 36 24 0 1 30 150 240 120 0 1 62 540 ] Fubini,A090582,RevTabl,[1 1 0 2 1 0 6 6 1 0 24 36 14 1 0 120 240 150 30 1 0 720 1800 1560 ] Fubini,,InvTabl,[] -Fubini,,InvRevTabl,[] Fubini,,RevInvTabl,[] +Fubini,,InvRevTabl,[] Fubini,,AccTabl,[1 0 1 0 1 3 0 1 7 13 0 1 15 51 75 0 1 31 181 421 541 0 1 63 603 ] Fubini,,RevAccTabl,[1 1 0 3 1 0 13 7 1 0 75 51 15 1 0 541 421 181 31 1 0 4683 3963 ] Fubini,,AccRevTabl,[1 1 1 2 3 3 6 12 13 13 24 60 74 75 75 120 360 510 540 541 541 720 ] Fubini,A344392,AntiDiagTabl,[1 0 0 1 0 1 0 1 2 0 1 6 0 1 14 6 0 1 30 36 0 1 62 150 24 0 1 126 ] -Fubini,,DiffxTabl,[1 0 2 0 2 6 0 2 18 24 0 2 42 144 120 0 2 90 600 1200 720 0 2 186 ] Fubini,A089072,BinTabl,[1 0 1 0 1 4 0 1 8 27 0 1 16 81 256 0 1 32 243 1024 3125 0 1 64 729 ] Fubini,,InvBinTabl,[1 0 1 0 1 0 0 1 4 -9 0 1 12 -3 -40 0 1 28 63 -184 125 0 1 60 357 ] +Fubini,,DiffxTabl,[1 0 2 0 2 6 0 2 18 24 0 2 42 144 120 0 2 90 600 1200 720 0 2 186 ] Fubini,A000670,RowSum,[1 1 3 13 75 541 4683 47293 545835 7087261 102247563 1622632573 ] Fubini,A052841,EvenSum,[1 0 2 6 38 270 2342 23646 272918 3543630 51123782 811316286 ] Fubini,A089677,OddSum,[0 1 1 7 37 271 2341 23647 272917 3543631 51123781 811316287 ] @@ -28,9 +28,9 @@ Fubini,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Fubini,A000142,ColRight,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] Fubini,A000312,BinConv,[1 1 4 27 256 3125 46656 823543 16777216 387420489 10000000000 ] Fubini,A344053,InvBinConv,[1 1 0 -9 -40 125 3444 18571 -241872 -5796711 -24387220 1132278191 ] -Fubini,A083411,TransSqrs,[0 1 9 79 765 8311 100989 1362439 20246445 328972471 5805917469 ] Fubini,A069321,TransNat0,[0 1 5 31 233 2071 21305 249271 3270713 47580151 760192505 ] Fubini,A005649,TransNat1,[1 2 8 44 308 2612 25988 296564 3816548 54667412 862440068 ] +Fubini,A083411,TransSqrs,[0 1 9 79 765 8311 100989 1362439 20246445 328972471 5805917469 ] Fubini,A180119,DiagRow1,[0 1 6 36 240 1800 15120 141120 1451520 16329600 199584000 ] Fubini,A037960,DiagRow2,[0 1 14 150 1560 16800 191520 2328480 30240000 419126400 6187104000 ] Fubini,A037961,DiagRow3,[0 1 30 540 8400 126000 1905120 29635200 479001600 8083152000 ] diff --git a/data/csv/FussCatalan.csv b/data/csv/FussCatalan.csv index 8cd3a27..7c1aa05 100644 --- a/data/csv/FussCatalan.csv +++ b/data/csv/FussCatalan.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence FussCatalan,A355173,Tabl,[1 0 1 0 1 2 0 1 3 5 0 1 4 9 14 0 1 5 14 28 42 0 1 6 20 48 90 132 0 ] FussCatalan,A054445,RevTabl,[1 1 0 2 1 0 5 3 1 0 14 9 4 1 0 42 28 14 5 1 0 132 90 48 20 6 1 0 ] FussCatalan,,InvTabl,[] -FussCatalan,,InvRevTabl,[] FussCatalan,,RevInvTabl,[] +FussCatalan,,InvRevTabl,[] FussCatalan,,AccTabl,[1 0 1 0 1 3 0 1 4 9 0 1 5 14 28 0 1 6 20 48 90 0 1 7 27 75 165 297 ] FussCatalan,,RevAccTabl,[1 1 0 3 1 0 9 4 1 0 28 14 5 1 0 90 48 20 6 1 0 297 165 75 27 7 1 0 ] FussCatalan,,AccRevTabl,[1 1 1 2 3 3 5 8 9 9 14 23 27 28 28 42 70 84 89 90 90 132 222 270 ] FussCatalan,A289871,AntiDiagTabl,[1 0 0 1 0 1 0 1 2 0 1 3 0 1 4 5 0 1 5 9 0 1 6 14 14 0 1 7 20 28 0 ] -FussCatalan,,DiffxTabl,[1 0 2 0 2 6 0 2 9 20 0 2 12 36 70 0 2 15 56 140 252 0 2 18 80 240 ] FussCatalan,,BinTabl,[1 0 1 0 1 4 0 1 5 17 0 1 6 24 78 0 1 7 32 118 377 0 1 8 41 168 595 ] FussCatalan,,InvBinTabl,[1 0 1 0 1 0 0 1 1 -1 0 1 2 0 -2 0 1 3 2 -2 -3 0 1 4 5 0 -5 -4 0 1 ] +FussCatalan,,DiffxTabl,[1 0 2 0 2 6 0 2 9 20 0 2 12 36 70 0 2 15 56 140 252 0 2 18 80 240 ] FussCatalan,A071724,RowSum,[1 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888 1931540 ] FussCatalan,,EvenSum,[1 0 2 3 18 33 186 379 2120 4596 25724 58082 325878 757259 4260282 ] FussCatalan,,OddSum,[0 1 1 6 10 57 111 622 1312 7338 16266 91144 209010 1174281 2760123 ] @@ -28,9 +28,9 @@ FussCatalan,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] FussCatalan,A000108,ColRight,[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 ] FussCatalan,,BinConv,[1 1 4 17 78 377 1888 9697 50746 269393 1446300 7835313 42762822 ] FussCatalan,A097045,InvBinConv,[1 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 ] -FussCatalan,,TransSqrs,[0 1 9 58 322 1645 7975 37310 170144 761226 3355970 14624148 ] FussCatalan,,TransNat0,[0 1 5 22 92 375 1507 6006 23816 94146 371450 1463836 5764904 ] FussCatalan,,TransNat1,[1 2 8 31 120 465 1804 7007 27248 106080 413440 1613062 6299792 ] +FussCatalan,,TransSqrs,[0 1 9 58 322 1645 7975 37310 170144 761226 3355970 14624148 ] FussCatalan,A000245,DiagRow1,[0 1 3 9 28 90 297 1001 3432 11934 41990 149226 534888 1931540 ] FussCatalan,A099376,DiagRow2,[0 1 4 14 48 165 572 2002 7072 25194 90440 326876 1188640 4345965 ] FussCatalan,A115144,DiagRow3,[0 1 5 20 75 275 1001 3640 13260 48450 177650 653752 2414425 ] diff --git a/data/csv/Gaussq2.csv b/data/csv/Gaussq2.csv index e858c6b..c66337e 100644 --- a/data/csv/Gaussq2.csv +++ b/data/csv/Gaussq2.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Gaussq2,A022166,Tabl,[1 1 1 1 3 1 1 7 7 1 1 15 35 15 1 1 31 155 155 31 1 1 63 651 1395 ] Gaussq2,A022166,RevTabl,[1 1 1 1 3 1 1 7 7 1 1 15 35 15 1 1 31 155 155 31 1 1 63 651 1395 ] Gaussq2,A135950,InvTabl,[1 -1 1 2 -3 1 -8 14 -7 1 64 -120 70 -15 1 -1024 1984 -1240 310 -31 ] -Gaussq2,A135950,InvRevTabl,[1 -1 1 2 -3 1 -8 14 -7 1 64 -120 70 -15 1 -1024 1984 -1240 310 -31 ] Gaussq2,A158474,RevInvTabl,[1 1 -1 1 -3 2 1 -7 14 -8 1 -15 70 -120 64 1 -31 310 -1240 1984 ] +Gaussq2,A135950,InvRevTabl,[1 -1 1 2 -3 1 -8 14 -7 1 64 -120 70 -15 1 -1024 1984 -1240 310 -31 ] Gaussq2,,AccTabl,[1 1 2 1 4 5 1 8 15 16 1 16 51 66 67 1 32 187 342 373 374 1 64 715 ] Gaussq2,,RevAccTabl,[1 2 1 5 4 1 16 15 8 1 67 66 51 16 1 374 373 342 187 32 1 2825 2824 ] Gaussq2,,AccRevTabl,[1 1 2 1 4 5 1 8 15 16 1 16 51 66 67 1 32 187 342 373 374 1 64 715 ] Gaussq2,,AntiDiagTabl,[1 1 1 1 1 3 1 7 1 1 15 7 1 31 35 1 1 63 155 15 1 127 651 155 1 1 ] -Gaussq2,,DiffxTabl,[1 1 2 1 6 3 1 14 21 4 1 30 105 60 5 1 62 465 620 155 6 1 126 1953 ] Gaussq2,,BinTabl,[1 1 2 1 4 8 1 8 22 44 1 16 66 166 332 1 32 218 714 1706 3412 1 64 ] Gaussq2,,InvBinTabl,[1 1 0 1 2 -4 1 6 -6 0 1 14 6 -46 92 1 30 94 -218 218 0 1 62 526 ] +Gaussq2,,DiffxTabl,[1 1 2 1 6 3 1 14 21 4 1 30 105 60 5 1 62 465 620 155 6 1 126 1953 ] Gaussq2,A006116,RowSum,[1 2 5 16 67 374 2825 29212 417199 8283458 229755605 8933488744 ] Gaussq2,A289541,EvenSum,[1 1 2 8 37 187 1304 14606 222379 4141729 107836478 4466744372 ] Gaussq2,,OddSum,[0 1 3 8 30 187 1521 14606 194820 4141729 121919127 4466744372 ] @@ -28,9 +28,9 @@ Gaussq2,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Gaussq2,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Gaussq2,,BinConv,[1 2 8 44 332 3412 48188 940564 25545052 969582644 51635485244 ] Gaussq2,,InvBinConv,[1 0 -4 0 92 0 -9124 0 3774172 0 -6459170980 0 45480123143132 0 ] -Gaussq2,,TransSqrs,[0 1 7 44 306 2567 27249 377366 6960476 173496553 5905329123 ] Gaussq2,,TransNat0,[0 1 5 24 134 935 8475 102242 1668796 37275561 1148778025 ] Gaussq2,,TransNat1,[1 3 10 40 201 1309 11300 131454 2085995 45559019 1378533630 ] +Gaussq2,,TransSqrs,[0 1 7 44 306 2567 27249 377366 6960476 173496553 5905329123 ] Gaussq2,A000225,DiagRow1,[1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 16383 32767 65535 ] Gaussq2,A006095,DiagRow2,[1 7 35 155 651 2667 10795 43435 174251 698027 2794155 11180715 ] Gaussq2,A006096,DiagRow3,[1 15 155 1395 11811 97155 788035 6347715 50955971 408345795 ] diff --git a/data/csv/Genocchi.csv b/data/csv/Genocchi.csv index 75acca2..d4f7292 100644 --- a/data/csv/Genocchi.csv +++ b/data/csv/Genocchi.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Genocchi,A297703,Tabl,[1 1 1 2 3 3 8 14 17 17 56 104 138 155 155 608 1160 1608 1918 2073 ] Genocchi,,RevTabl,[1 1 1 3 3 2 17 17 14 8 155 155 138 104 56 2073 2073 1918 1608 1160 ] Genocchi,,InvTabl,[] -Genocchi,,InvRevTabl,[] Genocchi,,RevInvTabl,[] +Genocchi,,InvRevTabl,[] Genocchi,,AccTabl,[1 1 2 2 5 8 8 22 39 56 56 160 298 453 608 608 1768 3376 5294 7367 ] Genocchi,,RevAccTabl,[1 2 1 8 5 2 56 39 22 8 608 453 298 160 56 9440 7367 5294 3376 1768 ] Genocchi,,AccRevTabl,[1 1 2 3 6 8 17 34 48 56 155 310 448 552 608 2073 4146 6064 7672 ] Genocchi,,AntiDiagTabl,[1 1 2 1 8 3 56 14 3 608 104 17 9440 1160 138 17 198272 18272 1608 ] -Genocchi,,DiffxTabl,[1 1 2 2 6 9 8 28 51 68 56 208 414 620 775 608 2320 4824 7672 10365 ] Genocchi,,BinTabl,[1 1 2 2 5 11 8 22 53 118 56 160 402 937 2075 608 1768 4536 10830 ] Genocchi,,InvBinTabl,[1 1 0 2 1 -1 8 6 -3 0 56 48 -14 -3 3 608 552 -104 -34 17 0 9440 ] +Genocchi,,DiffxTabl,[1 1 2 2 6 9 8 28 51 68 56 208 414 620 775 608 2320 4824 7672 10365 ] Genocchi,A005439,RowSum,[1 2 8 56 608 9440 198272 5410688 186043904 7867739648 401293838336 ] Genocchi,,EvenSum,[1 1 5 25 349 4289 109765 2500393 100852205 3681669649 214553157013 ] Genocchi,,OddSum,[0 1 3 31 259 5151 88507 2910295 85191699 4186069999 186740681323 ] @@ -28,9 +28,9 @@ Genocchi,A005439,ColLeft,[1 1 2 8 56 608 9440 198272 5410688 186043904 786773964 Genocchi,A001469,ColRight,[1 1 3 17 155 2073 38227 929569 28820619 1109652905 51943281731 ] Genocchi,,BinConv,[1 2 11 118 2075 54106 1958291 93874558 5754873035 439059245986 ] Genocchi,A036968,InvBinConv,[1 0 -1 0 3 0 -17 0 155 0 -2073 0 38227 0 -929569 0 28820619 0 ] -Genocchi,,TransSqrs,[0 1 15 235 4531 109847 3320431 123273403 5533969803 296088979327 ] Genocchi,,TransNat0,[0 1 9 99 1465 28787 731297 23409931 923609001 44075542083 ] Genocchi,A001469,TransNat1,[1 3 17 155 2073 38227 929569 28820619 1109652905 51943281731 ] +Genocchi,,TransSqrs,[0 1 15 235 4531 109847 3320431 123273403 5533969803 296088979327 ] Genocchi,A001469,DiagRow1,[1 3 17 155 2073 38227 929569 28820619 1109652905 51943281731 ] Genocchi,,DiagRow2,[2 14 138 1918 36154 891342 27891050 1080832286 50833628826 ] Genocchi,,DiagRow3,[8 104 1608 32008 814888 26031912 1023191048 48614323016 ] diff --git a/data/csv/Harmonic.csv b/data/csv/Harmonic.csv index ee9fb22..cdbac33 100644 --- a/data/csv/Harmonic.csv +++ b/data/csv/Harmonic.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Harmonic,A358694,Tabl,[1 0 1 0 2 1 0 6 4 1 0 24 18 7 1 0 120 96 46 11 1 0 720 600 326 101 ] Harmonic,A109822,RevTabl,[1 1 0 1 2 0 1 4 6 0 1 7 18 24 0 1 11 46 96 120 0 1 16 101 326 600 ] Harmonic,A227341,InvTabl,[1 0 1 0 -2 1 0 2 -4 1 0 -2 10 -7 1 0 2 -22 31 -11 1 0 -2 46 -115 ] -Harmonic,,InvRevTabl,[] Harmonic,,RevInvTabl,[1 1 0 1 -2 0 1 -4 2 0 1 -7 10 -2 0 1 -11 31 -22 2 0 1 -16 75 -115 ] +Harmonic,,InvRevTabl,[] Harmonic,,AccTabl,[1 0 1 0 2 3 0 6 10 11 0 24 42 49 50 0 120 216 262 273 274 0 720 ] Harmonic,,RevAccTabl,[1 1 0 3 2 0 11 10 6 0 50 49 42 24 0 274 273 262 216 120 0 1764 ] Harmonic,,AccRevTabl,[1 1 1 1 3 3 1 5 11 11 1 8 26 50 50 1 12 58 154 274 274 1 17 118 ] Harmonic,,AntiDiagTabl,[1 0 0 1 0 2 0 6 1 0 24 4 0 120 18 1 0 720 96 7 0 5040 600 46 1 0 ] -Harmonic,,DiffxTabl,[1 0 2 0 4 3 0 12 12 4 0 48 54 28 5 0 240 288 184 55 6 0 1440 1800 ] Harmonic,,BinTabl,[1 0 1 0 2 5 0 6 16 31 0 24 66 133 233 0 120 336 694 1251 2076 0 ] Harmonic,,InvBinTabl,[1 0 1 0 2 -3 0 6 -8 7 0 24 -30 25 -15 0 120 -144 118 -77 46 0 720 ] +Harmonic,,DiffxTabl,[1 0 2 0 4 3 0 12 12 4 0 48 54 28 5 0 240 288 184 55 6 0 1440 1800 ] Harmonic,A000254,RowSum,[1 1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] Harmonic,A182541,EvenSum,[1 0 1 4 19 107 702 5274 44712 422568 4407120 50292720 623471040 ] Harmonic,,OddSum,[0 1 2 7 31 167 1062 7794 64872 604008 6221520 70251120 862971840 ] @@ -28,9 +28,9 @@ Harmonic,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Harmonic,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Harmonic,,BinConv,[1 1 5 31 233 2076 21452 252372 3329349 48644302 779147639 ] Harmonic,,InvBinConv,[1 1 -3 7 -15 46 -420 5384 -62811 656620 -6213537 52114085 ] -Harmonic,,TransSqrs,[0 1 6 31 175 1119 8106 66002 598524 5992036 65708368 783813920 ] Harmonic,A081052,TransNat0,[0 1 4 17 85 499 3388 26200 227708 2199276 23382216 271461816 ] Harmonic,,TransNat1,[1 2 7 28 135 773 5152 39268 337292 3225852 34010856 392005656 ] +Harmonic,,TransSqrs,[0 1 6 31 175 1119 8106 66002 598524 5992036 65708368 783813920 ] Harmonic,A000124,DiagRow1,[0 2 4 7 11 16 22 29 37 46 56 67 79 92 106 121 137 154 172 191 ] Harmonic,A308305,DiagRow2,[0 6 18 46 101 197 351 583 916 1376 1992 2796 3823 5111 6701 8637 ] Harmonic,,DiagRow3,[0 24 96 326 932 2311 5119 10366 19526 34662 58566 94914 148436 ] diff --git a/data/csv/HermiteE.csv b/data/csv/HermiteE.csv index 87f13dc..0440af3 100644 --- a/data/csv/HermiteE.csv +++ b/data/csv/HermiteE.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -HermiteE,A066325,Tabl,[1 0 1 1 0 1 0 3 0 1 3 0 6 0 1 0 15 0 10 0 1 15 0 45 0 15 0 1 0 105 ] +HermiteE,A099174,Tabl,[1 0 1 1 0 1 0 3 0 1 3 0 6 0 1 0 15 0 10 0 1 15 0 45 0 15 0 1 0 105 ] HermiteE,A073278,RevTabl,[1 1 0 1 0 1 1 0 3 0 1 0 6 0 3 1 0 10 0 15 0 1 0 15 0 45 0 15 1 0 ] HermiteE,A066325,InvTabl,[1 0 1 -1 0 1 0 -3 0 1 3 0 -6 0 1 0 15 0 -10 0 1 -15 0 45 0 -15 0 1 ] -HermiteE,,InvRevTabl,[] HermiteE,A073278,RevInvTabl,[1 1 0 1 0 -1 1 0 -3 0 1 0 -6 0 3 1 0 -10 0 15 0 1 0 -15 0 45 0 -15 ] +HermiteE,,InvRevTabl,[] HermiteE,,AccTabl,[1 0 1 1 1 2 0 3 3 4 3 3 9 9 10 0 15 15 25 25 26 15 15 60 60 75 75 ] HermiteE,,RevAccTabl,[1 1 0 2 1 1 4 3 3 0 10 9 9 3 3 26 25 25 15 15 0 76 75 75 60 60 15 ] HermiteE,,AccRevTabl,[1 1 1 1 1 2 1 1 4 4 1 1 7 7 10 1 1 11 11 26 26 1 1 16 16 61 61 76 ] HermiteE,A001497,AntiDiagTabl,[1 0 1 1 0 0 3 3 1 0 0 0 15 15 6 1 0 0 0 0 105 105 45 10 1 0 0 0 0 ] -HermiteE,,DiffxTabl,[1 0 2 1 0 3 0 6 0 4 3 0 18 0 5 0 30 0 40 0 6 15 0 135 0 75 0 7 0 ] HermiteE,,BinTabl,[1 0 1 1 1 2 0 3 6 10 3 3 9 21 40 0 15 30 55 100 176 15 15 60 150 ] HermiteE,,InvBinTabl,[1 0 1 1 -1 2 0 3 -6 10 3 -3 9 -21 40 0 15 -30 55 -100 176 15 -15 ] +HermiteE,,DiffxTabl,[1 0 2 1 0 3 0 6 0 4 3 0 18 0 5 0 30 0 40 0 6 15 0 135 0 75 0 7 0 ] HermiteE,A000085,RowSum,[1 1 2 4 10 26 76 232 764 2620 9496 35696 140152 568504 2390480 ] HermiteE,A066223,EvenSum,[1 0 2 0 10 0 76 0 764 0 9496 0 140152 0 2390480 0 46206736 0 ] HermiteE,A066224,OddSum,[0 1 0 4 0 26 0 232 0 2620 0 35696 0 568504 0 10349536 0 211799312 ] @@ -28,9 +28,9 @@ HermiteE,A123022,ColLeft,[1 0 1 0 3 0 15 0 105 0 945 0 10395 0 135135 0 2027025 HermiteE,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] HermiteE,A344501,BinConv,[1 1 2 10 40 176 916 4852 27350 163270 1009396 6504356 43400512 ] HermiteE,A344501,InvBinConv,[1 1 2 10 40 176 916 4852 27350 163270 1009396 6504356 43400512 ] -HermiteE,,TransSqrs,[0 1 4 12 40 130 456 1624 6112 23580 94960 392656 1681824 7390552 ] HermiteE,A013989,TransNat0,[0 1 2 6 16 50 156 532 1856 6876 26200 104456 428352 1821976 ] HermiteE,A000085,TransNat1,[1 2 4 10 26 76 232 764 2620 9496 35696 140152 568504 2390480 ] +HermiteE,,TransSqrs,[0 1 4 12 40 130 456 1624 6112 23580 94960 392656 1681824 7390552 ] HermiteE,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] HermiteE,A000217,DiagRow2,[1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 ] HermiteE,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/HermiteH.csv b/data/csv/HermiteH.csv index 500703a..456a413 100644 --- a/data/csv/HermiteH.csv +++ b/data/csv/HermiteH.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence HermiteH,A060821,Tabl,[1 0 2 2 0 4 0 12 0 8 12 0 48 0 16 0 120 0 160 0 32 120 0 720 0 480 ] HermiteH,,RevTabl,[1 2 0 4 0 2 8 0 12 0 16 0 48 0 12 32 0 160 0 120 0 64 0 480 0 720 ] HermiteH,,InvTabl,[] -HermiteH,,InvRevTabl,[] HermiteH,,RevInvTabl,[] +HermiteH,,InvRevTabl,[] HermiteH,,AccTabl,[1 0 2 2 2 6 0 12 12 20 12 12 60 60 76 0 120 120 280 280 312 120 ] HermiteH,,RevAccTabl,[1 2 0 6 2 2 20 12 12 0 76 60 60 12 12 312 280 280 120 120 0 1384 ] HermiteH,,AccRevTabl,[1 2 2 4 4 6 8 8 20 20 16 16 64 64 76 32 32 192 192 312 312 64 64 ] HermiteH,A109767,AntiDiagTabl,[1 0 2 2 0 0 12 12 4 0 0 0 120 120 48 8 0 0 0 0 1680 1680 720 160 ] -HermiteH,,DiffxTabl,[1 0 4 2 0 12 0 24 0 32 12 0 144 0 80 0 240 0 640 0 192 120 0 2160 ] HermiteH,,BinTabl,[1 0 2 2 2 6 0 12 24 44 12 12 60 156 316 0 120 240 520 1120 2232 ] HermiteH,,InvBinTabl,[1 0 2 2 -2 6 0 12 -24 44 12 -12 60 -156 316 0 120 -240 520 -1120 ] +HermiteH,,DiffxTabl,[1 0 4 2 0 12 0 24 0 32 12 0 144 0 80 0 240 0 640 0 192 120 0 2160 ] HermiteH,A000898,RowSum,[1 2 6 20 76 312 1384 6512 32400 168992 921184 5222208 30710464 ] HermiteH,,EvenSum,[1 0 6 0 76 0 1384 0 32400 0 921184 0 30710464 0 1171979904 0 ] HermiteH,,OddSum,[0 2 0 20 0 312 0 6512 0 168992 0 5222208 0 186753920 0 7573069568 ] @@ -28,9 +28,9 @@ HermiteH,A067994,ColLeft,[1 0 2 0 12 0 120 0 1680 0 30240 0 665280 0 17297280 0 HermiteH,A000079,ColRight,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] HermiteH,,BinConv,[1 2 6 44 316 2232 18184 157712 1419408 13474592 133216864 ] HermiteH,,InvBinConv,[1 2 6 44 316 2232 18184 157712 1419408 13474592 133216864 ] -HermiteH,,TransSqrs,[0 2 16 84 448 2360 12864 71792 414208 2458656 15043840 94622528 ] HermiteH,,TransNat0,[0 2 8 36 160 760 3744 19376 104192 583200 3379840 20266048 ] HermiteH,,TransNat1,[1 4 14 56 236 1072 5128 25888 136592 752192 4301024 25488256 ] +HermiteH,,TransSqrs,[0 2 16 84 448 2360 12864 71792 414208 2458656 15043840 94622528 ] HermiteH,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] HermiteH,A001815,DiagRow2,[2 12 48 160 480 1344 3584 9216 23040 56320 135168 319488 745472 ] HermiteH,A000004,DiagRow3,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/Laguerre.csv b/data/csv/Laguerre.csv index 8f32970..a645912 100644 --- a/data/csv/Laguerre.csv +++ b/data/csv/Laguerre.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Laguerre,A021009,Tabl,[1 1 1 2 4 1 6 18 9 1 24 96 72 16 1 120 600 600 200 25 1 720 4320 ] Laguerre,A021010,RevTabl,[1 1 1 1 4 2 1 9 18 6 1 16 72 96 24 1 25 200 600 600 120 1 36 450 ] Laguerre,A021009,InvTabl,[1 -1 1 2 -4 1 -6 18 -9 1 24 -96 72 -16 1 -120 600 -600 200 -25 1 ] -Laguerre,,InvRevTabl,[] Laguerre,A021010,RevInvTabl,[1 1 -1 1 -4 2 1 -9 18 -6 1 -16 72 -96 24 1 -25 200 -600 600 -120 1 ] +Laguerre,,InvRevTabl,[] Laguerre,,AccTabl,[1 1 2 2 6 7 6 24 33 34 24 120 192 208 209 120 720 1320 1520 1545 ] Laguerre,,RevAccTabl,[1 2 1 7 6 2 34 33 24 6 209 208 192 120 24 1546 1545 1520 1320 720 ] Laguerre,,AccRevTabl,[1 1 2 1 5 7 1 10 28 34 1 17 89 185 209 1 26 226 826 1426 1546 1 37 ] Laguerre,A084950,AntiDiagTabl,[1 1 2 1 6 4 24 18 1 120 96 9 720 600 72 1 5040 4320 600 16 40320 ] -Laguerre,,DiffxTabl,[1 1 2 2 8 3 6 36 27 4 24 192 216 64 5 120 1200 1800 800 125 6 720 ] Laguerre,,BinTabl,[1 1 2 2 6 11 6 24 51 88 24 120 288 544 905 120 720 1920 3920 6945 ] Laguerre,,InvBinTabl,[1 1 0 2 2 -5 6 12 -21 22 24 72 -96 64 9 120 480 -480 80 545 -1244 ] +Laguerre,,DiffxTabl,[1 1 2 2 8 3 6 36 27 4 24 192 216 64 5 120 1200 1800 800 125 6 720 ] Laguerre,A002720,RowSum,[1 2 7 34 209 1546 13327 130922 1441729 17572114 234662231 ] Laguerre,A331325,EvenSum,[1 1 3 15 97 745 6571 65359 723969 8842257 118091251 1712261551 ] Laguerre,A331326,OddSum,[0 1 4 19 112 801 6756 65563 717760 8729857 116570980 1693096131 ] @@ -28,9 +28,9 @@ Laguerre,A000142,ColLeft,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 Laguerre,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Laguerre,A216831,BinConv,[1 2 11 88 905 11246 162607 2668436 48830273 983353690 21570885011 ] Laguerre,A343840,InvBinConv,[1 0 -5 22 9 -1244 14335 -79470 -586943 25131304 -434574909 ] -Laguerre,A105219,TransSqrs,[0 1 8 63 544 5225 55656 653023 8379008 116780049 1757211400 ] Laguerre,A103194,TransNat0,[0 1 6 39 292 2505 24306 263431 3154824 41368977 589410910 ] Laguerre,A000262,TransNat1,[1 3 13 73 501 4051 37633 394353 4596553 58941091 824073141 ] +Laguerre,A105219,TransSqrs,[0 1 8 63 544 5225 55656 653023 8379008 116780049 1757211400 ] Laguerre,A000290,DiagRow1,[1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 ] Laguerre,A163102,DiagRow2,[2 18 72 200 450 882 1568 2592 4050 6050 8712 12168 16562 22050 ] Laguerre,A179058,DiagRow3,[6 96 600 2400 7350 18816 42336 86400 163350 290400 490776 794976 ] diff --git a/data/csv/Lah.csv b/data/csv/Lah.csv index 689a041..5896951 100644 --- a/data/csv/Lah.csv +++ b/data/csv/Lah.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -Lah,A111596,Tabl,[1 0 1 0 2 1 0 6 6 1 0 24 36 12 1 0 120 240 120 20 1 0 720 1800 ] +Lah,A271703,Tabl,[1 0 1 0 2 1 0 6 6 1 0 24 36 12 1 0 120 240 120 20 1 0 720 1800 ] Lah,A089231,RevTabl,[1 1 0 1 2 0 1 6 6 0 1 12 36 24 0 1 20 120 240 120 0 1 30 300 1200 ] Lah,A111596,InvTabl,[1 0 1 0 -2 1 0 6 -6 1 0 -24 36 -12 1 0 120 -240 120 -20 1 0 -720 ] -Lah,,InvRevTabl,[] Lah,A089231,RevInvTabl,[1 1 0 1 -2 0 1 -6 6 0 1 -12 36 -24 0 1 -20 120 -240 120 0 1 -30 ] +Lah,,InvRevTabl,[] Lah,A349776,AccTabl,[1 0 1 0 2 3 0 6 12 13 0 24 60 72 73 0 120 360 480 500 501 0 720 ] Lah,,RevAccTabl,[1 1 0 3 2 0 13 12 6 0 73 72 60 24 0 501 500 480 360 120 0 4051 ] Lah,,AccRevTabl,[1 1 1 1 3 3 1 7 13 13 1 13 49 73 73 1 21 141 381 501 501 1 31 331 ] Lah,A330609,AntiDiagTabl,[1 0 0 1 0 2 0 6 1 0 24 6 0 120 36 1 0 720 240 12 0 5040 1800 120 1 ] -Lah,A360205,DiffxTabl,[1 0 2 0 4 3 0 12 18 4 0 48 108 48 5 0 240 720 480 100 6 0 1440 ] Lah,,BinTabl,[1 0 1 0 2 5 0 6 18 37 0 24 84 192 361 0 120 480 1200 2420 4301 0 ] Lah,,InvBinTabl,[1 0 1 0 2 -3 0 6 -6 1 0 24 -12 -24 73 0 120 0 -240 500 -699 0 720 ] +Lah,A360205,DiffxTabl,[1 0 2 0 4 3 0 12 18 4 0 48 108 48 5 0 240 720 480 100 6 0 1440 ] Lah,A000262,RowSum,[1 1 3 13 73 501 4051 37633 394353 4596553 58941091 824073141 ] Lah,A088312,EvenSum,[1 0 1 6 37 260 2101 19362 201097 2326536 29668681 413257790 ] Lah,A088313,OddSum,[0 1 2 7 36 241 1950 18271 193256 2270017 29272410 410815351 ] @@ -28,9 +28,9 @@ Lah,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Lah,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Lah,A344051,BinConv,[1 1 5 37 361 4301 60001 954325 16984577 333572041 7151967181 ] Lah,A344050,InvBinConv,[1 1 -3 1 73 -699 3001 24697 -783999 10946233 -80958779 -656003919 ] -Lah,A103194,TransSqrs,[0 1 6 39 292 2505 24306 263431 3154824 41368977 589410910 ] Lah,A052852,TransNat0,[0 1 4 21 136 1045 9276 93289 1047376 12975561 175721140 2581284541 ] Lah,A002720,TransNat1,[1 2 7 34 209 1546 13327 130922 1441729 17572114 234662231 ] +Lah,A103194,TransSqrs,[0 1 6 39 292 2505 24306 263431 3154824 41368977 589410910 ] Lah,A002378,DiagRow1,[0 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] Lah,A083374,DiagRow2,[0 6 36 120 300 630 1176 2016 3240 4950 7260 10296 14196 19110 ] Lah,A253285,DiagRow3,[0 24 240 1200 4200 11760 28224 60480 118800 217800 377520 624624 ] diff --git a/data/csv/Lehmer.csv b/data/csv/Lehmer.csv index 0fe0167..f4c6217 100644 --- a/data/csv/Lehmer.csv +++ b/data/csv/Lehmer.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Lehmer,A354794,Tabl,[1 0 1 0 1 1 0 4 3 1 0 27 19 6 1 0 256 175 55 10 1 0 3125 2101 660 ] Lehmer,,RevTabl,[1 1 0 1 1 0 1 3 4 0 1 6 19 27 0 1 10 55 175 256 0 1 15 125 660 ] Lehmer,A354795,InvTabl,[1 0 1 0 -1 1 0 -1 -3 1 0 -2 -1 -6 1 0 -6 0 5 -10 1 0 -24 4 15 25 ] -Lehmer,,InvRevTabl,[] Lehmer,,RevInvTabl,[1 1 0 1 -1 0 1 -3 -1 0 1 -6 -1 -2 0 1 -10 5 0 -6 0 1 -15 25 15 4 ] +Lehmer,,InvRevTabl,[] Lehmer,,AccTabl,[1 0 1 0 1 2 0 4 7 8 0 27 46 52 53 0 256 431 486 496 497 0 3125 ] Lehmer,,RevAccTabl,[1 1 0 2 1 0 8 7 4 0 53 52 46 27 0 497 496 486 431 256 0 6027 6026 ] Lehmer,,AccRevTabl,[1 1 1 1 2 2 1 4 8 8 1 7 26 53 53 1 11 66 241 497 497 1 16 141 801 ] Lehmer,,AntiDiagTabl,[1 0 0 1 0 1 0 4 1 0 27 3 0 256 19 1 0 3125 175 6 0 46656 2101 55 1 ] -Lehmer,,DiffxTabl,[1 0 2 0 2 3 0 8 9 4 0 54 57 24 5 0 512 525 220 50 6 0 6250 6303 ] Lehmer,,BinTabl,[1 0 1 0 1 3 0 4 11 22 0 27 73 144 247 0 256 687 1348 2304 3631 0 ] Lehmer,,InvBinTabl,[1 0 1 0 1 -1 0 4 -5 4 0 27 -35 30 -17 0 256 -337 298 -184 31 0 ] +Lehmer,,DiffxTabl,[1 0 2 0 2 3 0 8 9 4 0 54 57 24 5 0 512 525 220 50 6 0 6250 6303 ] Lehmer,A195979,RowSum,[1 1 2 8 53 497 6027 89595 1576682 32047986 738772383 19042778713 ] Lehmer,,EvenSum,[1 0 1 3 20 185 2227 32942 577663 11710659 269400564 6932743719 ] Lehmer,,OddSum,[0 1 1 5 33 312 3800 56653 999019 20337327 469371819 12110034994 ] @@ -28,9 +28,9 @@ Lehmer,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Lehmer,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Lehmer,,BinConv,[1 1 3 22 247 3631 65431 1390971 34000275 938046703 28802563636 ] Lehmer,,InvBinConv,[1 1 -1 4 -17 31 1351 -44925 1172963 -29981273 788395396 ] -Lehmer,,TransSqrs,[0 1 5 25 173 1636 19880 295709 5204799 105796675 2438745735 ] Lehmer,,TransNat0,[0 1 3 13 87 816 9888 146889 2583493 52489327 1209554233 ] Lehmer,,TransNat1,[1 2 5 21 140 1313 15915 236484 4160175 84537313 1948326616 ] +Lehmer,,TransSqrs,[0 1 5 25 173 1636 19880 295709 5204799 105796675 2438745735 ] Lehmer,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] Lehmer,A215862,DiagRow2,[0 4 19 55 125 245 434 714 1110 1650 2365 3289 4459 5915 7700 9860 ] Lehmer,,DiagRow3,[0 27 175 660 1890 4550 9702 18900 34320 58905 96525 152152 232050 ] diff --git a/data/csv/Leibniz.csv b/data/csv/Leibniz.csv index 72acf65..aca7c7c 100644 --- a/data/csv/Leibniz.csv +++ b/data/csv/Leibniz.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Leibniz,A003506,Tabl,[1 2 2 3 6 3 4 12 12 4 5 20 30 20 5 6 30 60 60 30 6 7 42 105 140 ] Leibniz,A003506,RevTabl,[1 2 2 3 6 3 4 12 12 4 5 20 30 20 5 6 30 60 60 30 6 7 42 105 140 ] Leibniz,,InvTabl,[] -Leibniz,,InvRevTabl,[] Leibniz,,RevInvTabl,[] +Leibniz,,InvRevTabl,[] Leibniz,,AccTabl,[1 2 4 3 9 12 4 16 28 32 5 25 55 75 80 6 36 96 156 186 192 7 49 154 ] Leibniz,,RevAccTabl,[1 4 2 12 9 3 32 28 16 4 80 75 55 25 5 192 186 156 96 36 6 448 441 ] Leibniz,,AccRevTabl,[1 2 4 3 9 12 4 16 28 32 5 25 55 75 80 6 36 96 156 186 192 7 49 154 ] Leibniz,A128502,AntiDiagTabl,[1 2 3 2 4 6 5 12 3 6 20 12 7 30 30 4 8 42 60 20 9 56 105 60 5 10 ] -Leibniz,,DiffxTabl,[1 2 4 3 12 9 4 24 36 16 5 40 90 80 25 6 60 180 240 150 36 7 84 315 ] Leibniz,,BinTabl,[1 2 4 3 9 18 4 16 40 80 5 25 75 175 350 6 36 126 336 756 1512 7 49 ] Leibniz,,InvBinTabl,[1 2 0 3 3 -6 4 8 -8 0 5 15 -5 -15 30 6 24 6 -36 36 0 7 35 28 -56 ] +Leibniz,,DiffxTabl,[1 2 4 3 12 9 4 24 36 16 5 40 90 80 25 6 60 180 240 150 36 7 84 315 ] Leibniz,A001787,RowSum,[1 4 12 32 80 192 448 1024 2304 5120 11264 24576 53248 114688 ] Leibniz,A057711,EvenSum,[1 2 6 16 40 96 224 512 1152 2560 5632 12288 26624 57344 122880 ] Leibniz,A057711,OddSum,[0 2 6 16 40 96 224 512 1152 2560 5632 12288 26624 57344 122880 ] @@ -28,9 +28,9 @@ Leibniz,A000027,ColLeft,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] Leibniz,A000027,ColRight,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] Leibniz,A037965,BinConv,[1 4 18 80 350 1512 6468 27456 115830 486200 2032316 8465184 ] Leibniz,A002457,InvBinConv,[1 0 -6 0 30 0 -140 0 630 0 -2772 0 12012 0 -51480 0 218790 0 ] -Leibniz,,TransSqrs,[0 2 18 96 400 1440 4704 14336 41472 115200 309760 811008 2076672 ] Leibniz,A001815,TransNat0,[0 2 12 48 160 480 1344 3584 9216 23040 56320 135168 319488 745472 ] Leibniz,A001788,TransNat1,[1 6 24 80 240 672 1792 4608 11520 28160 67584 159744 372736 860160 ] +Leibniz,,TransSqrs,[0 2 18 96 400 1440 4704 14336 41472 115200 309760 811008 2076672 ] Leibniz,A002378,DiagRow1,[2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] Leibniz,A027480,DiagRow2,[3 12 30 60 105 168 252 360 495 660 858 1092 1365 1680 2040 2448 ] Leibniz,A033488,DiagRow3,[4 20 60 140 280 504 840 1320 1980 2860 4004 5460 7280 9520 12240 ] diff --git a/data/csv/Levin.csv b/data/csv/Levin.csv index 72787fe..c12f42b 100644 --- a/data/csv/Levin.csv +++ b/data/csv/Levin.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Levin,A356546,Tabl,[1 2 2 6 12 6 20 60 60 20 70 280 420 280 70 252 1260 2520 2520 1260 ] Levin,A356546,RevTabl,[1 2 2 6 12 6 20 60 60 20 70 280 420 280 70 252 1260 2520 2520 1260 ] Levin,,InvTabl,[] -Levin,,InvRevTabl,[] Levin,,RevInvTabl,[] +Levin,,InvRevTabl,[] Levin,,AccTabl,[1 2 4 6 18 24 20 80 140 160 70 350 770 1050 1120 252 1512 4032 ] Levin,,RevAccTabl,[1 4 2 24 18 6 160 140 80 20 1120 1050 770 350 70 8064 7812 6552 ] Levin,,AccRevTabl,[1 2 4 6 18 24 20 80 140 160 70 350 770 1050 1120 252 1512 4032 ] Levin,A008556,AntiDiagTabl,[1 2 6 2 20 12 70 60 6 252 280 60 924 1260 420 20 3432 5544 2520 ] -Levin,,DiffxTabl,[1 2 4 6 24 18 20 120 180 80 70 560 1260 1120 350 252 2520 7560 ] Levin,,BinTabl,[1 2 4 6 18 36 20 80 200 400 70 350 1050 2450 4900 252 1512 5292 ] Levin,,InvBinTabl,[1 2 0 6 6 -12 20 40 -40 0 70 210 -70 -210 420 252 1008 252 -1512 ] +Levin,,DiffxTabl,[1 2 4 6 24 18 20 120 180 80 70 560 1260 1120 350 252 2520 7560 ] Levin,A059304,RowSum,[1 4 24 160 1120 8064 59136 439296 3294720 24893440 189190144 ] Levin,A069723,EvenSum,[1 2 12 80 560 4032 29568 219648 1647360 12446720 94595072 ] Levin,A069723,OddSum,[0 2 12 80 560 4032 29568 219648 1647360 12446720 94595072 ] @@ -28,9 +28,9 @@ Levin,A000984,ColLeft,[1 2 6 20 70 252 924 3432 12870 48620 184756 705432 270415 Levin,A000984,ColRight,[1 2 6 20 70 252 924 3432 12870 48620 184756 705432 2704156 ] Levin,A002894,BinConv,[1 4 36 400 4900 63504 853776 11778624 165636900 2363904400 ] Levin,A000897,InvBinConv,[1 0 -12 0 420 0 -18480 0 900900 0 -46558512 0 2498640144 0 ] -Levin,,TransSqrs,[0 2 36 480 5600 60480 620928 6150144 59304960 560102400 5202728960 ] Levin,,TransNat0,[0 2 24 240 2240 20160 177408 1537536 13178880 112020480 945950720 ] Levin,,TransNat1,[1 6 48 400 3360 28224 236544 1976832 16473600 136913920 1135140864 ] +Levin,,TransSqrs,[0 2 36 480 5600 60480 620928 6150144 59304960 560102400 5202728960 ] Levin,A005430,DiagRow1,[2 12 60 280 1260 5544 24024 102960 437580 1847560 7759752 32449872 ] Levin,A000911,DiagRow2,[6 60 420 2520 13860 72072 360360 1750320 8314020 38798760 ] Levin,,DiagRow3,[20 280 2520 18480 120120 720720 4084080 22170720 116396280 ] diff --git a/data/csv/Lozanic.csv b/data/csv/Lozanic.csv index 987183a..c9dbb95 100644 --- a/data/csv/Lozanic.csv +++ b/data/csv/Lozanic.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Lozanic,A034851,Tabl,[1 1 1 1 1 1 1 2 2 1 1 2 4 2 1 1 3 6 6 3 1 1 3 9 10 9 3 1 1 4 12 19 ] Lozanic,A034851,RevTabl,[1 1 1 1 1 1 1 2 2 1 1 2 4 2 1 1 3 6 6 3 1 1 3 9 10 9 3 1 1 4 12 19 ] Lozanic,A055138,InvTabl,[1 -1 1 0 -1 1 1 0 -2 1 -1 2 0 -2 1 -1 -3 6 0 -3 1 4 -3 -7 8 0 -3 1 ] -Lozanic,A055138,InvRevTabl,[1 -1 1 0 -1 1 1 0 -2 1 -1 2 0 -2 1 -1 -3 6 0 -3 1 4 -3 -7 8 0 -3 1 ] Lozanic,,RevInvTabl,[1 1 -1 1 -1 0 1 -2 0 1 1 -2 0 2 -1 1 -3 0 6 -3 -1 1 -3 0 8 -7 -3 4 ] +Lozanic,A055138,InvRevTabl,[1 -1 1 0 -1 1 1 0 -2 1 -1 2 0 -2 1 -1 -3 6 0 -3 1 4 -3 -7 8 0 -3 1 ] Lozanic,,AccTabl,[1 1 2 1 2 3 1 3 5 6 1 3 7 9 10 1 4 10 16 19 20 1 4 13 23 32 35 36 ] Lozanic,,RevAccTabl,[1 2 1 3 2 1 6 5 3 1 10 9 7 3 1 20 19 16 10 4 1 36 35 32 23 13 4 1 ] Lozanic,,AccRevTabl,[1 1 2 1 2 3 1 3 5 6 1 3 7 9 10 1 4 10 16 19 20 1 4 13 23 32 35 36 ] Lozanic,A102541,AntiDiagTabl,[1 1 1 1 1 1 1 2 1 1 2 2 1 3 4 1 1 3 6 2 1 4 9 6 1 1 4 12 10 3 1 5 ] -Lozanic,,DiffxTabl,[1 1 2 1 2 3 1 4 6 4 1 4 12 8 5 1 6 18 24 15 6 1 6 27 40 45 18 7 1 ] Lozanic,,BinTabl,[1 1 2 1 2 4 1 3 7 14 1 3 9 21 42 1 4 13 34 76 152 1 4 16 47 116 ] Lozanic,,InvBinTabl,[1 1 0 1 0 0 1 1 -1 0 1 1 1 -5 10 1 2 1 -4 4 0 1 2 4 -9 12 -18 36 1 ] +Lozanic,,DiffxTabl,[1 1 2 1 2 3 1 4 6 4 1 4 12 8 5 1 6 18 24 15 6 1 6 27 40 45 18 7 1 ] Lozanic,A005418,RowSum,[1 2 3 6 10 20 36 72 136 272 528 1056 2080 4160 8256 16512 32896 ] Lozanic,A005418,EvenSum,[1 1 2 3 6 10 20 36 72 136 272 528 1056 2080 4160 8256 16512 32896 ] Lozanic,A051437,OddSum,[0 1 1 3 4 10 16 36 64 136 256 528 1024 2080 4096 8256 16384 32896 ] @@ -28,9 +28,9 @@ Lozanic,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Lozanic,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Lozanic,,BinConv,[1 2 4 14 42 152 508 1892 6758 25556 94704 361748 1369140 5266848 ] Lozanic,,InvBinConv,[1 0 0 0 10 0 36 0 358 0 2200 0 17524 0 125048 0 957254 0 7160112 0 ] -Lozanic,,TransSqrs,[0 1 5 19 52 154 384 1020 2464 6152 14560 34928 81216 189472 433664 ] Lozanic,,TransNat0,[0 1 3 9 20 50 108 252 544 1224 2640 5808 12480 27040 57792 123840 ] Lozanic,,TransNat1,[1 3 6 15 30 70 144 324 680 1496 3168 6864 14560 31200 66048 140352 ] +Lozanic,,TransSqrs,[0 1 5 19 52 154 384 1020 2464 6152 14560 34928 81216 189472 433664 ] Lozanic,A001057,DiagRow1,[1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 ] Lozanic,A002620,DiagRow2,[1 2 4 6 9 12 16 20 25 30 36 42 49 56 64 72 81 90 100 110 ] Lozanic,A005993,DiagRow3,[1 2 6 10 19 28 44 60 85 110 146 182 231 280 344 408 489 570 670 ] diff --git a/data/csv/Motzkin.csv b/data/csv/Motzkin.csv index 854dc64..49b11e2 100644 --- a/data/csv/Motzkin.csv +++ b/data/csv/Motzkin.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Motzkin,A359364,Tabl,[1 1 0 1 0 1 1 0 3 0 1 0 6 0 2 1 0 10 0 10 0 1 0 15 0 30 0 5 1 0 21 ] Motzkin,A097610,RevTabl,[1 0 1 1 0 1 0 3 0 1 2 0 6 0 1 0 10 0 10 0 1 5 0 30 0 15 0 1 0 35 0 ] Motzkin,,InvTabl,[] -Motzkin,,InvRevTabl,[1 0 1 -1 0 1 0 -3 0 1 4 0 -6 0 1 0 20 0 -10 0 1 -35 0 60 0 -15 0 1 ] Motzkin,,RevInvTabl,[] +Motzkin,,InvRevTabl,[1 0 1 -1 0 1 0 -3 0 1 4 0 -6 0 1 0 20 0 -10 0 1 -35 0 60 0 -15 0 1 ] Motzkin,,AccTabl,[1 1 1 1 1 2 1 1 4 4 1 1 7 7 9 1 1 11 11 21 21 1 1 16 16 46 46 51 1 ] Motzkin,,RevAccTabl,[1 1 1 2 1 1 4 4 1 1 9 7 7 1 1 21 21 11 11 1 1 51 46 46 16 16 1 1 ] Motzkin,,AccRevTabl,[1 0 1 1 1 2 0 3 3 4 2 2 8 8 9 0 10 10 20 20 21 5 5 35 35 50 50 51 ] Motzkin,,AntiDiagTabl,[1 1 1 0 1 0 1 0 1 1 0 3 1 0 6 0 1 0 10 0 1 0 15 0 2 1 0 21 0 10 1 ] -Motzkin,,DiffxTabl,[1 1 0 1 0 3 1 0 9 0 1 0 18 0 10 1 0 30 0 50 0 1 0 45 0 150 0 35 1 ] Motzkin,,BinTabl,[1 1 1 1 1 2 1 1 4 10 1 1 7 19 39 1 1 11 31 71 151 1 1 16 46 121 ] Motzkin,,InvBinTabl,[1 1 -1 1 -1 2 1 -1 4 -10 1 -1 7 -19 39 1 -1 11 -31 71 -151 1 -1 16 ] +Motzkin,,DiffxTabl,[1 1 0 1 0 3 1 0 9 0 1 0 18 0 10 1 0 30 0 50 0 1 0 45 0 150 0 35 1 ] Motzkin,A001006,RowSum,[1 1 2 4 9 21 51 127 323 835 2188 5798 15511 41835 113634 310572 ] Motzkin,A001006,EvenSum,[1 1 2 4 9 21 51 127 323 835 2188 5798 15511 41835 113634 310572 ] Motzkin,A000004,OddSum,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] @@ -28,9 +28,9 @@ Motzkin,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Motzkin,A090192,ColRight,[1 0 1 0 2 0 5 0 14 0 42 0 132 0 429 0 1430 0 4862 0 ] Motzkin,,BinConv,[1 1 2 10 39 151 681 3137 14519 69463 339118 1674278 8376721 ] Motzkin,,InvBinConv,[1 -1 2 -10 39 -151 681 -3137 14519 -69463 339118 -1674278 8376721 ] -Motzkin,,TransSqrs,[0 0 4 12 56 200 720 2464 8288 27360 89220 287980 922152 2933528 ] Motzkin,,TransNat0,[0 0 2 6 20 60 180 532 1568 4608 13530 39710 116556 342212 1005186 ] Motzkin,A025179,TransNat1,[1 1 4 10 29 81 231 659 1891 5443 15718 45508 132067 384047 1118820 ] +Motzkin,,TransSqrs,[0 0 4 12 56 200 720 2464 8288 27360 89220 287980 922152 2933528 ] Motzkin,A138364,DiagRow1,[1 0 3 0 10 0 35 0 126 0 462 0 1716 0 6435 0 24310 0 92378 0 ] Motzkin,A002457,DiagRow2,[1 0 6 0 30 0 140 0 630 0 2772 0 12012 0 51480 0 218790 0 923780 0 ] Motzkin,A002802,DiagRow3,[1 0 10 0 70 0 420 0 2310 0 12012 0 60060 0 291720 0 1385670 0 ] diff --git a/data/csv/MotzkinGF.csv b/data/csv/MotzkinGF.csv index 40a63a2..9a73084 100644 --- a/data/csv/MotzkinGF.csv +++ b/data/csv/MotzkinGF.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence MotzkinGF,A064189,Tabl,[1 1 1 2 2 1 4 5 3 1 9 12 9 4 1 21 30 25 14 5 1 51 76 69 44 20 6 1 ] MotzkinGF,A026300,RevTabl,[1 1 1 1 2 2 1 3 5 4 1 4 9 12 9 1 5 14 25 30 21 1 6 20 44 69 76 51 ] MotzkinGF,A101950,InvTabl,[1 -1 1 0 -2 1 1 1 -3 1 -1 2 3 -4 1 0 -4 2 6 -5 1 1 2 -9 0 10 -6 1 ] -MotzkinGF,,InvRevTabl,[] MotzkinGF,,RevInvTabl,[1 1 -1 1 -2 0 1 -3 1 1 1 -4 3 2 -1 1 -5 6 2 -4 0 1 -6 10 0 -9 2 1 ] +MotzkinGF,,InvRevTabl,[] MotzkinGF,,AccTabl,[1 1 2 2 4 5 4 9 12 13 9 21 30 34 35 21 51 76 90 95 96 51 127 196 ] MotzkinGF,,RevAccTabl,[1 2 1 5 4 2 13 12 9 4 35 34 30 21 9 96 95 90 76 51 21 267 266 260 ] MotzkinGF,,AccRevTabl,[1 1 2 1 3 5 1 4 9 13 1 5 14 26 35 1 6 20 45 75 96 1 7 27 71 140 ] MotzkinGF,A106489,AntiDiagTabl,[1 1 2 1 4 2 9 5 1 21 12 3 51 30 9 1 127 76 25 4 323 196 69 14 1 ] -MotzkinGF,,DiffxTabl,[1 1 2 2 4 3 4 10 9 4 9 24 27 16 5 21 60 75 56 25 6 51 152 207 176 ] MotzkinGF,,BinTabl,[1 1 2 2 4 7 4 9 17 29 9 21 42 76 128 21 51 106 200 352 587 51 127 ] MotzkinGF,,InvBinTabl,[1 1 0 2 0 -1 4 1 -3 3 9 3 -6 4 0 21 9 -14 8 0 -5 51 25 -32 14 5 ] +MotzkinGF,,DiffxTabl,[1 1 2 2 4 3 4 10 9 4 9 24 27 16 5 21 60 75 56 25 6 51 152 207 176 ] MotzkinGF,A005773,RowSum,[1 2 5 13 35 96 267 750 2123 6046 17303 49721 143365 414584 1201917 ] MotzkinGF,A002426,EvenSum,[1 1 3 7 19 51 141 393 1107 3139 8953 25653 73789 212941 616227 ] MotzkinGF,A005717,OddSum,[0 1 2 6 16 45 126 357 1016 2907 8350 24068 69576 201643 585690 ] @@ -28,9 +28,9 @@ MotzkinGF,A001006,ColLeft,[1 1 2 4 9 21 51 127 323 835 2188 5798 15511 41835 113 MotzkinGF,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] MotzkinGF,A344502,BinConv,[1 2 7 29 128 587 2759 13190 63844 311948 1535488 7602971 37829455 ] MotzkinGF,A344503,InvBinConv,[1 0 -1 3 0 -5 15 0 -28 84 0 -165 495 0 -1001 3003 0 -6188 18564 0 ] -MotzkinGF,A344504,TransSqrs,[0 1 6 26 100 361 1254 4245 14108 46247 149998 482412 1540880 ] MotzkinGF,A330796,TransNat0,[0 1 4 14 46 147 462 1437 4438 13637 41746 127426 388076 1179739 ] MotzkinGF,A000244,TransNat1,[1 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 ] +MotzkinGF,A344504,TransSqrs,[0 1 6 26 100 361 1254 4245 14108 46247 149998 482412 1540880 ] MotzkinGF,A000027,DiagRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] MotzkinGF,A000096,DiagRow2,[2 5 9 14 20 27 35 44 54 65 77 90 104 119 135 152 170 189 209 230 ] MotzkinGF,A000297,DiagRow3,[4 12 25 44 70 104 147 200 264 340 429 532 650 784 935 1104 1292 ] diff --git a/data/csv/Narayana.csv b/data/csv/Narayana.csv index fb7f81b..979dce4 100644 --- a/data/csv/Narayana.csv +++ b/data/csv/Narayana.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Narayana,A090181,Tabl,[1 0 1 0 1 1 0 1 3 1 0 1 6 6 1 0 1 10 20 10 1 0 1 15 50 50 15 1 0 1 ] Narayana,A131198,RevTabl,[1 1 0 1 1 0 1 3 1 0 1 6 6 1 0 1 10 20 10 1 0 1 15 50 50 15 1 0 1 ] Narayana,A103364,InvTabl,[1 0 1 0 -1 1 0 2 -3 1 0 -7 12 -6 1 0 39 -70 40 -10 1 0 -321 585 ] -Narayana,,InvRevTabl,[] Narayana,,RevInvTabl,[1 1 0 1 -1 0 1 -3 2 0 1 -6 12 -7 0 1 -10 40 -70 39 0 1 -15 100 ] +Narayana,,InvRevTabl,[] Narayana,A349740,AccTabl,[1 0 1 0 1 2 0 1 4 5 0 1 7 13 14 0 1 11 31 41 42 0 1 16 66 116 131 ] Narayana,A104710,RevAccTabl,[1 1 0 2 1 0 5 4 1 0 14 13 7 1 0 42 41 31 11 1 0 132 131 116 66 16 ] Narayana,,AccRevTabl,[1 1 1 1 2 2 1 4 5 5 1 7 13 14 14 1 11 31 41 42 42 1 16 66 116 131 ] Narayana,A089732,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 1 3 0 1 6 1 0 1 10 6 0 1 15 20 1 0 1 21 50 10 ] -Narayana,,DiffxTabl,[1 0 2 0 2 3 0 2 9 4 0 2 18 24 5 0 2 30 80 50 6 0 2 45 200 250 90 7 ] Narayana,,BinTabl,[1 0 1 0 1 3 0 1 5 13 0 1 8 27 65 0 1 12 53 154 356 0 1 17 98 344 ] Narayana,,InvBinTabl,[1 0 1 0 1 -1 0 1 1 -5 0 1 4 -9 9 0 1 8 -7 -14 56 0 1 13 8 -64 120 ] +Narayana,,DiffxTabl,[1 0 2 0 2 3 0 2 9 4 0 2 18 24 5 0 2 30 80 50 6 0 2 45 200 250 90 7 ] Narayana,A000108,RowSum,[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 ] Narayana,A071688,EvenSum,[1 0 1 3 7 20 66 217 715 2424 8398 29414 104006 371384 1337220 ] Narayana,A071684,OddSum,[0 1 1 2 7 22 66 212 715 2438 8398 29372 104006 371516 1337220 ] @@ -28,9 +28,9 @@ Narayana,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Narayana,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Narayana,,BinConv,[1 1 3 13 65 356 2072 12601 79221 511174 3368090 22577160 153534680 ] Narayana,,InvBinConv,[1 1 -1 -5 9 56 -120 -825 1925 14014 -34398 -259896 659736 5116320 ] -Narayana,A141222,TransSqrs,[0 1 5 22 95 406 1722 7260 30459 127270 529958 2200276 9111830 ] Narayana,A001700,TransNat0,[0 1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 ] Narayana,A189176,TransNat1,[1 2 5 15 49 168 594 2145 7865 29172 109174 411502 1560090 5943200 ] +Narayana,A141222,TransSqrs,[0 1 5 22 95 406 1722 7260 30459 127270 529958 2200276 9111830 ] Narayana,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] Narayana,A002415,DiagRow2,[0 1 6 20 50 105 196 336 540 825 1210 1716 2366 3185 4200 5440 6936 ] Narayana,A006542,DiagRow3,[0 1 10 50 175 490 1176 2520 4950 9075 15730 26026 41405 63700 ] diff --git a/data/csv/Nicomachus.csv b/data/csv/Nicomachus.csv index 0031d89..4757865 100644 --- a/data/csv/Nicomachus.csv +++ b/data/csv/Nicomachus.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Nicomachus,A036561,Tabl,[1 2 3 4 6 9 8 12 18 27 16 24 36 54 81 32 48 72 108 162 243 64 96 ] Nicomachus,A175840,RevTabl,[1 3 2 9 6 4 27 18 12 8 81 54 36 24 16 243 162 108 72 48 32 729 486 ] Nicomachus,,InvTabl,[] -Nicomachus,,InvRevTabl,[] Nicomachus,,RevInvTabl,[] +Nicomachus,,InvRevTabl,[] Nicomachus,A230435,AccTabl,[1 2 5 4 10 19 8 20 38 65 16 40 76 130 211 32 80 152 260 422 665 64 ] Nicomachus,,RevAccTabl,[1 5 2 19 10 4 65 38 20 8 211 130 76 40 16 665 422 260 152 80 32 ] Nicomachus,,AccRevTabl,[1 3 5 9 15 19 27 45 57 65 81 135 171 195 211 243 405 513 585 633 ] Nicomachus,,AntiDiagTabl,[1 2 4 3 8 6 16 12 9 32 24 18 64 48 36 27 128 96 72 54 256 192 144 ] -Nicomachus,,DiffxTabl,[1 2 6 4 12 27 8 24 54 108 16 48 108 216 405 32 96 216 432 810 1458 ] Nicomachus,A232646,BinTabl,[1 2 5 4 10 25 8 20 50 125 16 40 100 250 625 32 80 200 500 1250 ] Nicomachus,A130321,InvBinTabl,[1 2 1 4 2 1 8 4 2 1 16 8 4 2 1 32 16 8 4 2 1 64 32 16 8 4 2 1 128 ] +Nicomachus,,DiffxTabl,[1 2 6 4 12 27 8 24 54 108 16 48 108 216 405 32 96 216 432 810 1458 ] Nicomachus,A001047,RowSum,[1 5 19 65 211 665 2059 6305 19171 58025 175099 527345 1586131 ] Nicomachus,,EvenSum,[1 2 13 26 133 266 1261 2522 11605 23210 105469 210938 953317 ] Nicomachus,,OddSum,[0 3 6 39 78 399 798 3783 7566 34815 69630 316407 632814 2859951 ] @@ -28,9 +28,9 @@ Nicomachus,A000079,ColLeft,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 163 Nicomachus,A000244,ColRight,[1 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 ] Nicomachus,A000351,BinConv,[1 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 ] Nicomachus,A000012,InvBinConv,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -Nicomachus,,TransSqrs,[0 3 42 327 1950 9975 46194 199551 819006 3232335 12369570 46173927 ] Nicomachus,,TransNat0,[0 3 24 129 582 2379 9132 33573 119634 416415 1423320 4795257 ] Nicomachus,,TransNat1,[1 8 43 194 793 3044 11191 39878 138805 474440 1598419 5322602 ] +Nicomachus,,TransSqrs,[0 3 42 327 1950 9975 46194 199551 819006 3232335 12369570 46173927 ] Nicomachus,A008776,DiagRow1,[2 6 18 54 162 486 1458 4374 13122 39366 118098 354294 1062882 ] Nicomachus,A003946,DiagRow2,[4 12 36 108 324 972 2916 8748 26244 78732 236196 708588 2125764 ] Nicomachus,A005051,DiagRow3,[8 24 72 216 648 1944 5832 17496 52488 157464 472392 1417176 ] diff --git a/data/csv/One.csv b/data/csv/One.csv index c6f3837..caaae14 100644 --- a/data/csv/One.csv +++ b/data/csv/One.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence One,A000012,Tabl,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A000012,RevTabl,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A059778,InvTabl,[1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 -1 1 0 0 ] -One,A059778,InvRevTabl,[1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 -1 1 0 0 ] One,A103451,RevInvTabl,[1 1 -1 1 -1 0 1 -1 0 0 1 -1 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0 0 0 1 -1 ] +One,A059778,InvRevTabl,[1 -1 1 0 -1 1 0 0 -1 1 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 -1 1 0 0 ] One,A002260,AccTabl,[1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 ] One,A004736,RevAccTabl,[1 2 1 3 2 1 4 3 2 1 5 4 3 2 1 6 5 4 3 2 1 7 6 5 4 3 2 1 8 7 6 5 4 ] One,A002260,AccRevTabl,[1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 ] One,A000012,AntiDiagTabl,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -One,A002260,DiffxTabl,[1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 ] One,A059268,BinTabl,[1 1 2 1 2 4 1 2 4 8 1 2 4 8 16 1 2 4 8 16 32 1 2 4 8 16 32 64 1 2 ] One,A010054,InvBinTabl,[1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 ] +One,A002260,DiffxTabl,[1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 ] One,A000027,RowSum,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] One,A001057,EvenSum,[1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 ] One,A001057,OddSum,[0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 ] @@ -28,9 +28,9 @@ One,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A000079,BinConv,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] One,A000007,InvBinConv,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -One,A000330,TransSqrs,[0 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 ] One,A000217,TransNat0,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] One,A000217,TransNat1,[1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 ] +One,A000330,TransSqrs,[0 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 ] One,A000012,DiagRow1,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A000012,DiagRow2,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] One,A000012,DiagRow3,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] diff --git a/data/csv/OrderedCyc.csv b/data/csv/OrderedCyc.csv deleted file mode 100644 index cedf187..0000000 --- a/data/csv/OrderedCyc.csv +++ /dev/null @@ -1,52 +0,0 @@ -Triangle,Trait,ANumber,Sequence -OEIS Similars: A048594 A075181 A225479,,, -ordered_cycle,Triangle ,A225479,[1 0 1 0 1 2 0 2 6 6 0 6 22 36 24 0 24 100 210 240 120 0 120 548 ] -ordered_cycle,TriRev ,A000000,[1 1 0 2 1 0 6 6 2 0 24 36 22 6 0 120 240 210 100 24 0 720 1800 ] -ordered_cycle,TriAcc ,A000000,[1 0 1 0 1 3 0 2 8 14 0 6 28 64 88 0 24 124 334 574 694 0 120 668 ] -ordered_cycle,TriRevAcc,A000000,[1 1 0 3 1 0 14 8 2 0 88 64 28 6 0 694 574 334 124 24 0 6578 5858 ] -ordered_cycle,TriAccRev,A000000,[1 1 1 2 3 3 6 12 14 14 24 60 82 88 88 120 360 570 670 694 694 720 ] -ordered_cycle,TriDiag ,A000000,[1 0 0 1 0 1 0 2 2 0 6 6 0 24 22 6 0 120 100 36 0 720 548 210 24 0 ] -ordered_cycle,TriPoly ,A000000,[1 0 1 0 1 1 0 3 2 1 0 14 10 3 1 0 88 76 21 4 1 0 694 772 222 36 5 ] -ordered_cycle,TriConv ,A000000,[1 0 1 0 1 5 0 2 14 76 0 6 50 360 2392 0 24 224 1908 15664 126676 0 ] -ordered_cycle,TriBin ,A000000,[1 0 1 0 1 4 0 2 10 30 0 6 34 120 324 0 24 148 582 1776 4540 0 120 ] -ordered_cycle,TriInvBin,A000000,[1 0 1 0 1 0 0 2 2 -6 0 6 10 -12 -12 0 24 52 -18 -96 140 0 120 308 ] -ordered_cycle,Sum ,A007840,[1 1 3 14 88 694 6578 72792 920904 13109088 207360912 3608233056 ] -ordered_cycle,EvenSum ,A052811,[1 0 2 6 46 340 3308 36288 460752 6551424 103685232 1803956880 ] -ordered_cycle,OddSum ,A000000,[0 1 1 8 42 354 3270 36504 460152 6557664 103675680 1804276176 ] -ordered_cycle,AltSum ,A006252,[1 -1 1 -2 4 -14 38 -216 600 -6240 9552 -319296 -519312 -28108560 ] -ordered_cycle,AccSum ,A000000,[1 1 4 24 186 1750 19300 243768 3467184 54820848 953707008 ] -ordered_cycle,AccRevSum,A000000,[1 1 4 24 186 1750 19300 243768 3467184 54820848 953707008 ] -ordered_cycle,RevAccSum,A052801,[1 2 8 46 342 3108 33324 411360 5741856 89379120 1534623936 ] -ordered_cycle,DiagSum ,A129841,[1 0 1 1 4 12 52 256 1502 10158 78360 680280 6574872 70075416 ] -ordered_cycle,RowLcm ,A000000,[1 1 2 6 792 8400 25153200 76734000 763351047043200 ] -ordered_cycle,RowGcd ,A000000,[1 1 2 2 2 2 2 24 24 24 24 48 48 48 48 384 1152 1152 1152 2304 2304 ] -ordered_cycle,RowMax ,A058583,[1 1 2 6 36 240 2040 21000 235200 3265920 47628000 795175920 ] -ordered_cycle,Middle ,A344498,[1 0 1 2 22 100 1350 9744 162456 1614816 32319000 410031600 ] -ordered_cycle,Central ,A000000,[1 1 22 1350 162456 32319000 9604465200 3986353491120 ] -ordered_cycle,LeftSide ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -ordered_cycle,RightSide,A000142,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] -ordered_cycle,PosHalf ,A227917,[1 1 4 26 232 2624 35888 575280 10569984 218911872 5044346112 ] -ordered_cycle,NegHalf ,A354237,[1 1 0 2 -8 64 -592 6768 -90624 1395840 -24292608 471453696 ] -ordered_cycle,Bin ,A277759,[1 1 4 30 324 4540 78060 1589448 37388400 997513200 29759790240 ] -ordered_cycle,InvBin ,A308565,[1 1 0 -6 -12 140 1020 -5208 -117264 -2448 17756640 117905040 ] -ordered_cycle,TransSqrs,A000000,[0 1 9 80 802 9154 118022 1701048 27139608 475183584 9062172768 ] -ordered_cycle,TransNat0,A215916,[0 1 5 32 254 2414 26746 338568 4820952 76270032 1327263024 ] -ordered_cycle,TransNat1,A052801,[1 2 8 46 342 3108 33324 411360 5741856 89379120 1534623936 ] -ordered_cycle,DiagRow0 ,A000142,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] -ordered_cycle,DiagRow1 ,A180119,[0 1 6 36 240 1800 15120 141120 1451520 16329600 199584000 ] -ordered_cycle,DiagRow2 ,A000000,[0 2 22 210 2040 21000 231840 2751840 35078400 479001600 6985440000 ] -ordered_cycle,DiagRow3 ,A000000,[0 6 100 1350 17640 235200 3265920 47628000 731808000 11855289600 ] -ordered_cycle,DiagCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -ordered_cycle,DiagCol1 ,A000142,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] -ordered_cycle,DiagCol2 ,A052517,[2 6 22 100 548 3528 26136 219168 2053152 21257280 241087680 ] -ordered_cycle,DiagCol3 ,A052748,[6 36 210 1350 9744 78792 708744 7036200 76521456 905507856 ] -ordered_cycle,PolyRow0 ,A000012,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -ordered_cycle,PolyRow1 ,A001477,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ] -ordered_cycle,PolyRow2 ,A014105,[0 3 10 21 36 55 78 105 136 171 210 253 300 351 406 465 528 595 666 ] -ordered_cycle,PolyRow3 ,A000000,[0 14 76 222 488 910 1524 2366 3472 4878 6620 8734 11256 14222 ] -ordered_cycle,PolyCol0 ,A000007,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -ordered_cycle,PolyCol1 ,A007840,[1 1 3 14 88 694 6578 72792 920904 13109088 207360912 3608233056 ] -ordered_cycle,PolyCol2 ,A088500,[1 2 10 76 772 9808 149552 2660544 54093696 1237306560 31446049728 ] -ordered_cycle,PolyCol3 ,A354263,[1 3 21 222 3132 55242 1169262 28873800 814870584 25871762016 ] -ordered_cycle,PolyDiag ,A317171,[1 1 10 222 8824 553870 50545008 6328330344 1041597412224 ] -# ordered_cycle, 49 traits, 30 A-numbers,61% \ No newline at end of file diff --git a/data/csv/Ordinals.csv b/data/csv/Ordinals.csv index 38d1208..8f3eba4 100644 --- a/data/csv/Ordinals.csv +++ b/data/csv/Ordinals.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Ordinals,A002262,Tabl,[0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 ] Ordinals,A025581,RevTabl,[0 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1 0 6 5 4 3 2 1 0 7 6 5 4 3 ] Ordinals,,InvTabl,[] -Ordinals,,InvRevTabl,[] Ordinals,,RevInvTabl,[] +Ordinals,,InvRevTabl,[] Ordinals,A112367,AccTabl,[0 0 1 0 1 3 0 1 3 6 0 1 3 6 10 0 1 3 6 10 15 0 1 3 6 10 15 21 0 1 ] Ordinals,A122432,RevAccTabl,[0 1 0 3 1 0 6 3 1 0 10 6 3 1 0 15 10 6 3 1 0 21 15 10 6 3 1 0 28 ] Ordinals,A141418,AccRevTabl,[0 1 1 2 3 3 3 5 6 6 4 7 9 10 10 5 9 12 14 15 15 6 11 15 18 20 21 ] Ordinals,A025685,AntiDiagTabl,[0 0 0 1 0 1 0 1 2 0 1 2 0 1 2 3 0 1 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 ] -Ordinals,,DiffxTabl,[0 0 2 0 2 6 0 2 6 12 0 2 6 12 20 0 2 6 12 20 30 0 2 6 12 20 30 42 ] Ordinals,,BinTabl,[0 0 1 0 1 4 0 1 4 12 0 1 4 12 32 0 1 4 12 32 80 0 1 4 12 32 80 192 ] Ordinals,A073424,InvBinTabl,[0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ] +Ordinals,,DiffxTabl,[0 0 2 0 2 6 0 2 6 12 0 2 6 12 20 0 2 6 12 20 30 0 2 6 12 20 30 42 ] Ordinals,A000217,RowSum,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] Ordinals,A110660,EvenSum,[0 0 2 2 6 6 12 12 20 20 30 30 42 42 56 56 72 72 90 90 ] Ordinals,A008794,OddSum,[0 1 1 4 4 9 9 16 16 25 25 36 36 49 49 64 64 81 81 100 ] @@ -28,9 +28,9 @@ Ordinals,A000004,ColLeft,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Ordinals,A001477,ColRight,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] Ordinals,A001787,BinConv,[0 1 4 12 32 80 192 448 1024 2304 5120 11264 24576 53248 114688 ] Ordinals,A003982,InvBinConv,[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -Ordinals,A000537,TransSqrs,[0 1 9 36 100 225 441 784 1296 2025 3025 4356 6084 8281 11025 14400 ] Ordinals,A000330,TransNat0,[0 1 5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 ] Ordinals,A007290,TransNat1,[0 2 8 20 40 70 112 168 240 330 440 572 728 910 1120 1360 1632 1938 ] +Ordinals,A000537,TransSqrs,[0 1 9 36 100 225 441 784 1296 2025 3025 4356 6084 8281 11025 14400 ] Ordinals,A001477,DiagRow1,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] Ordinals,A001477,DiagRow2,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] Ordinals,A001477,DiagRow3,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] diff --git a/data/csv/Partition.csv b/data/csv/Partition.csv index 5269263..49c5b4c 100644 --- a/data/csv/Partition.csv +++ b/data/csv/Partition.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Partition,A072233,Tabl,[1 0 1 0 1 1 0 1 1 1 0 1 2 1 1 0 1 2 2 1 1 0 1 3 3 2 1 1 0 1 3 4 3 ] Partition,A058398,RevTabl,[1 1 0 1 1 0 1 1 1 0 1 1 2 1 0 1 1 2 2 1 0 1 1 2 3 3 1 0 1 1 2 3 4 ] Partition,,InvTabl,[1 0 1 0 -1 1 0 0 -1 1 0 1 -1 -1 1 0 0 1 -1 -1 1 0 0 1 0 -1 -1 1 0 ] -Partition,,InvRevTabl,[] Partition,A139128,RevInvTabl,[1 1 0 1 -1 0 1 -1 0 0 1 -1 -1 1 0 1 -1 -1 1 0 0 1 -1 -1 0 1 0 0 1 ] +Partition,,InvRevTabl,[] Partition,A026820,AccTabl,[1 0 1 0 1 2 0 1 2 3 0 1 3 4 5 0 1 3 5 6 7 0 1 4 7 9 10 11 0 1 4 8 ] Partition,A058400,RevAccTabl,[1 1 0 2 1 0 3 2 1 0 5 4 3 1 0 7 6 5 3 1 0 11 10 9 7 4 1 0 15 14 13 ] Partition,,AccRevTabl,[1 1 1 1 2 2 1 2 3 3 1 2 4 5 5 1 2 4 6 7 7 1 2 4 7 10 11 11 1 2 4 7 ] Partition,A145574,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 1 1 0 1 2 1 0 1 2 1 0 1 3 2 1 0 1 3 3 1 0 1 4 ] -Partition,,DiffxTabl,[1 0 2 0 2 3 0 2 3 4 0 2 6 4 5 0 2 6 8 5 6 0 2 9 12 10 6 7 0 2 9 16 ] Partition,,BinTabl,[1 0 1 0 1 3 0 1 3 7 0 1 4 10 21 0 1 4 11 25 51 0 1 5 15 36 76 148 ] Partition,,InvBinTabl,[1 0 1 0 1 -1 0 1 -1 1 0 1 0 -2 5 0 1 0 -1 1 1 0 1 1 -3 4 -4 4 0 1 ] +Partition,,DiffxTabl,[1 0 2 0 2 3 0 2 3 4 0 2 6 4 5 0 2 6 8 5 6 0 2 9 12 10 6 7 0 2 9 16 ] Partition,A000041,RowSum,[1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 ] Partition,A027187,EvenSum,[1 0 1 1 3 3 6 7 12 14 22 27 40 49 69 86 118 146 195 242 ] Partition,A027193,OddSum,[0 1 1 2 2 4 5 8 10 16 20 29 37 52 66 90 113 151 190 248 ] @@ -28,9 +28,9 @@ Partition,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Partition,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Partition,A098545,BinConv,[1 1 3 7 21 51 148 365 983 2461 6360 15687 39757 97033 240425 ] Partition,,InvBinConv,[1 1 -1 1 5 1 4 15 55 139 152 397 1429 2549 3457 9330 26105 48791 ] -Partition,A296010,TransSqrs,[0 1 5 14 34 68 133 232 402 652 1048 1609 2465 3640 5358 7694 10993 ] Partition,A006128,TransNat0,[0 1 3 6 12 20 35 54 86 128 192 275 399 556 780 1068 1463 1965 2644 ] Partition,A093694,TransNat1,[1 2 5 9 17 27 46 69 108 158 234 331 476 657 915 1244 1694 2262 ] +Partition,A296010,TransSqrs,[0 1 5 14 34 68 133 232 402 652 1048 1609 2465 3640 5358 7694 10993 ] Partition,A000520,DiagRow1,[0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Partition,A065685,DiagRow2,[0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ] Partition,A065683,DiagRow3,[0 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ] diff --git a/data/csv/PartitionMax.csv b/data/csv/PartitionMax.csv index 1f3e4a1..7872507 100644 --- a/data/csv/PartitionMax.csv +++ b/data/csv/PartitionMax.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence PartitionMax,A026820,Tabl,[1 0 1 0 1 2 0 1 2 3 0 1 3 4 5 0 1 3 5 6 7 0 1 4 7 9 10 11 0 1 4 8 ] PartitionMax,A058400,RevTabl,[1 1 0 2 1 0 3 2 1 0 5 4 3 1 0 7 6 5 3 1 0 11 10 9 7 4 1 0 15 14 13 ] PartitionMax,,InvTabl,[] -PartitionMax,,InvRevTabl,[] PartitionMax,,RevInvTabl,[] +PartitionMax,,InvRevTabl,[] PartitionMax,,AccTabl,[1 0 1 0 1 3 0 1 3 6 0 1 4 8 13 0 1 4 9 15 22 0 1 5 12 21 31 42 0 1 ] PartitionMax,,RevAccTabl,[1 1 0 3 1 0 6 3 1 0 13 8 4 1 0 22 15 9 4 1 0 42 31 21 12 5 1 0 66 ] PartitionMax,,AccRevTabl,[1 1 1 2 3 3 3 5 6 6 5 9 12 13 13 7 13 18 21 22 22 11 21 30 37 41 ] PartitionMax,,AntiDiagTabl,[1 0 0 1 0 1 0 1 2 0 1 2 0 1 3 3 0 1 3 4 0 1 4 5 5 0 1 4 7 6 0 1 5 ] -PartitionMax,,DiffxTabl,[1 0 2 0 2 6 0 2 6 12 0 2 9 16 25 0 2 9 20 30 42 0 2 12 28 45 60 77 ] PartitionMax,,BinTabl,[1 0 1 0 1 4 0 1 4 12 0 1 5 16 43 0 1 5 17 48 122 0 1 6 22 65 170 ] PartitionMax,,InvBinTabl,[1 0 1 0 1 0 0 1 0 0 0 1 1 -2 3 0 1 1 -1 0 2 0 1 2 -2 1 0 0 0 1 2 ] +PartitionMax,,DiffxTabl,[1 0 2 0 2 6 0 2 6 12 0 2 9 16 25 0 2 9 20 30 42 0 2 12 28 45 60 77 ] PartitionMax,A058397,RowSum,[1 1 3 6 13 22 42 66 112 172 270 397 602 858 1245 1748 2464 3381 ] PartitionMax,,EvenSum,[1 0 2 2 8 9 24 29 62 78 146 184 321 403 657 829 1291 1615 2433 ] PartitionMax,,OddSum,[0 1 1 4 5 13 18 37 50 94 124 213 281 455 588 919 1173 1766 2238 ] @@ -28,9 +28,9 @@ PartitionMax,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] PartitionMax,A000041,ColRight,[1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 ] PartitionMax,,BinConv,[1 1 4 12 43 122 412 1142 3516 9846 28512 77694 220429 588975 ] PartitionMax,,InvBinConv,[1 1 0 0 3 2 0 8 28 60 72 188 693 1271 1703 4466 12579 23502 39916 ] -PartitionMax,,TransSqrs,[0 1 9 36 129 329 870 1829 3958 7586 14472 25489 45369 75299 125432 ] PartitionMax,,TransNat0,[0 1 5 14 39 81 182 331 634 1088 1882 3029 4973 7649 11886 17807 ] PartitionMax,,TransNat1,[1 2 8 20 52 103 224 397 746 1260 2152 3426 5575 8507 13131 19555 ] +PartitionMax,,TransSqrs,[0 1 9 36 129 329 870 1829 3958 7586 14472 25489 45369 75299 125432 ] PartitionMax,A000065,DiagRow1,[0 1 2 4 6 10 14 21 29 41 55 76 100 134 175 230 296 384 489 626 ] PartitionMax,A007042,DiagRow2,[0 1 3 5 9 13 20 28 40 54 75 99 133 174 229 295 383 488 625 790 ] PartitionMax,A335323,DiagRow3,[0 1 3 7 11 18 26 38 52 73 97 131 172 227 293 381 486 623 788 998 ] diff --git a/data/csv/Polygonal.csv b/data/csv/Polygonal.csv index f94f7c4..4208b65 100644 --- a/data/csv/Polygonal.csv +++ b/data/csv/Polygonal.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Polygonal,A139600,Tabl,[0 0 1 0 1 2 0 1 3 3 0 1 4 6 4 0 1 5 9 10 5 0 1 6 12 16 15 6 0 1 7 ] Polygonal,A134394,RevTabl,[0 1 0 2 1 0 3 3 1 0 4 6 4 1 0 5 10 9 5 1 0 6 15 16 12 6 1 0 7 21 ] Polygonal,,InvTabl,[] -Polygonal,,InvRevTabl,[] Polygonal,,RevInvTabl,[] +Polygonal,,InvRevTabl,[] Polygonal,,AccTabl,[0 0 1 0 1 3 0 1 4 7 0 1 5 11 15 0 1 6 15 25 30 0 1 7 19 35 50 56 0 ] Polygonal,,RevAccTabl,[0 1 0 3 1 0 7 4 1 0 15 11 5 1 0 30 25 15 6 1 0 56 50 35 19 7 1 0 ] Polygonal,,AccRevTabl,[0 1 1 2 3 3 3 6 7 7 4 10 14 15 15 5 15 24 29 30 30 6 21 37 49 55 ] Polygonal,,AntiDiagTabl,[0 0 0 1 0 1 0 1 2 0 1 3 0 1 4 3 0 1 5 6 0 1 6 9 4 0 1 7 12 10 0 1 ] -Polygonal,,DiffxTabl,[0 0 2 0 2 6 0 2 9 12 0 2 12 24 20 0 2 15 36 50 30 0 2 18 48 80 90 ] Polygonal,,BinTabl,[0 0 1 0 1 4 0 1 5 15 0 1 6 21 56 0 1 7 27 80 200 0 1 8 33 104 280 ] Polygonal,,InvBinTabl,[0 0 1 0 1 0 0 1 1 -3 0 1 2 -3 0 0 1 3 -3 0 0 0 1 4 -3 0 0 0 0 1 5 ] +Polygonal,,DiffxTabl,[0 0 2 0 2 6 0 2 9 12 0 2 12 24 20 0 2 15 36 50 30 0 2 18 48 80 90 ] Polygonal,A055795,RowSum,[0 1 3 7 15 30 56 98 162 255 385 561 793 1092 1470 1940 2516 3213 ] Polygonal,,EvenSum,[0 0 2 3 8 15 28 50 80 130 190 285 392 553 728 980 1248 1620 2010 ] Polygonal,,OddSum,[0 1 1 4 7 15 28 48 82 125 195 276 401 539 742 960 1268 1593 2037 ] @@ -28,9 +28,9 @@ Polygonal,A000004,ColLeft,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Polygonal,A001477,ColRight,[0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ] Polygonal,,BinConv,[0 1 4 15 56 200 672 2128 6400 18432 51200 137984 362496 931840 ] Polygonal,A103775,InvBinConv,[0 1 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] -Polygonal,,TransSqrs,[0 1 9 40 135 387 980 2240 4698 9165 16819 29304 48841 78351 121590 ] Polygonal,,TransNat0,[0 1 5 16 43 103 224 448 834 1461 2431 3872 5941 8827 12754 17984 ] Polygonal,,TransNat1,[0 2 8 23 58 133 280 546 996 1716 2816 4433 6734 9919 14224 19924 ] +Polygonal,,TransSqrs,[0 1 9 40 135 387 980 2240 4698 9165 16819 29304 48841 78351 121590 ] Polygonal,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] Polygonal,A000290,DiagRow2,[0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 ] Polygonal,A000326,DiagRow3,[0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 376 425 477 ] @@ -44,5 +44,5 @@ Polygonal,A249354,PolyRow3,[0 7 38 111 244 455 762 1183 1736 2439 3310 4367 5628 Polygonal,,PolyCol2,[0 2 10 38 130 414 1242 3542 9682 25550 65482 163782 401346 966590 ] Polygonal,,PolyCol3,[0 3 21 111 525 2316 9696 38946 151302 572025 2114679 7672665 ] Polygonal,,PolyDiag,[0 1 10 111 1476 23130 420126 8713922 203646472 5300831655 ] -Polygonal,,PosHalf,[0 1 4 13 40 117 324 853 2152 5245 12436 28845 65736 147685 327940 ] -Polygonal,,NegHalf,[0 1 0 1 0 -3 8 -31 72 -195 448 -1071 2416 -5475 12120 -26719 58232 ] +Polygonal,A360606,PosHalf,[0 1 4 13 40 117 324 853 2152 5245 12436 28845 65736 147685 327940 ] +Polygonal,A360605,NegHalf,[0 1 0 1 0 -3 8 -31 72 -195 448 -1071 2416 -5475 12120 -26719 58232 ] diff --git a/data/csv/PowLaguerre.csv b/data/csv/PowLaguerre.csv index 879df50..e729ee1 100644 --- a/data/csv/PowLaguerre.csv +++ b/data/csv/PowLaguerre.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -PowLaguerre,A021012,Tabl,[1 1 1 2 4 2 6 18 18 6 24 96 144 96 24 120 600 1200 1200 600 120 ] +PowLaguerre,A196347,Tabl,[1 1 1 2 4 2 6 18 18 6 24 96 144 96 24 120 600 1200 1200 600 120 ] PowLaguerre,A021012,RevTabl,[1 1 1 2 4 2 6 18 18 6 24 96 144 96 24 120 600 1200 1200 600 120 ] PowLaguerre,,InvTabl,[] -PowLaguerre,,InvRevTabl,[] PowLaguerre,,RevInvTabl,[] +PowLaguerre,,InvRevTabl,[] PowLaguerre,,AccTabl,[1 1 2 2 6 8 6 24 42 48 24 120 264 360 384 120 720 1920 3120 3720 ] PowLaguerre,,RevAccTabl,[1 2 1 8 6 2 48 42 24 6 384 360 264 120 24 3840 3720 3120 1920 720 ] PowLaguerre,,AccRevTabl,[1 1 2 2 6 8 6 24 42 48 24 120 264 360 384 120 720 1920 3120 3720 ] PowLaguerre,,AntiDiagTabl,[1 1 2 1 6 4 24 18 2 120 96 18 720 600 144 6 5040 4320 1200 96 ] -PowLaguerre,,DiffxTabl,[1 1 2 2 8 6 6 36 54 24 24 192 432 384 120 120 1200 3600 4800 3000 ] PowLaguerre,,BinTabl,[1 1 2 2 6 12 6 24 60 120 24 120 360 840 1680 120 720 2520 6720 ] PowLaguerre,,InvBinTabl,[1 1 0 2 2 -4 6 12 -12 0 24 72 -24 -72 144 120 480 120 -720 720 0 ] +PowLaguerre,,DiffxTabl,[1 1 2 2 8 6 6 36 54 24 24 192 432 384 120 120 1200 3600 4800 3000 ] PowLaguerre,A000165,RowSum,[1 2 8 48 384 3840 46080 645120 10321920 185794560 3715891200 ] PowLaguerre,A002866,EvenSum,[1 1 4 24 192 1920 23040 322560 5160960 92897280 1857945600 ] PowLaguerre,A002866,OddSum,[0 1 4 24 192 1920 23040 322560 5160960 92897280 1857945600 ] @@ -23,14 +23,14 @@ PowLaguerre,A000142,RowGcd,[1 1 2 6 24 120 720 5040 40320 362880 3628800 3991680 PowLaguerre,A059837,RowMax,[1 1 4 18 144 1200 14400 176400 2822400 45722880 914457600 ] PowLaguerre,A059837,ColMiddle,[1 1 4 18 144 1200 14400 176400 2822400 45722880 914457600 ] PowLaguerre,A122747,ColECentral,[1 4 144 14400 2822400 914457600 442597478400 299195895398400 ] -PowLaguerre,,ColOCentral,[1 18 1200 176400 45722880 18441561600 10685567692800 ] +PowLaguerre,A360602,ColOCentral,[1 18 1200 176400 45722880 18441561600 10685567692800 ] PowLaguerre,A000142,ColLeft,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] PowLaguerre,A000142,ColRight,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] PowLaguerre,A001813,BinConv,[1 2 12 120 1680 30240 665280 17297280 518918400 17643225600 ] PowLaguerre,A122747,InvBinConv,[1 0 -4 0 144 0 -14400 0 2822400 0 -914457600 0 442597478400 0 ] -PowLaguerre,,TransSqrs,[0 1 12 144 1920 28800 483840 9031680 185794560 4180377600 ] PowLaguerre,A014479,TransNat0,[0 1 8 72 768 9600 138240 2257920 41287680 836075520 18579456000 ] PowLaguerre,A187735,TransNat1,[1 3 16 120 1152 13440 184320 2903040 51609600 1021870080 ] +PowLaguerre,,TransSqrs,[0 1 12 144 1920 28800 483840 9031680 185794560 4180377600 ] PowLaguerre,A001563,DiagRow1,[1 4 18 96 600 4320 35280 322560 3265920 36288000 439084800 ] PowLaguerre,A001804,DiagRow2,[2 18 144 1200 10800 105840 1128960 13063680 163296000 2195424000 ] PowLaguerre,A001805,DiagRow3,[6 96 1200 14400 176400 2257920 30481920 435456000 6586272000 ] diff --git a/data/csv/Rencontres.csv b/data/csv/Rencontres.csv index 046bca0..e406145 100644 --- a/data/csv/Rencontres.csv +++ b/data/csv/Rencontres.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Rencontres,A008290,Tabl,[1 0 1 1 0 1 2 3 0 1 9 8 6 0 1 44 45 20 10 0 1 265 264 135 40 15 0 ] Rencontres,A098825,RevTabl,[1 1 0 1 0 1 1 0 3 2 1 0 6 8 9 1 0 10 20 45 44 1 0 15 40 135 264 ] Rencontres,A055137,InvTabl,[1 0 1 -1 0 1 -2 -3 0 1 -3 -8 -6 0 1 -4 -15 -20 -10 0 1 -5 -24 -45 ] -Rencontres,,InvRevTabl,[] Rencontres,,RevInvTabl,[1 1 0 1 0 -1 1 0 -3 -2 1 0 -6 -8 -3 1 0 -10 -20 -15 -4 1 0 -15 -40 ] +Rencontres,,InvRevTabl,[] Rencontres,,AccTabl,[1 0 1 1 1 2 2 5 5 6 9 17 23 23 24 44 89 109 119 119 120 265 529 ] Rencontres,,RevAccTabl,[1 1 0 2 1 1 6 5 5 2 24 23 23 17 9 120 119 119 109 89 44 720 719 ] Rencontres,,AccRevTabl,[1 1 1 1 1 2 1 1 4 6 1 1 7 15 24 1 1 11 31 76 120 1 1 16 56 191 455 ] Rencontres,,AntiDiagTabl,[1 0 1 1 2 0 9 3 1 44 8 0 265 45 6 1 1854 264 20 0 14833 1855 135 ] -Rencontres,,DiffxTabl,[1 0 2 1 0 3 2 6 0 4 9 16 18 0 5 44 90 60 40 0 6 265 528 405 160 75 ] Rencontres,,BinTabl,[1 0 1 1 1 2 2 5 8 12 9 17 31 51 78 44 89 154 249 384 570 265 529 ] Rencontres,,InvBinTabl,[1 0 1 1 -1 2 2 1 -4 8 9 -1 -1 -3 14 44 1 -26 41 -56 82 265 -1 -128 ] +Rencontres,,DiffxTabl,[1 0 2 1 0 3 2 6 0 4 9 16 18 0 5 44 90 60 40 0 6 265 528 405 160 75 ] Rencontres,A000142,RowSum,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] Rencontres,A062282,EvenSum,[1 0 2 2 16 64 416 2848 22912 205952 2060032 22659328 271913984 ] Rencontres,A063083,OddSum,[0 1 0 4 8 56 304 2192 17408 156928 1568768 17257472 207087616 ] @@ -28,9 +28,9 @@ Rencontres,A000166,ColLeft,[1 0 1 2 9 44 265 1854 14833 133496 1334961 14684570 Rencontres,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Rencontres,,BinConv,[1 1 2 12 78 570 4900 48160 530390 6464430 86327388 1252761048 ] Rencontres,,InvBinConv,[1 1 2 8 14 82 132 744 4566 -33442 587036 -6952064 87792652 ] -Rencontres,A052849,TransSqrs,[0 1 4 12 48 240 1440 10080 80640 725760 7257600 79833600 958003200 ] Rencontres,A000142,TransNat0,[0 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] Rencontres,A098558,TransNat1,[1 2 4 12 48 240 1440 10080 80640 725760 7257600 79833600 958003200 ] +Rencontres,A052849,TransSqrs,[0 1 4 12 48 240 1440 10080 80640 725760 7257600 79833600 958003200 ] Rencontres,A000004,DiagRow1,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Rencontres,A000217,DiagRow2,[1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 ] Rencontres,A007290,DiagRow3,[2 8 20 40 70 112 168 240 330 440 572 728 910 1120 1360 1632 1938 ] diff --git a/data/csv/RisingFact.csv b/data/csv/RisingFact.csv index f697fb7..d84aef1 100644 --- a/data/csv/RisingFact.csv +++ b/data/csv/RisingFact.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -RisingFact,A094587,Tabl,[1 1 1 2 2 1 6 6 3 1 24 24 12 4 1 120 120 60 20 5 1 720 720 360 120 ] +RisingFact,A008279,Tabl,[1 1 1 2 2 1 6 6 3 1 24 24 12 4 1 120 120 60 20 5 1 720 720 360 120 ] RisingFact,A008279,RevTabl,[1 1 1 1 2 2 1 3 6 6 1 4 12 24 24 1 5 20 60 120 120 1 6 30 120 360 ] RisingFact,A128229,InvTabl,[1 -1 1 0 -2 1 0 0 -3 1 0 0 0 -4 1 0 0 0 0 -5 1 0 0 0 0 0 -6 1 0 0 ] -RisingFact,,InvRevTabl,[] RisingFact,A158821,RevInvTabl,[1 1 -1 1 -2 0 1 -3 0 0 1 -4 0 0 0 1 -5 0 0 0 0 1 -6 0 0 0 0 0 1 -7 ] +RisingFact,,InvRevTabl,[] RisingFact,,AccTabl,[1 1 2 2 4 5 6 12 15 16 24 48 60 64 65 120 240 300 320 325 326 720 ] RisingFact,,RevAccTabl,[1 2 1 5 4 2 16 15 12 6 65 64 60 48 24 326 325 320 300 240 120 1957 ] RisingFact,A347667,AccRevTabl,[1 1 2 1 3 5 1 4 10 16 1 5 17 41 65 1 6 26 86 206 326 1 7 37 157 ] RisingFact,,AntiDiagTabl,[1 1 2 1 6 2 24 6 1 120 24 3 720 120 12 1 5040 720 60 4 40320 5040 ] -RisingFact,,DiffxTabl,[1 1 2 2 4 3 6 12 9 4 24 48 36 16 5 120 240 180 80 25 6 720 1440 ] RisingFact,,BinTabl,[1 1 2 2 4 7 6 12 21 34 24 48 84 136 209 120 240 420 680 1045 1546 ] RisingFact,,InvBinTabl,[1 1 0 2 0 -1 6 0 -3 4 24 0 -12 16 -15 120 0 -60 80 -75 56 720 0 ] +RisingFact,,DiffxTabl,[1 1 2 2 4 3 6 12 9 4 24 48 36 16 5 120 240 180 80 25 6 720 1440 ] RisingFact,A000522,RowSum,[1 2 5 16 65 326 1957 13700 109601 986410 9864101 108505112 ] RisingFact,A009179,EvenSum,[1 1 3 9 37 185 1111 7777 62217 559953 5599531 61594841 739138093 ] RisingFact,A009628,OddSum,[0 1 2 7 28 141 846 5923 47384 426457 4264570 46910271 562923252 ] @@ -28,9 +28,9 @@ RisingFact,A000142,ColLeft,[1 1 2 6 24 120 720 5040 40320 362880 3628800 3991680 RisingFact,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] RisingFact,A002720,BinConv,[1 2 7 34 209 1546 13327 130922 1441729 17572114 234662231 ] RisingFact,A009940,InvBinConv,[1 0 -1 4 -15 56 -185 204 6209 -112400 1520271 -19165420 237686449 ] -RisingFact,A030297,TransSqrs,[0 1 6 27 124 645 3906 27391 219192 1972809 19728190 217010211 ] RisingFact,A007526,TransNat0,[0 1 4 15 64 325 1956 13699 109600 986409 9864100 108505111 ] RisingFact,A111063,TransNat1,[1 3 9 31 129 651 3913 27399 219201 1972819 19728201 217010223 ] +RisingFact,A030297,TransSqrs,[0 1 6 27 124 645 3906 27391 219192 1972809 19728190 217010211 ] RisingFact,A000027,DiagRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] RisingFact,A002378,DiagRow2,[2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] RisingFact,A007531,DiagRow3,[6 24 60 120 210 336 504 720 990 1320 1716 2184 2730 3360 4080 4896 ] diff --git a/data/csv/Schroeder.csv b/data/csv/Schroeder.csv index b9ec67f..7091a80 100644 --- a/data/csv/Schroeder.csv +++ b/data/csv/Schroeder.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Schroeder,A122538,Tabl,[1 0 1 0 2 1 0 6 4 1 0 22 16 6 1 0 90 68 30 8 1 0 394 304 146 48 10 ] Schroeder,A106579,RevTabl,[1 1 0 1 2 0 1 4 6 0 1 6 16 22 0 1 8 30 68 90 0 1 10 48 146 304 394 ] Schroeder,A122542,InvTabl,[1 0 1 0 -2 1 0 2 -4 1 0 -2 8 -6 1 0 2 -12 18 -8 1 0 -2 16 -38 32 ] -Schroeder,,InvRevTabl,[] Schroeder,A266213,RevInvTabl,[1 1 0 1 -2 0 1 -4 2 0 1 -6 8 -2 0 1 -8 18 -12 2 0 1 -10 32 -38 16 ] +Schroeder,,InvRevTabl,[] Schroeder,,AccTabl,[1 0 1 0 2 3 0 6 10 11 0 22 38 44 45 0 90 158 188 196 197 0 394 698 ] Schroeder,,RevAccTabl,[1 1 0 3 2 0 11 10 6 0 45 44 38 22 0 197 196 188 158 90 0 903 902 ] Schroeder,A144944,AccRevTabl,[1 1 1 1 3 3 1 5 11 11 1 7 23 45 45 1 9 39 107 197 197 1 11 59 205 ] Schroeder,,AntiDiagTabl,[1 0 0 1 0 2 0 6 1 0 22 4 0 90 16 1 0 394 68 6 0 1806 304 30 1 0 ] -Schroeder,,DiffxTabl,[1 0 2 0 4 3 0 12 12 4 0 44 48 24 5 0 180 204 120 40 6 0 788 912 ] Schroeder,,BinTabl,[1 0 1 0 2 5 0 6 16 31 0 22 60 120 209 0 90 248 504 896 1471 0 394 ] Schroeder,,InvBinTabl,[1 0 1 0 2 -3 0 6 -8 7 0 22 -28 24 -15 0 90 -112 96 -64 31 0 394 ] +Schroeder,,DiffxTabl,[1 0 2 0 4 3 0 12 12 4 0 44 48 24 5 0 180 204 120 40 6 0 788 912 ] Schroeder,A001003,RowSum,[1 1 3 11 45 197 903 4279 20793 103049 518859 2646723 13648869 ] Schroeder,A239204,EvenSum,[1 0 1 4 17 76 353 1688 8257 41128 207905 1063932 5501073 28695252 ] Schroeder,A010683,OddSum,[0 1 2 7 28 121 550 2591 12536 61921 310954 1582791 8147796 ] @@ -28,9 +28,9 @@ Schroeder,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Schroeder,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Schroeder,A178792,BinConv,[1 1 5 31 209 1471 10625 78079 580865 4361215 32978945 250806271 ] Schroeder,A255047,InvBinConv,[1 1 -3 7 -15 31 -63 127 -255 511 -1023 2047 -4095 8191 -16383 ] -Schroeder,A065096,TransSqrs,[0 1 6 31 156 785 3978 20335 104856 545073 2854350 15046383 ] Schroeder,A239204,TransNat0,[0 1 4 17 76 353 1688 8257 41128 207905 1063932 5501073 28695252 ] Schroeder,A010683,TransNat1,[1 2 7 28 121 550 2591 12536 61921 310954 1582791 8147796 42344121 ] +Schroeder,A065096,TransSqrs,[0 1 6 31 156 785 3978 20335 104856 545073 2854350 15046383 ] Schroeder,A005843,DiagRow1,[0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 ] Schroeder,A054000,DiagRow2,[0 6 16 30 48 70 96 126 160 198 240 286 336 390 448 510 576 646 720 ] Schroeder,,DiagRow3,[0 22 68 146 264 430 652 938 1296 1734 2260 2882 3608 4446 5404 ] diff --git a/data/csv/SchroederP.csv b/data/csv/SchroederP.csv index 045a71a..0a97838 100644 --- a/data/csv/SchroederP.csv +++ b/data/csv/SchroederP.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence SchroederP,A104684,Tabl,[1 2 1 6 6 1 20 30 12 1 70 140 90 20 1 252 630 560 210 30 1 924 ] SchroederP,A063007,RevTabl,[1 1 2 1 6 6 1 12 30 20 1 20 90 140 70 1 30 210 560 630 252 1 42 ] SchroederP,,InvTabl,[1 -2 1 6 -6 1 -32 42 -12 1 310 -440 150 -20 1 -4932 7110 -2540 390 ] -SchroederP,,InvRevTabl,[] SchroederP,,RevInvTabl,[1 1 -2 1 -6 6 1 -12 42 -32 1 -20 150 -440 310 1 -30 390 -2540 7110 ] +SchroederP,,InvRevTabl,[] SchroederP,,AccTabl,[1 2 3 6 12 13 20 50 62 63 70 210 300 320 321 252 882 1442 1652 ] SchroederP,,RevAccTabl,[1 3 2 13 12 6 63 62 50 20 321 320 300 210 70 1683 1682 1652 1442 ] SchroederP,,AccRevTabl,[1 1 3 1 7 13 1 13 43 63 1 21 111 251 321 1 31 241 801 1431 1683 1 ] SchroederP,,AntiDiagTabl,[1 2 6 1 20 6 70 30 1 252 140 12 924 630 90 1 3432 2772 560 20 ] -SchroederP,,DiffxTabl,[1 2 2 6 12 3 20 60 36 4 70 280 270 80 5 252 1260 1680 840 150 6 ] SchroederP,,BinTabl,[1 2 3 6 12 19 20 50 92 147 70 210 440 780 1251 252 882 2072 4032 ] SchroederP,,InvBinTabl,[1 2 -1 6 0 -5 20 10 -28 35 70 70 -120 100 -29 252 378 -448 168 282 ] +SchroederP,,DiffxTabl,[1 2 2 6 12 3 20 60 36 4 70 280 270 80 5 252 1260 1680 840 150 6 ] SchroederP,A001850,RowSum,[1 3 13 63 321 1683 8989 48639 265729 1462563 8097453 45046719 ] SchroederP,A226994,EvenSum,[1 2 7 32 161 842 4495 24320 132865 731282 4048727 22523360 ] SchroederP,A047665,OddSum,[0 1 6 31 160 841 4494 24319 132864 731281 4048726 22523359 ] @@ -28,9 +28,9 @@ SchroederP,A000984,ColLeft,[1 2 6 20 70 252 924 3432 12870 48620 184756 705432 2 SchroederP,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] SchroederP,A005258,BinConv,[1 3 19 147 1251 11253 104959 1004307 9793891 96918753 970336269 ] SchroederP,,InvBinConv,[1 -1 -5 35 -29 -751 3991 4115 -137885 495269 2114245 -25786795 ] -SchroederP,,TransSqrs,[0 1 10 87 696 5265 38298 270655 1870816 12706353 85078170 ] SchroederP,A108666,TransNat0,[0 1 8 57 384 2505 16008 100849 628736 3888657 23900040 146146473 ] SchroederP,,TransNat1,[1 4 21 120 705 4188 24997 149488 894465 5351220 31997493 191193192 ] +SchroederP,,TransSqrs,[0 1 10 87 696 5265 38298 270655 1870816 12706353 85078170 ] SchroederP,A002378,DiagRow1,[2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] SchroederP,A033487,DiagRow2,[6 30 90 210 420 756 1260 1980 2970 4290 6006 8190 10920 14280 ] SchroederP,A105939,DiagRow3,[20 140 560 1680 4200 9240 18480 34320 60060 100100 160160 247520 ] diff --git a/data/csv/Seidel.csv b/data/csv/Seidel.csv index 07c7ce9..2973076 100644 --- a/data/csv/Seidel.csv +++ b/data/csv/Seidel.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Seidel,A008281,Tabl,[1 0 1 0 1 1 0 1 2 2 0 2 4 5 5 0 5 10 14 16 16 0 16 32 46 56 61 61 ] Seidel,A010094,RevTabl,[1 1 0 1 1 0 2 2 1 0 5 5 4 2 0 16 16 14 10 5 0 61 61 56 46 32 16 0 ] Seidel,,InvTabl,[] -Seidel,,InvRevTabl,[] Seidel,,RevInvTabl,[] +Seidel,,InvRevTabl,[] Seidel,,AccTabl,[1 0 1 0 1 2 0 1 3 5 0 2 6 11 16 0 5 15 29 45 61 0 16 48 94 150 211 ] Seidel,,RevAccTabl,[1 1 0 2 1 0 5 3 1 0 16 11 6 2 0 61 45 29 15 5 0 272 211 150 94 48 ] Seidel,A008282,AccRevTabl,[1 1 1 1 2 2 2 4 5 5 5 10 14 16 16 16 32 46 56 61 61 61 122 178 224 ] Seidel,,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 2 2 0 5 4 2 0 16 10 5 0 61 32 14 5 0 272 122 ] -Seidel,,DiffxTabl,[1 0 2 0 2 3 0 2 6 8 0 4 12 20 25 0 10 30 56 80 96 0 32 96 184 280 ] Seidel,,BinTabl,[1 0 1 0 1 3 0 1 4 11 0 2 8 23 57 0 5 20 59 152 361 0 16 64 190 496 ] Seidel,,InvBinTabl,[1 0 1 0 1 -1 0 1 0 -1 0 2 0 -1 1 0 5 0 -1 0 1 0 16 0 -2 0 1 -1 0 ] +Seidel,,DiffxTabl,[1 0 2 0 2 3 0 2 6 8 0 4 12 20 25 0 10 30 56 80 96 0 32 96 184 280 ] Seidel,A000111,RowSum,[1 1 2 5 16 61 272 1385 7936 50521 353792 2702765 22368256 ] Seidel,,EvenSum,[1 0 1 2 9 26 149 618 4277 23122 188457 1257154 11812705 93774138 ] Seidel,,OddSum,[0 1 1 3 7 35 123 767 3659 27399 165335 1445611 10555551 105586843 ] @@ -28,9 +28,9 @@ Seidel,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Seidel,A000111,ColRight,[1 1 1 2 5 16 61 272 1385 7936 50521 353792 2702765 22368256 ] Seidel,A001586,BinConv,[1 1 3 11 57 361 2763 24611 250737 2873041 36581523 512343611 ] Seidel,A000012,InvBinConv,[1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 ] -Seidel,,TransSqrs,[0 1 5 27 143 827 5175 35255 260251 2074623 17780875 163187203 ] Seidel,,TransNat0,[0 1 3 11 45 211 1113 6551 42585 303271 2348973 19665491 176992725 ] Seidel,A000111,TransNat1,[1 2 5 16 61 272 1385 7936 50521 353792 2702765 22368256 199360981 ] +Seidel,,TransSqrs,[0 1 5 27 143 827 5175 35255 260251 2074623 17780875 163187203 ] Seidel,A000111,DiagRow1,[0 1 2 5 16 61 272 1385 7936 50521 353792 2702765 22368256 ] Seidel,A006212,DiagRow2,[0 1 4 14 56 256 1324 7664 49136 345856 2652244 22014464 196658216 ] Seidel,A006213,DiagRow3,[0 2 10 46 224 1202 7120 46366 329984 2551202 21306880 191252686 ] diff --git a/data/csv/SeidelBoust.csv b/data/csv/SeidelBoust.csv index b732298..b0132fa 100644 --- a/data/csv/SeidelBoust.csv +++ b/data/csv/SeidelBoust.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence SeidelBoust,A008280,Tabl,[1 0 1 1 1 0 0 1 2 2 5 5 4 2 0 0 5 10 14 16 16 61 61 56 46 32 16 0 ] SeidelBoust,A108040,RevTabl,[1 1 0 0 1 1 2 2 1 0 0 2 4 5 5 16 16 14 10 5 0 0 16 32 46 56 61 61 ] SeidelBoust,,InvTabl,[] -SeidelBoust,,InvRevTabl,[] SeidelBoust,,RevInvTabl,[] +SeidelBoust,,InvRevTabl,[] SeidelBoust,,AccTabl,[1 0 1 1 2 2 0 1 3 5 5 10 14 16 16 0 5 15 29 45 61 61 122 178 224 ] SeidelBoust,,RevAccTabl,[1 1 0 2 2 1 5 3 1 0 16 16 14 10 5 61 45 29 15 5 0 272 272 256 224 ] SeidelBoust,,AccRevTabl,[1 1 1 0 1 2 2 4 5 5 0 2 6 11 16 16 32 46 56 61 61 0 16 48 94 150 ] SeidelBoust,,AntiDiagTabl,[1 0 1 1 0 1 5 1 0 0 5 2 61 5 4 2 0 61 10 2 1385 61 56 14 0 0 1385 ] -SeidelBoust,,DiffxTabl,[1 0 2 1 2 0 0 2 6 8 5 10 12 8 0 0 10 30 56 80 96 61 122 168 184 ] SeidelBoust,,BinTabl,[1 0 1 1 2 3 0 1 4 11 5 10 19 34 57 0 5 20 59 152 361 61 122 239 ] SeidelBoust,,InvBinTabl,[1 0 1 1 0 -1 0 1 0 -1 5 0 -1 0 1 0 5 0 -1 0 1 61 0 -5 0 1 0 -1 0 ] +SeidelBoust,,DiffxTabl,[1 0 2 1 2 0 0 2 6 8 5 10 12 8 0 0 10 30 56 80 96 61 122 168 184 ] SeidelBoust,A000111,RowSum,[1 1 2 5 16 61 272 1385 7936 50521 353792 2702765 22368256 ] SeidelBoust,,EvenSum,[1 0 1 2 9 26 149 618 4277 23122 188457 1257154 11812705 93774138 ] SeidelBoust,,OddSum,[0 1 1 3 7 35 123 767 3659 27399 165335 1445611 10555551 105586843 ] @@ -28,9 +28,9 @@ SeidelBoust,A122045,ColLeft,[1 0 1 0 5 0 61 0 1385 0 50521 0 2702765 0 199360981 SeidelBoust,A009006,ColRight,[1 1 0 2 0 16 0 272 0 7936 0 353792 0 22368256 0 1903757312 0 ] SeidelBoust,A001586,BinConv,[1 1 3 11 57 361 2763 24611 250737 2873041 36581523 512343611 ] SeidelBoust,A000012,InvBinConv,[1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 ] -SeidelBoust,,TransSqrs,[0 1 1 27 39 827 1611 35255 86795 2074623 6180615 163187203 ] SeidelBoust,,TransNat0,[0 1 1 11 19 211 519 6551 20903 303271 1188947 19665491 91426347 ] SeidelBoust,,TransNat1,[1 2 3 16 35 272 791 7936 28839 353792 1542739 22368256 113794603 ] +SeidelBoust,,TransSqrs,[0 1 1 27 39 827 1611 35255 86795 2074623 6180615 163187203 ] SeidelBoust,,DiagRow1,[0 1 2 2 16 16 272 272 7936 7936 353792 353792 22368256 22368256 ] SeidelBoust,,DiagRow2,[1 1 4 14 32 256 544 7664 15872 345856 707584 22014464 44736512 ] SeidelBoust,,DiagRow3,[0 5 10 46 224 800 7120 23536 329984 1053440 21306880 66750976 ] diff --git a/data/csv/Sierpinski.csv b/data/csv/Sierpinski.csv index 5017395..a12efbd 100644 --- a/data/csv/Sierpinski.csv +++ b/data/csv/Sierpinski.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Sierpinski,A047999,Tabl,[1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 ] Sierpinski,A047999,RevTabl,[1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 ] Sierpinski,A047999,InvTabl,[1 -1 1 -1 0 1 1 -1 -1 1 -1 0 0 0 1 1 -1 0 0 -1 1 1 0 -1 0 -1 0 1 ] -Sierpinski,A047999,InvRevTabl,[1 -1 1 -1 0 1 1 -1 -1 1 -1 0 0 0 1 1 -1 0 0 -1 1 1 0 -1 0 -1 0 1 ] Sierpinski,A047999,RevInvTabl,[1 1 -1 1 0 -1 1 -1 -1 1 1 0 0 0 -1 1 -1 0 0 -1 1 1 0 -1 0 -1 0 1 1 ] +Sierpinski,A047999,InvRevTabl,[1 -1 1 -1 0 1 1 -1 -1 1 -1 0 0 0 1 1 -1 0 0 -1 1 1 0 -1 0 -1 0 1 ] Sierpinski,A261363,AccTabl,[1 1 2 1 1 2 1 2 3 4 1 1 1 1 2 1 2 2 2 3 4 1 1 2 2 3 3 4 1 2 3 4 5 ] Sierpinski,A358549,RevAccTabl,[1 2 1 2 1 1 4 3 2 1 2 1 1 1 1 4 3 2 2 2 1 4 3 3 2 2 1 1 8 7 6 5 4 ] Sierpinski,A261363,AccRevTabl,[1 1 2 1 1 2 1 2 3 4 1 1 1 1 2 1 2 2 2 3 4 1 1 2 2 3 3 4 1 2 3 4 5 ] Sierpinski,A000012,AntiDiagTabl,[1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 0 ] -Sierpinski,,DiffxTabl,[1 1 2 1 0 3 1 2 3 4 1 0 0 0 5 1 2 0 0 5 6 1 0 3 0 5 0 7 1 2 3 4 5 ] Sierpinski,,BinTabl,[1 1 2 1 1 2 1 2 4 8 1 1 1 1 2 1 2 3 4 6 12 1 1 2 4 8 16 32 1 2 4 8 ] Sierpinski,,InvBinTabl,[1 1 0 1 -1 2 1 0 0 0 1 -1 1 -1 2 1 0 -1 2 -2 0 1 -1 2 -4 8 -16 32 ] +Sierpinski,,DiffxTabl,[1 1 2 1 0 3 1 2 3 4 1 0 0 0 5 1 2 0 0 5 6 1 0 3 0 5 0 7 1 2 3 4 5 ] Sierpinski,A001316,RowSum,[1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16 2 4 4 8 ] Sierpinski,A060632,EvenSum,[1 1 2 2 2 2 4 4 2 2 4 4 4 4 8 8 2 2 4 4 ] Sierpinski,A001316,OddSum,[0 1 0 2 0 2 0 4 0 2 0 4 0 4 0 8 0 2 0 4 ] @@ -28,9 +28,9 @@ Sierpinski,A000012,ColLeft,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Sierpinski,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] Sierpinski,A088560,BinConv,[1 2 2 8 2 12 32 128 2 20 92 464 992 4032 8192 32768 2 36 308 2320 ] Sierpinski,,InvBinConv,[1 0 2 0 2 0 32 0 2 0 92 0 992 0 8192 0 2 0 308 0 ] -Sierpinski,,TransSqrs,[0 1 4 14 16 42 56 140 64 146 168 380 224 500 560 1240 256 546 584 ] Sierpinski,A335063,TransNat0,[0 1 2 6 4 10 12 28 8 18 20 44 24 52 56 120 16 34 36 76 ] Sierpinski,,TransNat1,[1 3 4 10 6 14 16 36 10 22 24 52 28 60 64 136 18 38 40 84 ] +Sierpinski,,TransSqrs,[0 1 4 14 16 42 56 140 64 146 168 380 224 500 560 1240 256 546 584 ] Sierpinski,A000035,DiagRow1,[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ] Sierpinski,A021913,DiagRow2,[1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 ] Sierpinski,A011765,DiagRow3,[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 ] diff --git a/data/csv/StirlingCyc.csv b/data/csv/StirlingCyc.csv index 901b473..e6bdc7b 100644 --- a/data/csv/StirlingCyc.csv +++ b/data/csv/StirlingCyc.csv @@ -1,16 +1,16 @@ Triangle,Anum,Trait,Sequence -StirlingCyc,A048994,Tabl,[1 0 1 0 1 1 0 2 3 1 0 6 11 6 1 0 24 50 35 10 1 0 120 274 225 85 15 ] +StirlingCyc,A132393,Tabl,[1 0 1 0 1 1 0 2 3 1 0 6 11 6 1 0 24 50 35 10 1 0 120 274 225 85 15 ] StirlingCyc,A054654,RevTabl,[1 1 0 1 1 0 1 3 2 0 1 6 11 6 0 1 10 35 50 24 0 1 15 85 225 274 120 ] StirlingCyc,A048993,InvTabl,[1 0 1 0 -1 1 0 1 -3 1 0 -1 7 -6 1 0 1 -15 25 -10 1 0 -1 31 -90 65 ] -StirlingCyc,,InvRevTabl,[] StirlingCyc,A106800,RevInvTabl,[1 1 0 1 -1 0 1 -3 1 0 1 -6 7 -1 0 1 -10 25 -15 1 0 1 -15 65 -90 31 ] +StirlingCyc,,InvRevTabl,[] StirlingCyc,A349782,AccTabl,[1 0 1 0 1 2 0 2 5 6 0 6 17 23 24 0 24 74 109 119 120 0 120 394 619 ] StirlingCyc,,RevAccTabl,[1 1 0 2 1 0 6 5 2 0 24 23 17 6 0 120 119 109 74 24 0 720 719 704 ] StirlingCyc,A096747,AccRevTabl,[1 1 1 1 2 2 1 4 6 6 1 7 18 24 24 1 11 46 96 120 120 1 16 101 326 ] StirlingCyc,A331327,AntiDiagTabl,[1 0 0 1 0 1 0 2 1 0 6 3 0 24 11 1 0 120 50 6 0 720 274 35 1 0 5040 ] -StirlingCyc,A360174,DiffxTabl,[1 0 2 0 2 3 0 4 9 4 0 12 33 24 5 0 48 150 140 50 6 0 240 822 900 ] StirlingCyc,,BinTabl,[1 0 1 0 1 3 0 2 7 16 0 6 23 57 115 0 24 98 257 546 1021 0 120 514 ] StirlingCyc,,InvBinTabl,[1 0 1 0 1 -1 0 2 -1 -2 0 6 -1 -9 19 0 24 2 -43 74 -79 0 120 34 ] +StirlingCyc,A360174,DiffxTabl,[1 0 2 0 2 3 0 4 9 4 0 12 33 24 5 0 48 150 140 50 6 0 240 822 900 ] StirlingCyc,A000142,RowSum,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] StirlingCyc,A105752,EvenSum,[1 0 1 3 12 60 360 2520 20160 181440 1814400 19958400 239500800 ] StirlingCyc,A001710,OddSum,[0 1 1 3 12 60 360 2520 20160 181440 1814400 19958400 239500800 ] @@ -28,9 +28,9 @@ StirlingCyc,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingCyc,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] StirlingCyc,A211210,BinConv,[1 1 3 16 115 1021 10696 128472 1734447 25937683 424852351 ] StirlingCyc,A317274,InvBinConv,[1 1 -1 -2 19 -79 76 2640 -36945 329371 -1861949 -4438774 355714228 ] -StirlingCyc,A151881,TransSqrs,[0 1 5 23 120 724 5012 39332 345832 3371976 36135792 422379792 ] StirlingCyc,A000254,TransNat0,[0 1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] StirlingCyc,A000774,TransNat1,[1 2 5 17 74 394 2484 18108 149904 1389456 14257440 160460640 ] +StirlingCyc,A151881,TransSqrs,[0 1 5 23 120 724 5012 39332 345832 3371976 36135792 422379792 ] StirlingCyc,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] StirlingCyc,A000914,DiagRow2,[0 2 11 35 85 175 322 546 870 1320 1925 2717 3731 5005 6580 8500 ] StirlingCyc,A001303,DiagRow3,[0 6 50 225 735 1960 4536 9450 18150 32670 55770 91091 143325 ] diff --git a/data/csv/StirlingCyc2.csv b/data/csv/StirlingCyc2.csv index d38fe1e..ae7b56b 100644 --- a/data/csv/StirlingCyc2.csv +++ b/data/csv/StirlingCyc2.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence StirlingCyc2,A358622,Tabl,[1 0 0 0 1 0 0 2 0 0 0 6 3 0 0 0 24 20 0 0 0 0 120 130 15 0 0 0 0 ] StirlingCyc2,,RevTabl,[1 0 0 0 1 0 0 0 2 0 0 0 3 6 0 0 0 0 20 24 0 0 0 0 15 130 120 0 0 0 ] StirlingCyc2,,InvTabl,[] -StirlingCyc2,,InvRevTabl,[] StirlingCyc2,,RevInvTabl,[] +StirlingCyc2,,InvRevTabl,[] StirlingCyc2,,AccTabl,[1 0 0 0 1 1 0 2 2 2 0 6 9 9 9 0 24 44 44 44 44 0 120 250 265 265 ] StirlingCyc2,,RevAccTabl,[1 0 0 1 1 0 2 2 2 0 9 9 9 6 0 44 44 44 44 24 0 265 265 265 265 250 ] StirlingCyc2,,AccRevTabl,[1 0 0 0 1 1 0 0 2 2 0 0 3 9 9 0 0 0 20 44 44 0 0 0 15 145 265 265 ] StirlingCyc2,,AntiDiagTabl,[1 0 0 0 0 1 0 2 0 0 6 0 0 24 3 0 0 120 20 0 0 720 130 0 0 0 5040 ] -StirlingCyc2,,DiffxTabl,[1 0 0 0 2 0 0 4 0 0 0 12 9 0 0 0 48 60 0 0 0 0 240 390 60 0 0 0 0 ] StirlingCyc2,,BinTabl,[1 0 0 0 1 2 0 2 4 6 0 6 15 27 42 0 24 68 132 216 320 0 120 370 765 ] StirlingCyc2,,InvBinTabl,[1 0 0 0 1 -2 0 2 -4 6 0 6 -9 9 -6 0 24 -28 12 24 -80 0 120 -110 ] +StirlingCyc2,,DiffxTabl,[1 0 0 0 2 0 0 4 0 0 0 12 9 0 0 0 48 60 0 0 0 0 240 390 60 0 0 0 0 ] StirlingCyc2,A000166,RowSum,[1 0 1 2 9 44 265 1854 14833 133496 1334961 14684570 176214841 ] StirlingCyc2,A216778,EvenSum,[1 0 0 0 3 20 130 924 7413 66744 667476 7342280 88107415 1145396460 ] StirlingCyc2,A216779,OddSum,[0 0 1 2 6 24 135 930 7420 66752 667485 7342290 88107426 1145396472 ] @@ -28,9 +28,9 @@ StirlingCyc2,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingCyc2,A000007,ColRight,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingCyc2,,BinConv,[1 0 2 6 42 320 2970 31794 385574 5212752 77619060 1260722100 ] StirlingCyc2,,InvBinConv,[1 0 -2 6 -6 -80 930 -7014 38374 -46416 -2993220 65145300 ] -StirlingCyc2,,TransSqrs,[0 0 1 2 18 104 775 6306 57372 575424 6318549 75437570 973440094 ] StirlingCyc2,A162973,TransNat0,[0 0 1 2 12 64 425 3198 27216 258144 2701737 30933770 384675148 ] StirlingCyc2,,TransNat1,[1 0 2 4 21 108 690 5052 42049 391640 4036698 45618340 560889989 ] +StirlingCyc2,,TransSqrs,[0 0 1 2 18 104 775 6306 57372 575424 6318549 75437570 973440094 ] StirlingCyc2,A003982,DiagRow1,[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingCyc2,A109135,DiagRow2,[0 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingCyc2,A007253,DiagRow3,[0 6 20 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/StirlingCycB.csv b/data/csv/StirlingCycB.csv index 7acbe96..ce84e30 100644 --- a/data/csv/StirlingCycB.csv +++ b/data/csv/StirlingCycB.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence StirlingCycB,A028338,Tabl,[1 1 1 3 4 1 15 23 9 1 105 176 86 16 1 945 1689 950 230 25 1 10395 ] StirlingCycB,A039758,RevTabl,[1 1 1 1 4 3 1 9 23 15 1 16 86 176 105 1 25 230 950 1689 945 1 36 ] StirlingCycB,A039755,InvTabl,[1 -1 1 1 -4 1 -1 13 -9 1 1 -40 58 -16 1 -1 121 -330 170 -25 1 1 ] -StirlingCycB,,InvRevTabl,[] StirlingCycB,A039756,RevInvTabl,[1 1 -1 1 -4 1 1 -9 13 -1 1 -16 58 -40 1 1 -25 170 -330 121 -1 1 ] +StirlingCycB,,InvRevTabl,[] StirlingCycB,,AccTabl,[1 1 2 3 7 8 15 38 47 48 105 281 367 383 384 945 2634 3584 3814 ] StirlingCycB,,RevAccTabl,[1 2 1 8 7 3 48 47 38 15 384 383 367 281 105 3840 3839 3814 3584 ] StirlingCycB,,AccRevTabl,[1 1 2 1 5 8 1 10 33 48 1 17 103 279 384 1 26 256 1206 2895 3840 1 ] StirlingCycB,,AntiDiagTabl,[1 1 3 1 15 4 105 23 1 945 176 9 10395 1689 86 1 135135 19524 950 ] -StirlingCycB,,DiffxTabl,[1 1 2 3 8 3 15 46 27 4 105 352 258 64 5 945 3378 2850 920 125 6 ] StirlingCycB,,BinTabl,[1 1 2 3 7 12 15 38 70 112 105 281 543 907 1390 945 2634 5273 9092 ] StirlingCycB,,InvBinTabl,[1 1 0 3 1 -4 15 8 -22 28 105 71 -161 181 -146 945 744 -1483 1502 ] +StirlingCycB,,DiffxTabl,[1 1 2 3 8 3 15 46 27 4 105 352 258 64 5 945 3378 2850 920 125 6 ] StirlingCycB,A000165,RowSum,[1 2 8 48 384 3840 46080 645120 10321920 185794560 3715891200 ] StirlingCycB,A002866,EvenSum,[1 1 4 24 192 1920 23040 322560 5160960 92897280 1857945600 ] StirlingCycB,A002866,OddSum,[0 1 4 24 192 1920 23040 322560 5160960 92897280 1857945600 ] @@ -28,9 +28,9 @@ StirlingCycB,A001147,ColLeft,[1 1 3 15 105 945 10395 135135 2027025 34459425 654 StirlingCycB,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] StirlingCycB,,BinConv,[1 2 12 112 1390 21316 387016 8089152 190810614 5004826108 ] StirlingCycB,,InvBinConv,[1 0 -4 28 -146 176 13096 -332856 6220278 -100254944 1315515816 ] -StirlingCycB,,TransSqrs,[0 1 8 68 680 7984 108416 1676800 29146624 563014656 11971694592 ] StirlingCycB,A203159,TransNat0,[0 1 6 44 400 4384 56448 836352 14026752 262803456 5441863680 ] StirlingCycB,,TransNat1,[1 3 14 92 784 8224 102528 1481472 24348672 448598016 9157754880 ] +StirlingCycB,,TransSqrs,[0 1 8 68 680 7984 108416 1676800 29146624 563014656 11971694592 ] StirlingCycB,A000290,DiagRow1,[1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 ] StirlingCycB,A024196,DiagRow2,[3 23 86 230 505 973 1708 2796 4335 6435 9218 12818 17381 23065 ] StirlingCycB,A024197,DiagRow3,[15 176 950 3480 10045 24640 53676 106800 197835 345840 576290 ] diff --git a/data/csv/StirlingSet.csv b/data/csv/StirlingSet.csv index 9e8e95f..1c3cde5 100644 --- a/data/csv/StirlingSet.csv +++ b/data/csv/StirlingSet.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence StirlingSet,A048993,Tabl,[1 0 1 0 1 1 0 1 3 1 0 1 7 6 1 0 1 15 25 10 1 0 1 31 90 65 15 1 0 1 ] StirlingSet,A106800,RevTabl,[1 1 0 1 1 0 1 3 1 0 1 6 7 1 0 1 10 25 15 1 0 1 15 65 90 31 1 0 1 ] StirlingSet,A048994,InvTabl,[1 0 1 0 -1 1 0 2 -3 1 0 -6 11 -6 1 0 24 -50 35 -10 1 0 -120 274 ] -StirlingSet,,InvRevTabl,[] StirlingSet,A054654,RevInvTabl,[1 1 0 1 -1 0 1 -3 2 0 1 -6 11 -6 0 1 -10 35 -50 24 0 1 -15 85 -225 ] +StirlingSet,,InvRevTabl,[] StirlingSet,A359107,AccTabl,[1 0 1 0 1 2 0 1 4 5 0 1 8 14 15 0 1 16 41 51 52 0 1 32 122 187 202 ] StirlingSet,,RevAccTabl,[1 1 0 2 1 0 5 4 1 0 15 14 8 1 0 52 51 41 16 1 0 203 202 187 122 32 ] StirlingSet,,AccRevTabl,[1 1 1 1 2 2 1 4 5 5 1 7 14 15 15 1 11 36 51 52 52 1 16 81 171 202 ] StirlingSet,A136011,AntiDiagTabl,[1 0 0 1 0 1 0 1 1 0 1 3 0 1 7 1 0 1 15 6 0 1 31 25 1 0 1 63 90 10 ] -StirlingSet,A069138,DiffxTabl,[1 0 2 0 2 3 0 2 9 4 0 2 21 24 5 0 2 45 100 50 6 0 2 93 360 325 90 ] StirlingSet,,BinTabl,[1 0 1 0 1 3 0 1 5 13 0 1 9 30 71 0 1 17 73 204 456 0 1 33 186 615 ] StirlingSet,,InvBinTabl,[1 0 1 0 1 -1 0 1 1 -5 0 1 5 -12 15 0 1 13 -17 -4 56 0 1 29 0 -113 ] +StirlingSet,A069138,DiffxTabl,[1 0 2 0 2 3 0 2 9 4 0 2 21 24 5 0 2 45 100 50 6 0 2 93 360 325 90 ] StirlingSet,A000110,RowSum,[1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 ] StirlingSet,A024430,EvenSum,[1 0 1 3 8 25 97 434 2095 10707 58194 338195 2097933 13796952 ] StirlingSet,A024429,OddSum,[0 1 1 2 7 27 106 443 2045 10440 57781 340375 2115664 13847485 ] @@ -28,9 +28,9 @@ StirlingSet,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingSet,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] StirlingSet,A122455,BinConv,[1 1 3 13 71 456 3337 27203 243203 2357356 24554426 272908736 ] StirlingSet,A343841,InvBinConv,[1 1 -1 -5 15 56 -455 -237 16947 -64220 -529494 6833608 -8606015 ] -StirlingSet,A033452,TransSqrs,[0 1 5 22 99 471 2386 12867 73681 446620 2856457 19217243 135610448 ] StirlingSet,A005493,TransNat0,[0 1 3 10 37 151 674 3263 17007 94828 562595 3535027 23430840 ] StirlingSet,A000110,TransNat1,[1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 ] +StirlingSet,A033452,TransSqrs,[0 1 5 22 99 471 2386 12867 73681 446620 2856457 19217243 135610448 ] StirlingSet,A000217,DiagRow1,[0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 ] StirlingSet,A001296,DiagRow2,[0 1 7 25 65 140 266 462 750 1155 1705 2431 3367 4550 6020 7820 ] StirlingSet,A001297,DiagRow3,[0 1 15 90 350 1050 2646 5880 11880 22275 39325 66066 106470 165620 ] diff --git a/data/csv/StirlingSet2.csv b/data/csv/StirlingSet2.csv index cc732df..43ae422 100644 --- a/data/csv/StirlingSet2.csv +++ b/data/csv/StirlingSet2.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence StirlingSet2,A358623,Tabl,[1 0 0 0 1 0 0 1 0 0 0 1 3 0 0 0 1 10 0 0 0 0 1 25 15 0 0 0 0 1 56 ] StirlingSet2,,RevTabl,[1 0 0 0 1 0 0 0 1 0 0 0 3 1 0 0 0 0 10 1 0 0 0 0 15 25 1 0 0 0 0 0 ] StirlingSet2,,InvTabl,[] -StirlingSet2,,InvRevTabl,[] StirlingSet2,,RevInvTabl,[] +StirlingSet2,,InvRevTabl,[] StirlingSet2,,AccTabl,[1 0 0 0 1 1 0 1 1 1 0 1 4 4 4 0 1 11 11 11 11 0 1 26 41 41 41 41 0 ] StirlingSet2,,RevAccTabl,[1 0 0 1 1 0 1 1 1 0 4 4 4 1 0 11 11 11 11 1 0 41 41 41 41 26 1 0 ] StirlingSet2,,AccRevTabl,[1 0 0 0 1 1 0 0 1 1 0 0 3 4 4 0 0 0 10 11 11 0 0 0 15 40 41 41 0 0 ] StirlingSet2,,AntiDiagTabl,[1 0 0 0 0 1 0 1 0 0 1 0 0 1 3 0 0 1 10 0 0 1 25 0 0 0 1 56 15 0 0 ] -StirlingSet2,,DiffxTabl,[1 0 0 0 2 0 0 2 0 0 0 2 9 0 0 0 2 30 0 0 0 0 2 75 60 0 0 0 0 2 168 ] StirlingSet2,,BinTabl,[1 0 0 0 1 2 0 1 2 3 0 1 5 12 22 0 1 12 33 64 105 0 1 27 93 214 405 ] StirlingSet2,,InvBinTabl,[1 0 0 0 1 -2 0 1 -2 3 0 1 1 -6 14 0 1 8 -27 56 -95 0 1 23 -57 86 ] +StirlingSet2,,DiffxTabl,[1 0 0 0 2 0 0 2 0 0 0 2 9 0 0 0 2 30 0 0 0 0 2 75 60 0 0 0 0 2 168 ] StirlingSet2,A000296,RowSum,[1 0 1 1 4 11 41 162 715 3425 17722 98253 580317 3633280 24011157 ] StirlingSet2,A097763,EvenSum,[1 0 0 0 3 10 25 56 224 1506 9951 57992 315425 1761552 11022180 ] StirlingSet2,A097762,OddSum,[0 0 1 1 1 1 16 106 491 1919 7771 40261 264892 1871728 12988977 ] @@ -28,9 +28,9 @@ StirlingSet2,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingSet2,A000007,ColRight,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingSet2,,BinConv,[1 0 2 3 22 105 681 4858 38130 328737 3064195 30647496 327006747 ] StirlingSet2,,InvBinConv,[1 0 -2 3 14 -95 69 2506 -16766 -6495 949895 -7070624 -7563117 ] -StirlingSet2,,TransSqrs,[0 0 1 1 13 41 236 1170 6567 38407 238255 1555269 10663500 76572952 ] StirlingSet2,,TransNat0,[0 0 1 1 7 21 96 428 2129 11287 64003 385375 2454244 16467036 ] StirlingSet2,,TransNat1,[1 0 2 2 11 32 137 590 2844 14712 81725 483628 3034561 20100316 ] +StirlingSet2,,TransSqrs,[0 0 1 1 13 41 236 1170 6567 38407 238255 1555269 10663500 76572952 ] StirlingSet2,A003982,DiagRow1,[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingSet2,A348019,DiagRow2,[0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] StirlingSet2,A004488,DiagRow3,[0 1 10 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] diff --git a/data/csv/StirlingSetB.csv b/data/csv/StirlingSetB.csv index 6bd5e56..b54233e 100644 --- a/data/csv/StirlingSetB.csv +++ b/data/csv/StirlingSetB.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence StirlingSetB,A154602,Tabl,[1 1 1 3 4 1 11 19 9 1 49 104 70 16 1 257 641 550 190 25 1 1539 ] StirlingSetB,,RevTabl,[1 1 1 1 4 3 1 9 19 11 1 16 70 104 49 1 25 190 550 641 257 1 36 425 ] StirlingSetB,,InvTabl,[1 -1 1 1 -4 1 -1 17 -9 1 1 -96 74 -16 1 -1 729 -690 210 -25 1 1 ] -StirlingSetB,,InvRevTabl,[] StirlingSetB,,RevInvTabl,[1 1 -1 1 -4 1 1 -9 17 -1 1 -16 74 -96 1 1 -25 210 -690 729 -1 1 ] +StirlingSetB,,InvRevTabl,[] StirlingSetB,,AccTabl,[1 1 2 3 7 8 11 30 39 40 49 153 223 239 240 257 898 1448 1638 1663 ] StirlingSetB,,RevAccTabl,[1 2 1 8 7 3 40 39 30 11 240 239 223 153 49 1664 1663 1638 1448 898 ] StirlingSetB,,AccRevTabl,[1 1 2 1 5 8 1 10 29 40 1 17 87 191 240 1 26 216 766 1407 1664 1 37 ] StirlingSetB,,AntiDiagTabl,[1 1 3 1 11 4 49 19 1 257 104 9 1539 641 70 1 10299 4380 550 16 ] -StirlingSetB,,DiffxTabl,[1 1 2 3 8 3 11 38 27 4 49 208 210 64 5 257 1282 1650 760 125 6 ] StirlingSetB,,BinTabl,[1 1 2 3 7 12 11 30 58 96 49 153 327 587 950 257 898 2089 4020 6906 ] StirlingSetB,,InvBinTabl,[1 1 0 3 1 -4 11 8 -18 20 49 55 -89 69 -10 257 384 -475 206 258 ] +StirlingSetB,,DiffxTabl,[1 1 2 3 8 3 11 38 27 4 49 208 210 64 5 257 1282 1650 760 125 6 ] StirlingSetB,A055882,RowSum,[1 2 8 40 240 1664 12992 112256 1059840 10827264 118758400 ] StirlingSetB,,EvenSum,[1 1 4 20 120 832 6496 56128 529920 5413632 59379200 694855680 ] StirlingSetB,,OddSum,[0 1 4 20 120 832 6496 56128 529920 5413632 59379200 694855680 ] @@ -28,9 +28,9 @@ StirlingSetB,A004211,ColLeft,[1 1 3 11 49 257 1539 10299 75905 609441 5284451 49 StirlingSetB,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] StirlingSetB,,BinConv,[1 2 12 96 950 10988 143976 2094032 33318534 573707692 10600961800 ] StirlingSetB,,InvBinConv,[1 0 -4 20 -10 -776 7784 -25432 -416634 8540368 -72453016 ] -StirlingSetB,,TransSqrs,[0 1 8 64 544 4976 48960 516160 5804032 69305344 875277312 ] StirlingSetB,,TransNat0,[0 1 6 40 296 2416 21568 208832 2176896 24275968 288048640 ] StirlingSetB,,TransNat1,[1 3 14 80 536 4080 34560 321088 3236736 35103232 406807040 ] +StirlingSetB,,TransSqrs,[0 1 8 64 544 4976 48960 516160 5804032 69305344 875277312 ] StirlingSetB,A000290,DiagRow1,[1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 ] StirlingSetB,,DiagRow2,[3 19 70 190 425 833 1484 2460 3855 5775 8338 11674 15925 21245 ] StirlingSetB,,DiagRow3,[11 104 550 2080 6265 16016 36204 74400 141735 253880 432146 704704 ] diff --git a/data/csv/Sylvester.csv b/data/csv/Sylvester.csv index 1b08c3f..1427cc1 100644 --- a/data/csv/Sylvester.csv +++ b/data/csv/Sylvester.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence Sylvester,A341101,Tabl,[1 0 2 0 1 4 0 2 6 8 0 6 19 24 16 0 24 80 110 80 32 0 120 418 615 ] Sylvester,,RevTabl,[1 2 0 4 1 0 8 6 2 0 16 24 19 6 0 32 80 110 80 24 0 64 240 500 615 ] Sylvester,,InvTabl,[] -Sylvester,,InvRevTabl,[] Sylvester,,RevInvTabl,[] +Sylvester,,InvRevTabl,[] Sylvester,,AccTabl,[1 0 2 0 1 5 0 2 8 16 0 6 25 49 65 0 24 104 214 294 326 0 120 538 ] Sylvester,,RevAccTabl,[1 2 0 5 1 0 16 8 2 0 65 49 25 6 0 326 294 214 104 24 0 1957 1893 ] Sylvester,,AccRevTabl,[1 2 2 4 5 5 8 14 16 16 16 40 59 65 65 32 112 222 302 326 326 64 ] Sylvester,,AntiDiagTabl,[1 0 0 2 0 1 0 2 4 0 6 6 0 24 19 8 0 120 80 24 0 720 418 110 16 0 ] -Sylvester,,DiffxTabl,[1 0 4 0 2 12 0 4 18 32 0 12 57 96 80 0 48 240 440 400 192 0 240 ] Sylvester,,BinTabl,[1 0 2 0 1 6 0 2 10 32 0 6 31 99 250 0 24 128 422 1096 2452 0 120 ] Sylvester,,InvBinTabl,[1 0 2 0 1 2 0 2 2 -4 0 6 7 -15 10 0 24 32 -58 24 52 0 120 178 -279 ] +Sylvester,,DiffxTabl,[1 0 4 0 2 12 0 4 18 32 0 12 57 96 80 0 48 240 440 400 192 0 240 ] Sylvester,A000522,RowSum,[1 2 5 16 65 326 1957 13700 109601 986410 9864101 108505112 ] Sylvester,,EvenSum,[1 0 4 6 35 160 982 6846 54805 493200 4932056 54252550 651030679 ] Sylvester,,OddSum,[0 2 1 10 30 166 975 6854 54796 493210 4932045 54252562 651030666 ] @@ -28,9 +28,9 @@ Sylvester,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] Sylvester,A000079,ColRight,[1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 ] Sylvester,,BinConv,[1 2 6 32 250 2452 28294 372276 5476902 88843388 1572182524 ] Sylvester,,InvBinConv,[1 2 2 -4 10 52 -626 3600 -4378 -209092 3583684 -37666352 228658456 ] -Sylvester,,TransSqrs,[0 2 17 98 554 3414 23631 184134 1604396 15497994 164595605 ] Sylvester,,TransNat0,[0 2 9 38 180 994 6385 47074 392672 3659730 37709393 425779518 ] Sylvester,,TransNat1,[1 4 14 54 245 1320 8342 60774 502273 4646140 47573494 534284630 ] +Sylvester,,TransSqrs,[0 2 17 98 554 3414 23631 184134 1604396 15497994 164595605 ] Sylvester,A001788,DiagRow1,[0 1 6 24 80 240 672 1792 4608 11520 28160 67584 159744 372736 ] Sylvester,,DiagRow2,[0 2 19 110 500 1960 6944 22848 71040 211200 605440 1683968 4566016 ] Sylvester,,DiagRow3,[0 6 80 615 3570 17360 74592 292320 1066560 3674880 12080640 ] diff --git a/data/csv/SymPoly.csv b/data/csv/SymPoly.csv index 3b81507..06ee14d 100644 --- a/data/csv/SymPoly.csv +++ b/data/csv/SymPoly.csv @@ -1,48 +1,48 @@ Triangle,Anum,Trait,Sequence -Sympoly,A165675,Tabl,[1 1 1 2 3 1 6 11 5 1 24 50 26 7 1 120 274 154 47 9 1 720 1764 1044 ] -Sympoly,A105954,RevTabl,[1 1 1 1 3 2 1 5 11 6 1 7 26 50 24 1 9 47 154 274 120 1 11 74 342 ] -Sympoly,,InvTabl,[1 -1 1 1 -3 1 0 4 -5 1 0 0 9 -7 1 0 0 0 16 -9 1 0 0 0 0 25 -11 1 0 ] -Sympoly,,InvRevTabl,[] -Sympoly,,RevInvTabl,[1 1 -1 1 -3 1 1 -5 4 0 1 -7 9 0 0 1 -9 16 0 0 0 1 -11 25 0 0 0 0 1 ] -Sympoly,,AccTabl,[1 1 2 2 5 6 6 17 22 23 24 74 100 107 108 120 394 548 595 604 605 ] -Sympoly,,RevAccTabl,[1 2 1 6 5 2 23 22 17 6 108 107 100 74 24 605 604 595 548 394 120 ] -Sympoly,,AccRevTabl,[1 1 2 1 4 6 1 6 17 23 1 8 34 84 108 1 10 57 211 485 605 1 12 86 ] -Sympoly,,AntiDiagTabl,[1 1 2 1 6 3 24 11 1 120 50 5 720 274 26 1 5040 1764 154 7 40320 ] -Sympoly,,DiffxTabl,[1 1 2 2 6 3 6 22 15 4 24 100 78 28 5 120 548 462 188 45 6 720 3528 ] -Sympoly,,BinTabl,[1 1 2 2 5 9 6 17 33 55 24 74 150 259 409 120 394 822 1451 2337 ] -Sympoly,,InvBinTabl,[1 1 0 2 1 -3 6 5 -11 13 24 26 -50 55 -47 120 154 -274 287 -231 136 ] -Sympoly,A093345,RowSum,[1 2 6 23 108 605 3956 29649 250892 2367629 24662700 281153801 ] -Sympoly,,EvenSum,[1 1 3 11 51 283 1839 13719 115675 1088459 11311047 128686087 ] -Sympoly,,OddSum,[0 1 3 12 57 322 2117 15930 135217 1279170 13351653 152467714 ] -Sympoly,,AltSum,[1 0 0 -1 -6 -39 -278 -2211 -19542 -190711 -2040606 -23781627 ] -Sympoly,,AntiDiagSum,[1 1 3 9 36 175 1021 6965 54480 480843 4727469 51234337 606792460 ] -Sympoly,,AccSum,[1 3 13 68 413 2866 22457 196634 1906165 20290322 235457201 ] -Sympoly,,AccRevSum,[1 3 11 47 235 1369 9191 70207 602755 5753597 60495199 695039491 ] -Sympoly,,RowLcm,[1 1 6 330 54600 178489080 7911786960 839338855795440 ] -Sympoly,A000012,RowGcd,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -Sympoly,A000254,RowMax,[1 1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] -Sympoly,,ColMiddle,[1 1 3 11 26 154 342 2754 5944 60216 127860 1557660 3272688 ] -Sympoly,,ColECentral,[1 3 26 342 5944 127860 3272688 97053936 3270729600 123418922400 ] -Sympoly,,ColOCentral,[1 11 154 2754 60216 1557660 46536624 1576890000 59753750400 ] -Sympoly,A000142,ColLeft,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] -Sympoly,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] -Sympoly,,BinConv,[1 2 9 55 409 3546 34981 386163 4711281 62903242 911810321 ] -Sympoly,,InvBinConv,[1 0 -3 13 -47 136 1 -5935 88481 -1060568 11963541 -131894883 ] -Sympoly,,TransSqrs,[0 1 7 40 233 1482 10513 83366 735417 7166746 76599497 891914958 ] -Sympoly,,TransNat0,[0 1 5 24 127 764 5235 40558 351863 3385968 35832499 413885690 ] -Sympoly,,TransNat1,[1 3 11 47 235 1369 9191 70207 602755 5753597 60495199 695039491 ] -Sympoly,A004273,DiagRow1,[1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 ] -Sympoly,A080663,DiagRow2,[2 11 26 47 74 107 146 191 242 299 362 431 506 587 674 767 866 971 ] -Sympoly,A165676,DiagRow3,[6 50 154 342 638 1066 1650 2414 3382 4578 6026 7750 9774 12122 ] -Sympoly,A000254,DiagCol1,[1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] -Sympoly,A001705,DiagCol2,[1 5 26 154 1044 8028 69264 663696 6999840 80627040 1007441280 ] -Sympoly,A001711,DiagCol3,[1 7 47 342 2754 24552 241128 2592720 30334320 383970240 5231113920 ] -Sympoly,,PolyTabl,[1 1 1 2 2 1 6 6 3 1 24 23 12 4 1 120 108 56 20 5 1 720 605 300 111 ] -Sympoly,A000027,PolyRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] -Sympoly,A002378,PolyRow2,[2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] -Sympoly,,PolyRow3,[6 23 56 111 194 311 468 671 926 1239 1616 2063 2586 3191 3884 4671 ] -Sympoly,,PolyCol2,[1 3 12 56 300 1836 12760 99912 873696 8458976 89952192 1043100480 ] -Sympoly,,PolyCol3,[1 4 20 111 678 4569 34038 280197 2541654 25295193 274793742 ] -Sympoly,,PolyDiag,[1 2 12 111 1344 19965 350856 7117677 163705344 4209578829 ] -Sympoly,,PosHalf,[1 3 15 103 903 9663 122287 1787991 29671479 551104591 11328415263 ] -Sympoly,,NegHalf,[1 -1 3 -13 75 -517 3875 -26301 29531 5729003 -227270157 7253745395 ] +SymPoly,A165675,Tabl,[1 1 1 2 3 1 6 11 5 1 24 50 26 7 1 120 274 154 47 9 1 720 1764 1044 ] +SymPoly,A105954,RevTabl,[1 1 1 1 3 2 1 5 11 6 1 7 26 50 24 1 9 47 154 274 120 1 11 74 342 ] +SymPoly,,InvTabl,[1 -1 1 1 -3 1 0 4 -5 1 0 0 9 -7 1 0 0 0 16 -9 1 0 0 0 0 25 -11 1 0 ] +SymPoly,,RevInvTabl,[1 1 -1 1 -3 1 1 -5 4 0 1 -7 9 0 0 1 -9 16 0 0 0 1 -11 25 0 0 0 0 1 ] +SymPoly,,InvRevTabl,[] +SymPoly,,AccTabl,[1 1 2 2 5 6 6 17 22 23 24 74 100 107 108 120 394 548 595 604 605 ] +SymPoly,,RevAccTabl,[1 2 1 6 5 2 23 22 17 6 108 107 100 74 24 605 604 595 548 394 120 ] +SymPoly,,AccRevTabl,[1 1 2 1 4 6 1 6 17 23 1 8 34 84 108 1 10 57 211 485 605 1 12 86 ] +SymPoly,,AntiDiagTabl,[1 1 2 1 6 3 24 11 1 120 50 5 720 274 26 1 5040 1764 154 7 40320 ] +SymPoly,,BinTabl,[1 1 2 2 5 9 6 17 33 55 24 74 150 259 409 120 394 822 1451 2337 ] +SymPoly,,InvBinTabl,[1 1 0 2 1 -3 6 5 -11 13 24 26 -50 55 -47 120 154 -274 287 -231 136 ] +SymPoly,,DiffxTabl,[1 1 2 2 6 3 6 22 15 4 24 100 78 28 5 120 548 462 188 45 6 720 3528 ] +SymPoly,A093345,RowSum,[1 2 6 23 108 605 3956 29649 250892 2367629 24662700 281153801 ] +SymPoly,,EvenSum,[1 1 3 11 51 283 1839 13719 115675 1088459 11311047 128686087 ] +SymPoly,,OddSum,[0 1 3 12 57 322 2117 15930 135217 1279170 13351653 152467714 ] +SymPoly,,AltSum,[1 0 0 -1 -6 -39 -278 -2211 -19542 -190711 -2040606 -23781627 ] +SymPoly,,AntiDiagSum,[1 1 3 9 36 175 1021 6965 54480 480843 4727469 51234337 606792460 ] +SymPoly,,AccSum,[1 3 13 68 413 2866 22457 196634 1906165 20290322 235457201 ] +SymPoly,,AccRevSum,[1 3 11 47 235 1369 9191 70207 602755 5753597 60495199 695039491 ] +SymPoly,,RowLcm,[1 1 6 330 54600 178489080 7911786960 839338855795440 ] +SymPoly,A000012,RowGcd,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] +SymPoly,A000254,RowMax,[1 1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] +SymPoly,,ColMiddle,[1 1 3 11 26 154 342 2754 5944 60216 127860 1557660 3272688 ] +SymPoly,,ColECentral,[1 3 26 342 5944 127860 3272688 97053936 3270729600 123418922400 ] +SymPoly,,ColOCentral,[1 11 154 2754 60216 1557660 46536624 1576890000 59753750400 ] +SymPoly,A000142,ColLeft,[1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600 ] +SymPoly,A000012,ColRight,[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] +SymPoly,,BinConv,[1 2 9 55 409 3546 34981 386163 4711281 62903242 911810321 ] +SymPoly,,InvBinConv,[1 0 -3 13 -47 136 1 -5935 88481 -1060568 11963541 -131894883 ] +SymPoly,,TransNat0,[0 1 5 24 127 764 5235 40558 351863 3385968 35832499 413885690 ] +SymPoly,,TransNat1,[1 3 11 47 235 1369 9191 70207 602755 5753597 60495199 695039491 ] +SymPoly,,TransSqrs,[0 1 7 40 233 1482 10513 83366 735417 7166746 76599497 891914958 ] +SymPoly,A004273,DiagRow1,[1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 ] +SymPoly,A080663,DiagRow2,[2 11 26 47 74 107 146 191 242 299 362 431 506 587 674 767 866 971 ] +SymPoly,A165676,DiagRow3,[6 50 154 342 638 1066 1650 2414 3382 4578 6026 7750 9774 12122 ] +SymPoly,A000254,DiagCol1,[1 3 11 50 274 1764 13068 109584 1026576 10628640 120543840 ] +SymPoly,A001705,DiagCol2,[1 5 26 154 1044 8028 69264 663696 6999840 80627040 1007441280 ] +SymPoly,A001711,DiagCol3,[1 7 47 342 2754 24552 241128 2592720 30334320 383970240 5231113920 ] +SymPoly,,PolyTabl,[1 1 1 2 2 1 6 6 3 1 24 23 12 4 1 120 108 56 20 5 1 720 605 300 111 ] +SymPoly,A000027,PolyRow1,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ] +SymPoly,A002378,PolyRow2,[2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 ] +SymPoly,,PolyRow3,[6 23 56 111 194 311 468 671 926 1239 1616 2063 2586 3191 3884 4671 ] +SymPoly,,PolyCol2,[1 3 12 56 300 1836 12760 99912 873696 8458976 89952192 1043100480 ] +SymPoly,,PolyCol3,[1 4 20 111 678 4569 34038 280197 2541654 25295193 274793742 ] +SymPoly,,PolyDiag,[1 2 12 111 1344 19965 350856 7117677 163705344 4209578829 ] +SymPoly,,PosHalf,[1 3 15 103 903 9663 122287 1787991 29671479 551104591 11328415263 ] +SymPoly,,NegHalf,[1 -1 3 -13 75 -517 3875 -26301 29531 5729003 -227270157 7253745395 ] diff --git a/data/csv/TernaryTrees.csv b/data/csv/TernaryTrees.csv index f7063da..3ff7fa9 100644 --- a/data/csv/TernaryTrees.csv +++ b/data/csv/TernaryTrees.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence TernaryTrees,A355172,Tabl,[1 0 1 0 1 3 0 1 5 12 0 1 7 25 55 0 1 9 42 130 273 0 1 11 63 245 ] TernaryTrees,,RevTabl,[1 1 0 3 1 0 12 5 1 0 55 25 7 1 0 273 130 42 9 1 0 1428 700 245 63 ] TernaryTrees,,InvTabl,[] -TernaryTrees,,InvRevTabl,[] TernaryTrees,,RevInvTabl,[] +TernaryTrees,,InvRevTabl,[] TernaryTrees,,AccTabl,[1 0 1 0 1 4 0 1 6 18 0 1 8 33 88 0 1 10 52 182 455 0 1 12 75 320 ] TernaryTrees,,RevAccTabl,[1 1 0 4 1 0 18 6 1 0 88 33 8 1 0 455 182 52 10 1 0 2448 1020 320 ] TernaryTrees,,AccRevTabl,[1 1 1 3 4 4 12 17 18 18 55 80 87 88 88 273 403 445 454 455 455 ] TernaryTrees,,AntiDiagTabl,[1 0 0 1 0 1 0 1 3 0 1 5 0 1 7 12 0 1 9 25 0 1 11 42 55 0 1 13 63 ] -TernaryTrees,,DiffxTabl,[1 0 2 0 2 9 0 2 15 48 0 2 21 100 275 0 2 27 168 650 1638 0 2 33 ] TernaryTrees,,BinTabl,[1 0 1 0 1 5 0 1 7 30 0 1 9 49 201 0 1 11 72 356 1438 0 1 13 99 567 ] TernaryTrees,,InvBinTabl,[1 0 1 0 1 1 0 1 3 0 0 1 5 7 -7 0 1 7 18 12 -42 0 1 9 33 55 0 -198 ] +TernaryTrees,,DiffxTabl,[1 0 2 0 2 9 0 2 15 48 0 2 21 100 275 0 2 27 168 650 1638 0 2 33 ] TernaryTrees,A006629,RowSum,[1 1 4 18 88 455 2448 13566 76912 444015 2601300 15426840 92431584 ] TernaryTrees,,EvenSum,[1 0 3 5 62 139 1684 4297 52284 143366 1755839 5040716 62096408 ] TernaryTrees,,OddSum,[0 1 1 13 26 316 764 9269 24628 300649 845461 10386124 30335176 ] @@ -28,9 +28,9 @@ TernaryTrees,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] TernaryTrees,A001764,ColRight,[1 1 3 12 55 273 1428 7752 43263 246675 1430715 8414640 50067108 ] TernaryTrees,,BinConv,[1 1 5 30 201 1438 10734 82512 647945 5170710 41782726 341013236 ] TernaryTrees,A246434,InvBinConv,[1 1 1 0 -7 -42 -198 -858 -3575 -14586 -58786 -235144 -936054 ] -TernaryTrees,,TransSqrs,[0 1 13 129 1134 9320 73440 562457 4221052 31202145 227981325 ] TernaryTrees,,TransNat0,[0 1 7 47 310 2030 13260 86583 565708 3700125 24231675 158896452 ] TernaryTrees,,TransNat1,[1 2 11 65 398 2485 15708 100149 642620 4144140 26832975 174323292 ] +TernaryTrees,,TransSqrs,[0 1 13 129 1134 9320 73440 562457 4221052 31202145 227981325 ] TernaryTrees,A102893,DiagRow1,[0 1 5 25 130 700 3876 21945 126500 740025 4382625 26225628 ] TernaryTrees,A102594,DiagRow2,[0 1 7 42 245 1428 8379 49588 296010 1781325 10798788 65900296 ] TernaryTrees,A230547,DiagRow3,[0 1 9 63 408 2565 15939 98670 610740 3786588 23535820 146710476 ] diff --git a/data/csv/WardSet.csv b/data/csv/WardSet.csv index 9c65920..d8e5d52 100644 --- a/data/csv/WardSet.csv +++ b/data/csv/WardSet.csv @@ -2,15 +2,15 @@ Triangle,Anum,Trait,Sequence WardSet,A269939,Tabl,[1 0 1 0 1 3 0 1 10 15 0 1 25 105 105 0 1 56 490 1260 945 0 1 119 ] WardSet,A181996,RevTabl,[1 1 0 3 1 0 15 10 1 0 105 105 25 1 0 945 1260 490 56 1 0 10395 ] WardSet,,InvTabl,[] -WardSet,,InvRevTabl,[] WardSet,,RevInvTabl,[] +WardSet,,InvRevTabl,[] WardSet,,AccTabl,[1 0 1 0 1 4 0 1 11 26 0 1 26 131 236 0 1 57 547 1807 2752 0 1 120 ] WardSet,,RevAccTabl,[1 1 0 4 1 0 26 11 1 0 236 131 26 1 0 2752 1807 547 57 1 0 39208 ] WardSet,,AccRevTabl,[1 1 1 3 4 4 15 25 26 26 105 210 235 236 236 945 2205 2695 2751 ] WardSet,A137375,AntiDiagTabl,[1 0 0 1 0 1 0 1 3 0 1 10 0 1 25 15 0 1 56 105 0 1 119 490 105 0 1 ] -WardSet,,DiffxTabl,[1 0 2 0 2 9 0 2 30 60 0 2 75 420 525 0 2 168 1960 6300 5670 0 2 ] WardSet,,BinTabl,[1 0 1 0 1 5 0 1 12 48 0 1 27 183 679 0 1 58 661 3560 12710 0 1 121 ] WardSet,,InvBinTabl,[1 0 1 0 1 1 0 1 8 -12 0 1 23 33 -169 0 1 54 325 -368 -1010 0 1 117 ] +WardSet,,DiffxTabl,[1 0 2 0 2 9 0 2 30 60 0 2 75 420 525 0 2 168 1960 6300 5670 0 2 ] WardSet,A000311,RowSum,[1 1 4 26 236 2752 39208 660032 12818912 282137824 6939897856 ] WardSet,,EvenSum,[1 0 3 10 130 1316 19964 327496 6429616 140887472 3471763328 ] WardSet,,OddSum,[0 1 1 16 106 1436 19244 332536 6389296 141250352 3468134528 ] @@ -28,9 +28,9 @@ WardSet,A000007,ColLeft,[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] WardSet,A001147,ColRight,[1 1 3 15 105 945 10395 135135 2027025 34459425 654729075 ] WardSet,,BinConv,[1 1 5 48 679 12710 296246 8267448 268799091 9978566310 ] WardSet,,InvBinConv,[1 1 1 -12 -169 -1010 11614 484736 7251635 -31586634 -5267648770 ] -WardSet,,TransSqrs,[0 1 13 176 2726 48420 976284 22089800 555025760 15345208528 ] WardSet,,TransNat0,[0 1 7 66 786 11348 192788 3769328 83383808 2059259808 56163653184 ] WardSet,,TransNat1,[1 2 11 92 1022 14100 231996 4429360 96202720 2341397632 ] +WardSet,,TransSqrs,[0 1 13 176 2726 48420 976284 22089800 555025760 15345208528 ] WardSet,A000457,DiagRow1,[0 1 10 105 1260 17325 270270 4729725 91891800 1964187225 ] WardSet,A000497,DiagRow2,[0 1 25 490 9450 190575 4099095 94594500 2343240900 62199262125 ] WardSet,A000504,DiagRow3,[0 1 56 1918 56980 1636635 47507460 1422280860 44346982680 ] diff --git a/data/md/Abel.md b/data/md/Abel.md index ffe5968..dff9663 100644 --- a/data/md/Abel.md +++ b/data/md/Abel.md @@ -1,5 +1,5 @@ # Abel -['A061356', 'A137452', 'A139526'] +['A137452', 'A061356', 'A139526'] Abel Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Abel Triangle view | Row7 | [0, 117649, 100842, 36015, 6860, 735, 42, 1] | | Row8 | [0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1] | -Abel Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 2, 1, 0, 9, 6, 1, 0, 64, 48, 12, 1, 0, 625, 500, 150, 20, 1, 0, 7776, 6480, 2160, 360, 30, 1, 0, 117649, 100842, 36015, 6860, 735, 42, 1, 0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1] | -| TriRev | [1, 1, 0, 1, 2, 0, 1, 6, 9, 0, 1, 12, 48, 64, 0, 1, 20, 150, 500, 625, 0, 1, 30, 360, 2160, 6480, 7776, 0, 1, 42, 735, 6860, 36015, 100842, 117649, 0, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 0] | -| TriAcc | [1, 0, 1, 0, 2, 3, 0, 9, 15, 16, 0, 64, 112, 124, 125, 0, 625, 1125, 1275, 1295, 1296, 0, 7776, 14256, 16416, 16776, 16806, 16807, 0, 117649, 218491, 254506, 261366, 262101, 262143, 262144, 0, 2097152, 3932160, 4620288, 4763648, 4781568, 4782912, 4782968, 4782969] | -| TriRevAcc | [1, 1, 0, 3, 2, 0, 16, 15, 9, 0, 125, 124, 112, 64, 0, 1296, 1295, 1275, 1125, 625, 0, 16807, 16806, 16776, 16416, 14256, 7776, 0, 262144, 262143, 262101, 261366, 254506, 218491, 117649, 0, 4782969, 4782968, 4782912, 4781568, 4763648, 4620288, 3932160, 2097152, 0] | -| TriAccRev | [1, 1, 1, 1, 3, 3, 1, 7, 16, 16, 1, 13, 61, 125, 125, 1, 21, 171, 671, 1296, 1296, 1, 31, 391, 2551, 9031, 16807, 16807, 1, 43, 778, 7638, 43653, 144495, 262144, 262144, 1, 57, 1401, 19321, 162681, 850809, 2685817, 4782969, 4782969] | -| TriDiag | [1, 0, 0, 1, 0, 2, 0, 9, 1, 0, 64, 6, 0, 625, 48, 1, 0, 7776, 500, 12] | +Abel Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 1], [0, 9, 6, 1], [0, 64, 48, 12, 1], [0, 625, 500, 150, 20, 1], [0, 7776, 6480, 2160, 360, 30, 1], [0, 117649, 100842, 36015, 6860, 735, 42, 1], [0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1]] | +| RevTabl | [[1], [1, 0], [1, 2, 0], [1, 6, 9, 0], [1, 12, 48, 64, 0], [1, 20, 150, 500, 625, 0], [1, 30, 360, 2160, 6480, 7776, 0], [1, 42, 735, 6860, 36015, 100842, 117649, 0], [1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 9, 1], [0, 64, 6], [0, 625, 48, 1], [0, 7776, 500, 12], [0, 117649, 6480, 150, 1]] | +| AccTabl | [[1], [0, 1], [0, 2, 3], [0, 9, 15, 16], [0, 64, 112, 124, 125], [0, 625, 1125, 1275, 1295, 1296], [0, 7776, 14256, 16416, 16776, 16806, 16807], [0, 117649, 218491, 254506, 261366, 262101, 262143, 262144], [0, 2097152, 3932160, 4620288, 4763648, 4781568, 4782912, 4782968, 4782969]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 0], [16, 15, 9, 0], [125, 124, 112, 64, 0], [1296, 1295, 1275, 1125, 625, 0], [16807, 16806, 16776, 16416, 14256, 7776, 0], [262144, 262143, 262101, 261366, 254506, 218491, 117649, 0], [4782969, 4782968, 4782912, 4781568, 4763648, 4620288, 3932160, 2097152, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 3, 3], [1, 7, 16, 16], [1, 13, 61, 125, 125], [1, 21, 171, 671, 1296, 1296], [1, 31, 391, 2551, 9031, 16807, 16807], [1, 43, 778, 7638, 43653, 144495, 262144, 262144], [1, 57, 1401, 19321, 162681, 850809, 2685817, 4782969, 4782969]] | Abel Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969] | -| EvenSum | [1, 0, 1, 6, 49, 520, 6841, 107744, 1979713] | -| OddSum | [0, 1, 2, 10, 76, 776, 9966, 154400, 2803256] | -| AltSum | [1, -1, -1, -4, -27, -256, -3125, -46656, -823543] | -| AccSum | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | -| AccRevSum | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | -| RevAccSum | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | -| DiagSum | [1, 0, 1, 2, 10, 70, 674, 8288] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969] | +| EvenSum | [1, 0, 1, 6, 49, 520, 6841, 107744, 1979713] | +| OddSum | [0, 1, 2, 10, 76, 776, 9966, 154400, 2803256] | +| AltSum | [1, -1, -1, -4, -27, -256, -3125, -46656, -823543] | +| AccSum | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | +| AccRevSum | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | +| AntiDiagSum | [1, 0, 1, 2, 10, 70, 674, 8288, 124280] | Abel Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 18, 192, 7500, 38880, 7058940, 220200960] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 9, 64, 625, 7776, 117649, 2097152] | -| Middle | [1, 0, 2, 9, 48, 500, 2160, 36015, 143360] | -| Central | [1, 2, 48, 2160, 143360] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 49, 729, 14641, 371293, 11390625, 410338673] | -| NegHalf | [1, 1, -3, 25, -343, 6561, -161051, 4826809, -170859375] | -| Bin | [1, 1, 5, 46, 593, 9726, 192637, 4457580, 117769409] | -| InvBin | [1, 1, -3, 10, -15, -474, 12565, -258572, 5136705] | -| TransSqrs | [0, 1, 6, 42, 380, 4320, 59682, 974848, 18423288] | -| TransNat0 | [0, 1, 4, 24, 200, 2160, 28812, 458752, 8503056] | -| TransNat1 | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 18, 192, 7500, 38880, 7058940, 220200960] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 9, 64, 625, 7776, 117649, 2097152] | +| ColMiddle | [1, 0, 2, 9, 48, 500, 2160, 36015, 143360] | +| ColECenter | [1, 2, 48, 2160, 143360] | +| ColOCenter | [0, 9, 500, 36015] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 42, 380, 4320, 59682, 974848, 18423288] | +| TransNat0 | [0, 1, 4, 24, 200, 2160, 28812, 458752, 8503056] | +| TransNat1 | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | Abel Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Abel Polynomial values as columns | PolyCol7 | [1, 7, 63, 700, 9317, 145152, 2599051, 52706752, 1196015625] | | PolyCol8 | [1, 8, 80, 968, 13824, 228488, 4302592, 91125000, 2147483648] | -# AbelInv -True +# Abel:Inv +[] -AbelInv Triangle view +Abel:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ AbelInv Triangle view | Row7 | [0, 7, -672, 2835, -2240, 525, -42, 1] | | Row8 | [0, -8, 1792, -13608, 17920, -7000, 1008, -56, 1] | -AbelInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -2, 1, 0, 3, -6, 1, 0, -4, 24, -12, 1, 0, 5, -80, 90, -20, 1, 0, -6, 240, -540, 240, -30, 1, 0, 7, -672, 2835, -2240, 525, -42, 1, 0, -8, 1792, -13608, 17920, -7000, 1008, -56, 1] | -| TriRev | [1, 1, 0, 1, -2, 0, 1, -6, 3, 0, 1, -12, 24, -4, 0, 1, -20, 90, -80, 5, 0, 1, -30, 240, -540, 240, -6, 0, 1, -42, 525, -2240, 2835, -672, 7, 0, 1, -56, 1008, -7000, 17920, -13608, 1792, -8, 0] | -| TriAcc | [1, 0, 1, 0, -2, -1, 0, 3, -3, -2, 0, -4, 20, 8, 9, 0, 5, -75, 15, -5, -4, 0, -6, 234, -306, -66, -96, -95, 0, 7, -665, 2170, -70, 455, 413, 414, 0, -8, 1784, -11824, 6096, -904, 104, 48, 49] | -| TriRevAcc | [1, 1, 0, -1, -2, 0, -2, -3, 3, 0, 9, 8, 20, -4, 0, -4, -5, 15, -75, 5, 0, -95, -96, -66, -306, 234, -6, 0, 414, 413, 455, -70, 2170, -665, 7, 0, 49, 48, 104, -904, 6096, -11824, 1784, -8, 0] | -| TriAccRev | [1, 1, 1, 1, -1, -1, 1, -5, -2, -2, 1, -11, 13, 9, 9, 1, -19, 71, -9, -4, -4, 1, -29, 211, -329, -89, -95, -95, 1, -41, 484, -1756, 1079, 407, 414, 414, 1, -55, 953, -6047, 11873, -1735, 57, 49, 49] | -| TriDiag | [1, 0, 0, 1, 0, -2, 0, 3, 1, 0, -4, -6, 0, 5, 24, 1, 0, -6, -80, -12] | - -AbelInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -2, 9, -4, -95, 414, 49] | -| EvenSum | [1, 0, 1, -6, 25, -100, 481, -2954, 20721] | -| OddSum | [0, 1, -2, 4, -16, 96, -576, 3368, -20672] | -| AltSum | [1, -1, 3, -10, 41, -196, 1057, -6322, 41393] | -| AccSum | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | -| AccRevSum | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | -| RevAccSum | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | -| DiagSum | [1, 0, 1, -2, 4, -10, 30, -98] | - -AbelInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 24, 720, 2160, 907200, 108864000] | -| RowGcd | [1, 1, -2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 90, 540, 2835, 17920] | -| Middle | [1, 0, -2, 3, 24, -80, -540, 2835, 17920] | -| Central | [1, -2, 24, -540, 17920] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, 1, 41, -239, 229, 8401, -87151] | -| NegHalf | [1, 1, 5, 25, 153, 1121, 9373, 87417, 898033] | -| Bin | [1, 1, -3, -8, 81, 26, -3815, 17494, 178241] | -| InvBin | [1, 1, 5, 28, 209, 1826, 18217, 203106, 2487361] | -| TransSqrs | [0, 1, 2, -12, 0, 200, -780, -1344, 30016] | -| TransNat0 | [0, 1, 0, -6, 12, 40, -330, 588, 5096] | -| TransNat1 | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | - -AbelInv Diagonals as rows +Abel:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -2, 1], [0, 3, -6, 1], [0, -4, 24, -12, 1], [0, 5, -80, 90, -20, 1], [0, -6, 240, -540, 240, -30, 1], [0, 7, -672, 2835, -2240, 525, -42, 1], [0, -8, 1792, -13608, 17920, -7000, 1008, -56, 1]] | +| RevTabl | [[1], [1, 0], [1, -2, 0], [1, -6, 3, 0], [1, -12, 24, -4, 0], [1, -20, 90, -80, 5, 0], [1, -30, 240, -540, 240, -6, 0], [1, -42, 525, -2240, 2835, -672, 7, 0], [1, -56, 1008, -7000, 17920, -13608, 1792, -8, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -2], [0, 3, 1], [0, -4, -6], [0, 5, 24, 1], [0, -6, -80, -12], [0, 7, 240, 90, 1]] | +| AccTabl | [[1], [0, 1], [0, -2, -1], [0, 3, -3, -2], [0, -4, 20, 8, 9], [0, 5, -75, 15, -5, -4], [0, -6, 234, -306, -66, -96, -95], [0, 7, -665, 2170, -70, 455, 413, 414], [0, -8, 1784, -11824, 6096, -904, 104, 48, 49]] | +| RevAccTabl | [[1], [1, 0], [-1, -2, 0], [-2, -3, 3, 0], [9, 8, 20, -4, 0], [-4, -5, 15, -75, 5, 0], [-95, -96, -66, -306, 234, -6, 0], [414, 413, 455, -70, 2170, -665, 7, 0], [49, 48, 104, -904, 6096, -11824, 1784, -8, 0]] | +| AccRevTabl | [[1], [1, 1], [1, -1, -1], [1, -5, -2, -2], [1, -11, 13, 9, 9], [1, -19, 71, -9, -4, -4], [1, -29, 211, -329, -89, -95, -95], [1, -41, 484, -1756, 1079, 407, 414, 414], [1, -55, 953, -6047, 11873, -1735, 57, 49, 49]] | + +Abel:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -2, 9, -4, -95, 414, 49] | +| EvenSum | [1, 0, 1, -6, 25, -100, 481, -2954, 20721] | +| OddSum | [0, 1, -2, 4, -16, 96, -576, 3368, -20672] | +| AltSum | [1, -1, 3, -10, 41, -196, 1057, -6322, 41393] | +| AccSum | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | +| AccRevSum | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | +| AntiDiagSum | [1, 0, 1, -2, 4, -10, 30, -98, 338] | + +Abel:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 720, 2160, 907200, 108864000] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 90, 540, 2835, 17920] | +| ColMiddle | [1, 0, -2, 3, 24, -80, -540, 2835, 17920] | +| ColECenter | [1, -2, 24, -540, 17920] | +| ColOCenter | [0, 3, -80, 2835] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -12, 0, 200, -780, -1344, 30016] | +| TransNat0 | [0, 1, 0, -6, 12, 40, -330, 588, 5096] | +| TransNat1 | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | + +Abel:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ AbelInv Diagonals as rows | DiagRow7 | [0, -8, -4608, -262440, -5406720, -61875000, -480370176, -2826399576, -13495173120]| | DiagRow8 | [0, 9, 11520, 1082565, 32440320, 502734375, 5043886848, 37096494435, 215922769920]| -AbelInv Diagonals as columns +Abel:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ AbelInv Diagonals as columns | DiagCol7 | [1, -56, 1764, -41160, 792330, -13311144, 201885684, -2826399576, 37096494435] | | DiagCol8 | [1, -72, 2880, -84480, 2027520, -42172416, 787218432, -13495173120, 215922769920] | -AbelInv Polynomial values as rows +Abel:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ AbelInv Polynomial values as rows | PolyRow7 | [0, 414, -1594, -11778, -20772, 110, 87954, 267246, 523832] | | PolyRow8 | [0, 49, 18608, 18129, -145952, -565615, -1050096, -966287, 749504] | -AbelInv Polynomial values as columns +Abel:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ AbelInv Polynomial values as columns | PolyCol7 | [1, 7, 35, 70, -567, -4228, 16177, 267246, -966287] | | PolyCol8 | [1, 8, 48, 152, -544, -8152, 976, 523832, 749504] | -# AbelRev -True +# Abel:Rev +[] -AbelRev Triangle view +Abel:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ AbelRev Triangle view | Row7 | [1, 42, 735, 6860, 36015, 100842, 117649, 0] | | Row8 | [1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 0] | -AbelRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 2, 0, 1, 6, 9, 0, 1, 12, 48, 64, 0, 1, 20, 150, 500, 625, 0, 1, 30, 360, 2160, 6480, 7776, 0, 1, 42, 735, 6860, 36015, 100842, 117649, 0, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 0] | -| TriRev | [1, 0, 1, 0, 2, 1, 0, 9, 6, 1, 0, 64, 48, 12, 1, 0, 625, 500, 150, 20, 1, 0, 7776, 6480, 2160, 360, 30, 1, 0, 117649, 100842, 36015, 6860, 735, 42, 1, 0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1] | -| TriAcc | [1, 1, 1, 1, 3, 3, 1, 7, 16, 16, 1, 13, 61, 125, 125, 1, 21, 171, 671, 1296, 1296, 1, 31, 391, 2551, 9031, 16807, 16807, 1, 43, 778, 7638, 43653, 144495, 262144, 262144, 1, 57, 1401, 19321, 162681, 850809, 2685817, 4782969, 4782969] | -| TriRevAcc | [1, 1, 1, 3, 3, 1, 16, 16, 7, 1, 125, 125, 61, 13, 1, 1296, 1296, 671, 171, 21, 1, 16807, 16807, 9031, 2551, 391, 31, 1, 262144, 262144, 144495, 43653, 7638, 778, 43, 1, 4782969, 4782969, 2685817, 850809, 162681, 19321, 1401, 57, 1] | -| TriAccRev | [1, 0, 1, 0, 2, 3, 0, 9, 15, 16, 0, 64, 112, 124, 125, 0, 625, 1125, 1275, 1295, 1296, 0, 7776, 14256, 16416, 16776, 16806, 16807, 0, 117649, 218491, 254506, 261366, 262101, 262143, 262144, 0, 2097152, 3932160, 4620288, 4763648, 4781568, 4782912, 4782968, 4782969] | -| TriDiag | [1, 1, 1, 0, 1, 2, 1, 6, 0, 1, 12, 9, 1, 20, 48, 0, 1, 30, 150, 64] | - -AbelRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969] | -| EvenSum | [1, 1, 1, 10, 49, 776, 6841, 154400, 1979713] | -| OddSum | [0, 0, 2, 6, 76, 520, 9966, 107744, 2803256] | -| AltSum | [1, 1, -1, 4, -27, 256, -3125, 46656, -823543] | -| AccSum | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | -| AccRevSum | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | -| RevAccSum | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | -| DiagSum | [1, 1, 1, 3, 7, 22, 69, 245] | - -AbelRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 18, 192, 7500, 38880, 7058940, 220200960] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 9, 64, 625, 7776, 117649, 2097152] | -| Middle | [1, 1, 2, 6, 48, 150, 2160, 6860, 143360] | -| Central | [1, 2, 48, 2160, 143360] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 8, 50, 432, 4802, 65536, 1062882, 20000000] | -| NegHalf | [1, -2, 0, -2, -16, -162, -2048, -31250, -559872] | -| Bin | [1, 1, 5, 46, 593, 9726, 192637, 4457580, 117769409] | -| InvBin | [1, -1, -3, -10, -15, 474, 12565, 258572, 5136705] | -| TransSqrs | [0, 0, 2, 42, 780, 15120, 318990, 7397376, 188484408] | -| TransNat0 | [0, 0, 2, 24, 300, 4320, 72030, 1376256, 29760696] | -| TransNat1 | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | - -AbelRev Diagonals as rows +Abel:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 2, 0], [1, 6, 9, 0], [1, 12, 48, 64, 0], [1, 20, 150, 500, 625, 0], [1, 30, 360, 2160, 6480, 7776, 0], [1, 42, 735, 6860, 36015, 100842, 117649, 0], [1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 1], [0, 9, 6, 1], [0, 64, 48, 12, 1], [0, 625, 500, 150, 20, 1], [0, 7776, 6480, 2160, 360, 30, 1], [0, 117649, 100842, 36015, 6860, 735, 42, 1], [0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 2], [1, 6, 0], [1, 12, 9], [1, 20, 48, 0], [1, 30, 150, 64], [1, 42, 360, 500, 0]] | +| AccTabl | [[1], [1, 1], [1, 3, 3], [1, 7, 16, 16], [1, 13, 61, 125, 125], [1, 21, 171, 671, 1296, 1296], [1, 31, 391, 2551, 9031, 16807, 16807], [1, 43, 778, 7638, 43653, 144495, 262144, 262144], [1, 57, 1401, 19321, 162681, 850809, 2685817, 4782969, 4782969]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 1], [16, 16, 7, 1], [125, 125, 61, 13, 1], [1296, 1296, 671, 171, 21, 1], [16807, 16807, 9031, 2551, 391, 31, 1], [262144, 262144, 144495, 43653, 7638, 778, 43, 1], [4782969, 4782969, 2685817, 850809, 162681, 19321, 1401, 57, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 3], [0, 9, 15, 16], [0, 64, 112, 124, 125], [0, 625, 1125, 1275, 1295, 1296], [0, 7776, 14256, 16416, 16776, 16806, 16807], [0, 117649, 218491, 254506, 261366, 262101, 262143, 262144], [0, 2097152, 3932160, 4620288, 4763648, 4781568, 4782912, 4782968, 4782969]] | + +Abel:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969] | +| EvenSum | [1, 1, 1, 10, 49, 776, 6841, 154400, 1979713] | +| OddSum | [0, 0, 2, 6, 76, 520, 9966, 107744, 2803256] | +| AltSum | [1, 1, -1, 4, -27, 256, -3125, 46656, -823543] | +| AccSum | [1, 2, 7, 40, 325, 3456, 45619, 720896, 13286025] | +| AccRevSum | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | +| AntiDiagSum | [1, 1, 1, 3, 7, 22, 69, 245, 903] | + +Abel:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 18, 192, 7500, 38880, 7058940, 220200960] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 9, 64, 625, 7776, 117649, 2097152] | +| ColMiddle | [1, 1, 2, 6, 48, 150, 2160, 6860, 143360] | +| ColECenter | [1, 2, 48, 2160, 143360] | +| ColOCenter | [1, 6, 150, 6860] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 42, 780, 15120, 318990, 7397376, 188484408] | +| TransNat0 | [0, 0, 2, 24, 300, 4320, 72030, 1376256, 29760696] | +| TransNat1 | [1, 1, 5, 40, 425, 5616, 88837, 1638400, 34543665] | + +Abel:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ AbelRev Diagonals as rows | DiagRow7 | [1, 56, 2268, 84000, 3074610, 114960384, 4459971516, 180889572864, 7696360546875]| | DiagRow8 | [1, 72, 3600, 159720, 6842880, 294064056, 12920683776, 586389375000, 27638114549760]| -AbelRev Diagonals as columns +Abel:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ AbelRev Diagonals as columns | DiagCol7 | [0, 2097152, 38263752, 360000000, 2338460520, 11824496640, 49696825464, 180889572864, 586389375000] | | DiagCol8 | [0, 43046721, 900000000, 9646149645, 70946979840, 403786706895, 1899340515072, 7696360546875, 27638114549760] | -AbelRev Polynomial values as rows +Abel:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ AbelRev Polynomial values as rows | PolyRow7 | [1, 262144, 11390625, 113379904, 594823321, 2176782336, 6321363049, 15625000000, 34296447249] | | PolyRow8 | [1, 4782969, 410338673, 6103515625, 42618442977, 194754273881, 678223072849, 1954897493193, 4902227890625] | -AbelRev Polynomial values as columns +Abel:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ AbelRev Polynomial values as columns | PolyCol7 | [1, 1, 15, 484, 24389, 1679616, 147008443, 15625000000, 1954897493193] | | PolyCol8 | [1, 1, 17, 625, 35937, 2825761, 282475249, 34296447249, 4902227890625] | -# AbelInvRev -True +# Abel:Inv:Rev +[] -AbelInvRev Triangle view +Abel:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ AbelInvRev Triangle view | Row7 | [1, -42, 525, -2240, 2835, -672, 7, 0] | | Row8 | [1, -56, 1008, -7000, 17920, -13608, 1792, -8, 0] | -AbelInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -2, 0, 1, -6, 3, 0, 1, -12, 24, -4, 0, 1, -20, 90, -80, 5, 0, 1, -30, 240, -540, 240, -6, 0, 1, -42, 525, -2240, 2835, -672, 7, 0, 1, -56, 1008, -7000, 17920, -13608, 1792, -8, 0] | -| TriRev | [1, 0, 1, 0, -2, 1, 0, 3, -6, 1, 0, -4, 24, -12, 1, 0, 5, -80, 90, -20, 1, 0, -6, 240, -540, 240, -30, 1, 0, 7, -672, 2835, -2240, 525, -42, 1, 0, -8, 1792, -13608, 17920, -7000, 1008, -56, 1] | -| TriAcc | [1, 1, 1, 1, -1, -1, 1, -5, -2, -2, 1, -11, 13, 9, 9, 1, -19, 71, -9, -4, -4, 1, -29, 211, -329, -89, -95, -95, 1, -41, 484, -1756, 1079, 407, 414, 414, 1, -55, 953, -6047, 11873, -1735, 57, 49, 49] | -| TriRevAcc | [1, 1, 1, -1, -1, 1, -2, -2, -5, 1, 9, 9, 13, -11, 1, -4, -4, -9, 71, -19, 1, -95, -95, -89, -329, 211, -29, 1, 414, 414, 407, 1079, -1756, 484, -41, 1, 49, 49, 57, -1735, 11873, -6047, 953, -55, 1] | -| TriAccRev | [1, 0, 1, 0, -2, -1, 0, 3, -3, -2, 0, -4, 20, 8, 9, 0, 5, -75, 15, -5, -4, 0, -6, 234, -306, -66, -96, -95, 0, 7, -665, 2170, -70, 455, 413, 414, 0, -8, 1784, -11824, 6096, -904, 104, 48, 49] | -| TriDiag | [1, 1, 1, 0, 1, -2, 1, -6, 0, 1, -12, 3, 1, -20, 24, 0, 1, -30, 90, -4] | - -AbelInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -2, 9, -4, -95, 414, 49] | -| EvenSum | [1, 1, 1, 4, 25, 96, 481, 3368, 20721] | -| OddSum | [0, 0, -2, -6, -16, -100, -576, -2954, -20672] | -| AltSum | [1, 1, 3, 10, 41, 196, 1057, 6322, 41393] | -| AccSum | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | -| AccRevSum | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | -| RevAccSum | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | -| DiagSum | [1, 1, 1, -1, -5, -8, 5, 57] | - -AbelInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 24, 720, 2160, 907200, 108864000] | -| RowGcd | [1, 1, -2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 90, 540, 2835, 17920] | -| Middle | [1, 1, -2, -6, 24, 90, -540, -2240, 17920] | -| Central | [1, -2, 24, -540, 17920] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 0, -10, 8, 122, -428, -1594, 18608] | -| NegHalf | [1, -2, 8, -38, 216, -1402, 10156, -80838, 698704] | -| Bin | [1, 1, -3, -8, 81, 26, -3815, 17494, 178241] | -| InvBin | [1, -1, 5, -28, 209, -1826, 18217, -203106, 2487361] | -| TransSqrs | [0, 0, -2, 6, 48, -300, -240, 10710, -48384] | -| TransNat0 | [0, 0, -2, 0, 24, -60, -240, 2310, -4704] | -| TransNat1 | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | - -AbelInvRev Diagonals as rows +Abel:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -2, 0], [1, -6, 3, 0], [1, -12, 24, -4, 0], [1, -20, 90, -80, 5, 0], [1, -30, 240, -540, 240, -6, 0], [1, -42, 525, -2240, 2835, -672, 7, 0], [1, -56, 1008, -7000, 17920, -13608, 1792, -8, 0]] | +| RevTabl | [[1], [0, 1], [0, -2, 1], [0, 3, -6, 1], [0, -4, 24, -12, 1], [0, 5, -80, 90, -20, 1], [0, -6, 240, -540, 240, -30, 1], [0, 7, -672, 2835, -2240, 525, -42, 1], [0, -8, 1792, -13608, 17920, -7000, 1008, -56, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -2], [1, -6, 0], [1, -12, 3], [1, -20, 24, 0], [1, -30, 90, -4], [1, -42, 240, -80, 0]] | +| AccTabl | [[1], [1, 1], [1, -1, -1], [1, -5, -2, -2], [1, -11, 13, 9, 9], [1, -19, 71, -9, -4, -4], [1, -29, 211, -329, -89, -95, -95], [1, -41, 484, -1756, 1079, 407, 414, 414], [1, -55, 953, -6047, 11873, -1735, 57, 49, 49]] | +| RevAccTabl | [[1], [1, 1], [-1, -1, 1], [-2, -2, -5, 1], [9, 9, 13, -11, 1], [-4, -4, -9, 71, -19, 1], [-95, -95, -89, -329, 211, -29, 1], [414, 414, 407, 1079, -1756, 484, -41, 1], [49, 49, 57, -1735, 11873, -6047, 953, -55, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -2, -1], [0, 3, -3, -2], [0, -4, 20, 8, 9], [0, 5, -75, 15, -5, -4], [0, -6, 234, -306, -66, -96, -95], [0, 7, -665, 2170, -70, 455, 413, 414], [0, -8, 1784, -11824, 6096, -904, 104, 48, 49]] | + +Abel:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -2, 9, -4, -95, 414, 49] | +| EvenSum | [1, 1, 1, 4, 25, 96, 481, 3368, 20721] | +| OddSum | [0, 0, -2, -6, -16, -100, -576, -2954, -20672] | +| AltSum | [1, 1, 3, 10, 41, 196, 1057, 6322, 41393] | +| AccSum | [1, 2, -1, -8, 21, 36, -425, 1002, 5145] | +| AccRevSum | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | +| AntiDiagSum | [1, 1, 1, -1, -5, -8, 5, 57, 119] | + +Abel:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 720, 2160, 907200, 108864000] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 90, 540, 2835, 17920] | +| ColMiddle | [1, 1, -2, -6, 24, 90, -540, -2240, 17920] | +| ColECenter | [1, -2, 24, -540, 17920] | +| ColOCenter | [1, -6, 90, -2240] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -2, 6, 48, -300, -240, 10710, -48384] | +| TransNat0 | [0, 0, -2, 0, 24, -60, -240, 2310, -4704] | +| TransNat1 | [1, 1, -3, -2, 33, -64, -335, 2724, -4655] | + +Abel:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ AbelInvRev Diagonals as rows | DiagRow7 | [1, -56, 1764, -41160, 792330, -13311144, 201885684, -2826399576, 37096494435]| | DiagRow8 | [1, -72, 2880, -84480, 2027520, -42172416, 787218432, -13495173120, 215922769920]| -AbelInvRev Diagonals as columns +Abel:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ AbelInvRev Diagonals as columns | DiagCol7 | [0, -8, -4608, -262440, -5406720, -61875000, -480370176, -2826399576, -13495173120] | | DiagCol8 | [0, 9, 11520, 1082565, 32440320, 502734375, 5043886848, 37096494435, 215922769920] | -AbelInvRev Polynomial values as rows +Abel:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ AbelInvRev Polynomial values as rows | PolyRow7 | [1, 414, 8401, 15562, -68823, -485834, -1689911, -4406814, -9686543] | | PolyRow8 | [1, 49, -87151, -746447, -2570207, -4800079, -2699471, 16202929, 76958273] | -AbelInvRev Polynomial values as columns +Abel:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Baxter.md b/data/md/Baxter.md index 1055b8b..e32299f 100644 --- a/data/md/Baxter.md +++ b/data/md/Baxter.md @@ -14,45 +14,41 @@ Baxter Triangle view | Row7 | [0, 1, 56, 490, 980, 490, 56, 1] | | Row8 | [0, 1, 84, 1176, 4116, 4116, 1176, 84, 1] | -Baxter Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 21, 22, 0, 1, 21, 71, 91, 92, 0, 1, 36, 211, 386, 421, 422, 0, 1, 57, 547, 1527, 2017, 2073, 2074, 0, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 6, 5, 1, 0, 22, 21, 11, 1, 0, 92, 91, 71, 21, 1, 0, 422, 421, 386, 211, 36, 1, 0, 2074, 2073, 2017, 1527, 547, 57, 1, 0, 10754, 10753, 10669, 9493, 5377, 1261, 85, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 5, 6, 6, 1, 11, 21, 22, 22, 1, 21, 71, 91, 92, 92, 1, 36, 211, 386, 421, 422, 422, 1, 57, 547, 1527, 2017, 2073, 2074, 2074, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754, 10754] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 4, 0, 1, 10, 1, 0, 1, 20, 10] | +Baxter Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 4, 1], [0, 1, 10, 10, 1], [0, 1, 20, 50, 20, 1], [0, 1, 35, 175, 175, 35, 1], [0, 1, 56, 490, 980, 490, 56, 1], [0, 1, 84, 1176, 4116, 4116, 1176, 84, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 4, 1, 0], [1, 10, 10, 1, 0], [1, 20, 50, 20, 1, 0], [1, 35, 175, 175, 35, 1, 0], [1, 56, 490, 980, 490, 56, 1, 0], [1, 84, 1176, 4116, 4116, 1176, 84, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 4], [0, 1, 10, 1], [0, 1, 20, 10], [0, 1, 35, 50, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 5, 6], [0, 1, 11, 21, 22], [0, 1, 21, 71, 91, 92], [0, 1, 36, 211, 386, 421, 422], [0, 1, 57, 547, 1527, 2017, 2073, 2074], [0, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [6, 5, 1, 0], [22, 21, 11, 1, 0], [92, 91, 71, 21, 1, 0], [422, 421, 386, 211, 36, 1, 0], [2074, 2073, 2017, 1527, 547, 57, 1, 0], [10754, 10753, 10669, 9493, 5377, 1261, 85, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 5, 6, 6], [1, 11, 21, 22, 22], [1, 21, 71, 91, 92, 92], [1, 36, 211, 386, 421, 422, 422], [1, 57, 547, 1527, 2017, 2073, 2074, 2074], [1, 85, 1261, 5377, 9493, 10669, 10753, 10754, 10754]] | Baxter Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 22, 92, 422, 2074, 10754] | -| EvenSum | [1, 0, 1, 4, 11, 40, 211, 1092, 5377] | -| OddSum | [0, 1, 1, 2, 11, 52, 211, 982, 5377] | -| AltSum | [1, -1, 0, 2, 0, -12, 0, 110, 0] | -| AccSum | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | -| AccRevSum | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | -| RevAccSum | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | -| DiagSum | [1, 0, 1, 1, 2, 5, 12, 31] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 22, 92, 422, 2074, 10754] | +| EvenSum | [1, 0, 1, 4, 11, 40, 211, 1092, 5377] | +| OddSum | [0, 1, 1, 2, 11, 52, 211, 982, 5377] | +| AltSum | [1, -1, 0, 2, 0, -12, 0, 110, 0] | +| AccSum | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | +| AccRevSum | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | +| AntiDiagSum | [1, 0, 1, 1, 2, 5, 12, 31, 87] | Baxter Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 4, 10, 100, 175, 1960, 8232] | -| RowGcd | [1, 1, 1, 4, 10, 10, 35, 14, 84] | -| RowMax | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | -| Middle | [1, 0, 1, 1, 10, 20, 175, 490, 4116] | -| Central | [1, 1, 10, 175, 4116] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 13, 69, 417, 2763, 19609, 146793] | -| NegHalf | [1, 1, -1, -3, 13, 17, -241, 121, 5081] | -| Bin | [1, 1, 3, 16, 105, 806, 6867, 63316, 620433] | -| InvBin | [1, 1, -1, -8, 17, 206, -565, -8420, 26369] | -| TransSqrs | [0, 1, 5, 26, 147, 876, 5427, 34630, 226193] | -| TransNat0 | [0, 1, 3, 12, 55, 276, 1477, 8296, 48393] | -| TransNat1 | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 10, 100, 175, 1960, 8232] | +| RowGcd | [1, 1, 1, 4, 10, 10, 35, 14, 84] | +| RowMax | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | +| ColMiddle | [1, 0, 1, 1, 10, 20, 175, 490, 4116] | +| ColECenter | [1, 1, 10, 175, 4116] | +| ColOCenter | [0, 1, 20, 490] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 26, 147, 876, 5427, 34630, 226193] | +| TransNat0 | [0, 1, 3, 12, 55, 276, 1477, 8296, 48393] | +| TransNat1 | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | Baxter Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Baxter Polynomial values as columns | PolyCol7 | [1, 7, 56, 546, 6328, 82964, 1187816, 18171118, 292765256] | | PolyCol8 | [1, 8, 72, 776, 9864, 141576, 2217672, 37102088, 653559048] | -# BaxterRev -True +# Baxter:Rev +[] -BaxterRev Triangle view +Baxter:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ BaxterRev Triangle view | Row7 | [1, 56, 490, 980, 490, 56, 1, 0] | | Row8 | [1, 84, 1176, 4116, 4116, 1176, 84, 1, 0] | -BaxterRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 5, 6, 6, 1, 11, 21, 22, 22, 1, 21, 71, 91, 92, 92, 1, 36, 211, 386, 421, 422, 422, 1, 57, 547, 1527, 2017, 2073, 2074, 2074, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754, 10754] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 6, 6, 5, 1, 22, 22, 21, 11, 1, 92, 92, 91, 71, 21, 1, 422, 422, 421, 386, 211, 36, 1, 2074, 2074, 2073, 2017, 1527, 547, 57, 1, 10754, 10754, 10753, 10669, 9493, 5377, 1261, 85, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 21, 22, 0, 1, 21, 71, 91, 92, 0, 1, 36, 211, 386, 421, 422, 0, 1, 57, 547, 1527, 2017, 2073, 2074, 0, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 10, 1, 1, 20, 10, 0, 1, 35, 50, 1] | - -BaxterRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 22, 92, 422, 2074, 10754] | -| EvenSum | [1, 1, 1, 2, 11, 52, 211, 982, 5377] | -| OddSum | [0, 0, 1, 4, 11, 40, 211, 1092, 5377] | -| AltSum | [1, 1, 0, -2, 0, 12, 0, -110, 0] | -| AccSum | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | -| AccRevSum | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | -| RevAccSum | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | -| DiagSum | [1, 1, 1, 2, 5, 12, 31, 87] | - -BaxterRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 4, 10, 100, 175, 1960, 8232] | -| RowGcd | [1, 1, 1, 4, 10, 10, 35, 14, 84] | -| RowMax | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | -| Middle | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | -| Central | [1, 1, 10, 175, 4116] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 26, 138, 834, 5526, 39218, 293586] | -| NegHalf | [1, -2, 2, 6, -26, -34, 482, -242, -10162] | -| Bin | [1, 1, 3, 16, 105, 806, 6867, 63316, 620433] | -| InvBin | [1, -1, -1, 8, 17, -206, -565, 8420, 26369] | -| TransSqrs | [0, 0, 1, 8, 59, 416, 2895, 20112, 140161] | -| TransNat0 | [0, 0, 1, 6, 33, 184, 1055, 6222, 37639] | -| TransNat1 | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | - -BaxterRev Diagonals as rows +Baxter:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 4, 1, 0], [1, 10, 10, 1, 0], [1, 20, 50, 20, 1, 0], [1, 35, 175, 175, 35, 1, 0], [1, 56, 490, 980, 490, 56, 1, 0], [1, 84, 1176, 4116, 4116, 1176, 84, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 4, 1], [0, 1, 10, 10, 1], [0, 1, 20, 50, 20, 1], [0, 1, 35, 175, 175, 35, 1], [0, 1, 56, 490, 980, 490, 56, 1], [0, 1, 84, 1176, 4116, 4116, 1176, 84, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 4, 0], [1, 10, 1], [1, 20, 10, 0], [1, 35, 50, 1], [1, 56, 175, 20, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 5, 6, 6], [1, 11, 21, 22, 22], [1, 21, 71, 91, 92, 92], [1, 36, 211, 386, 421, 422, 422], [1, 57, 547, 1527, 2017, 2073, 2074, 2074], [1, 85, 1261, 5377, 9493, 10669, 10753, 10754, 10754]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [6, 6, 5, 1], [22, 22, 21, 11, 1], [92, 92, 91, 71, 21, 1], [422, 422, 421, 386, 211, 36, 1], [2074, 2074, 2073, 2017, 1527, 547, 57, 1], [10754, 10754, 10753, 10669, 9493, 5377, 1261, 85, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 5, 6], [0, 1, 11, 21, 22], [0, 1, 21, 71, 91, 92], [0, 1, 36, 211, 386, 421, 422], [0, 1, 57, 547, 1527, 2017, 2073, 2074], [0, 1, 85, 1261, 5377, 9493, 10669, 10753, 10754]] | + +Baxter:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 22, 92, 422, 2074, 10754] | +| EvenSum | [1, 1, 1, 2, 11, 52, 211, 982, 5377] | +| OddSum | [0, 0, 1, 4, 11, 40, 211, 1092, 5377] | +| AltSum | [1, 1, 0, -2, 0, 12, 0, -110, 0] | +| AccSum | [1, 2, 5, 18, 77, 368, 1899, 10370, 59147] | +| AccRevSum | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | +| AntiDiagSum | [1, 1, 1, 2, 5, 12, 31, 87, 252] | + +Baxter:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 10, 100, 175, 1960, 8232] | +| RowGcd | [1, 1, 1, 4, 10, 10, 35, 14, 84] | +| RowMax | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | +| ColMiddle | [1, 1, 1, 4, 10, 50, 175, 980, 4116] | +| ColECenter | [1, 1, 10, 175, 4116] | +| ColOCenter | [1, 4, 50, 980] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 8, 59, 416, 2895, 20112, 140161] | +| TransNat0 | [0, 0, 1, 6, 33, 184, 1055, 6222, 37639] | +| TransNat1 | [1, 1, 3, 12, 55, 276, 1477, 8296, 48393] | + +Baxter:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ BaxterRev Diagonals as rows | DiagRow7 | [1, 84, 2520, 41580, 457380, 3737448, 24293412, 131589315, 614083470]| | DiagRow8 | [1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100, 4971151900]| -BaxterRev Diagonals as columns +Baxter:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -188,7 +180,7 @@ BaxterRev Diagonals as columns | DiagCol7 | [0, 1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100] | | DiagCol8 | [0, 1, 165, 9075, 259545, 4723719, 61408347, 614083470, 4971151900] | -BaxterRev Polynomial values as rows +Baxter:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ BaxterRev Polynomial values as rows | PolyRow7 | [1, 2074, 19609, 85066, 257665, 631906, 1346809, 2595874, 4637761] | | PolyRow8 | [1, 10754, 146793, 804556, 2900945, 8182446, 19609849, 41823608, 81694881] | -BaxterRev Polynomial values as columns +Baxter:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Bell.md b/data/md/Bell.md index 01ba269..838a7bd 100644 --- a/data/md/Bell.md +++ b/data/md/Bell.md @@ -14,45 +14,41 @@ Bell Triangle view | Row7 | [877, 1080, 1335, 1657, 2066, 2589, 3263, 4140] | | Row8 | [4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147] | -Bell Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 15, 20, 27, 37, 52, 52, 67, 87, 114, 151, 203, 203, 255, 322, 409, 523, 674, 877, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147] | -| TriRev | [1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140] | -| TriAcc | [1, 1, 3, 2, 5, 10, 5, 12, 22, 37, 15, 35, 62, 99, 151, 52, 119, 206, 320, 471, 674, 203, 458, 780, 1189, 1712, 2386, 3263, 877, 1957, 3292, 4949, 7015, 9604, 12867, 17007, 4140, 9157, 15254, 22686, 31775, 42930, 56674, 73681, 94828] | -| TriRevAcc | [1, 3, 1, 10, 5, 2, 37, 22, 12, 5, 151, 99, 62, 35, 15, 674, 471, 320, 206, 119, 52, 3263, 2386, 1712, 1189, 780, 458, 203, 17007, 12867, 9604, 7015, 4949, 3292, 1957, 877, 94828, 73681, 56674, 42930, 31775, 22686, 15254, 9157, 4140] | -| TriAccRev | [1, 2, 3, 5, 8, 10, 15, 25, 32, 37, 52, 89, 116, 136, 151, 203, 354, 468, 555, 622, 674, 877, 1551, 2074, 2483, 2805, 3060, 3263, 4140, 7403, 9992, 12058, 13715, 15050, 16130, 17007, 21147, 38154, 51898, 63053, 72142, 79574, 85671, 90688, 94828] | -| TriDiag | [1, 1, 2, 2, 5, 3, 15, 7, 5, 52, 20, 10, 203, 67, 27, 15, 877, 255, 87, 37] | +Bell Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 2], [2, 3, 5], [5, 7, 10, 15], [15, 20, 27, 37, 52], [52, 67, 87, 114, 151, 203], [203, 255, 322, 409, 523, 674, 877], [877, 1080, 1335, 1657, 2066, 2589, 3263, 4140], [4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]] | +| RevTabl | [[1], [2, 1], [5, 3, 2], [15, 10, 7, 5], [52, 37, 27, 20, 15], [203, 151, 114, 87, 67, 52], [877, 674, 523, 409, 322, 255, 203], [4140, 3263, 2589, 2066, 1657, 1335, 1080, 877], [21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140]] | +| AntiDiag | [[1], [1], [2, 2], [5, 3], [15, 7, 5], [52, 20, 10], [203, 67, 27, 15], [877, 255, 87, 37], [4140, 1080, 322, 114, 52]] | +| AccTabl | [[1], [1, 3], [2, 5, 10], [5, 12, 22, 37], [15, 35, 62, 99, 151], [52, 119, 206, 320, 471, 674], [203, 458, 780, 1189, 1712, 2386, 3263], [877, 1957, 3292, 4949, 7015, 9604, 12867, 17007], [4140, 9157, 15254, 22686, 31775, 42930, 56674, 73681, 94828]] | +| RevAccTabl | [[1], [3, 1], [10, 5, 2], [37, 22, 12, 5], [151, 99, 62, 35, 15], [674, 471, 320, 206, 119, 52], [3263, 2386, 1712, 1189, 780, 458, 203], [17007, 12867, 9604, 7015, 4949, 3292, 1957, 877], [94828, 73681, 56674, 42930, 31775, 22686, 15254, 9157, 4140]] | +| AccRevTabl | [[1], [2, 3], [5, 8, 10], [15, 25, 32, 37], [52, 89, 116, 136, 151], [203, 354, 468, 555, 622, 674], [877, 1551, 2074, 2483, 2805, 3060, 3263], [4140, 7403, 9992, 12058, 13715, 15050, 16130, 17007], [21147, 38154, 51898, 63053, 72142, 79574, 85671, 90688, 94828]] | Bell Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 3, 10, 37, 151, 674, 3263, 17007, 94828] | -| EvenSum | [1, 1, 7, 15, 94, 290, 1925, 7541, 54217] | -| OddSum | [0, 2, 3, 22, 57, 384, 1338, 9466, 40611] | -| AltSum | [1, -1, 4, -7, 37, -94, 587, -1925, 13606] | -| AccSum | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | -| AccRevSum | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | -| RevAccSum | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | -| DiagSum | [1, 1, 4, 8, 27, 82, 312, 1256] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 3, 10, 37, 151, 674, 3263, 17007, 94828] | +| EvenSum | [1, 1, 7, 15, 94, 290, 1925, 7541, 54217] | +| OddSum | [0, 2, 3, 22, 57, 384, 1338, 9466, 40611] | +| AltSum | [1, -1, 4, -7, 37, -94, 587, -1925, 13606] | +| AccSum | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | +| AccRevSum | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | +| AntiDiagSum | [1, 1, 4, 8, 27, 82, 312, 1256, 5708] | Bell Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 30, 210, 259740, 6087317964, 150538831664272170, 9345270808867942826280, 2034479281802115592718204963760] | -| RowGcd | [1, 2, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| Middle | [1, 1, 3, 7, 27, 87, 409, 1657, 9089] | -| Central | [1, 3, 27, 409, 9089] | -| LeftSide | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | -| RightSide | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| PosHalf | [1, 4, 19, 103, 634, 4393, 33893, 288158, 2674849] | -| NegHalf | [1, 0, 7, -17, 166, -931, 8333, -67902, 668341] | -| Bin | [1, 3, 13, 71, 457, 3355, 27509, 248127, 2434129] | -| InvBin | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TransSqrs | [0, 2, 23, 182, 1293, 8932, 62014, 439442, 3202127] | -| TransNat0 | [0, 2, 13, 72, 393, 2202, 12850, 78488, 502327] | -| TransNat1 | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 30, 210, 259740, 6087317964, 150538831664272170, 9345270808867942826280, 2034479281802115592718204963760] | +| RowGcd | [1, 2, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| ColMiddle | [1, 1, 3, 7, 27, 87, 409, 1657, 9089] | +| ColECenter | [1, 3, 27, 409, 9089] | +| ColOCenter | [1, 7, 87, 1657] | +| ColLeft | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | +| ColRight | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| TransSqrs | [0, 2, 23, 182, 1293, 8932, 62014, 439442, 3202127] | +| TransNat0 | [0, 2, 13, 72, 393, 2202, 12850, 78488, 502327] | +| TransNat1 | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | Bell Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Bell Polynomial values as columns | PolyCol7 | [1, 15, 268, 5689, 139021, 3818258, 115919867, 3842472699, 137743402366] | | PolyCol8 | [1, 17, 346, 8381, 233839, 7334924, 254360387, 9631827181, 394463953204] | -# BellRev -True +# Bell:Rev +[] -BellRev Triangle view +Bell:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ BellRev Triangle view | Row7 | [4140, 3263, 2589, 2066, 1657, 1335, 1080, 877] | | Row8 | [21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140] | -BellRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140] | -| TriRev | [1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 15, 20, 27, 37, 52, 52, 67, 87, 114, 151, 203, 203, 255, 322, 409, 523, 674, 877, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147] | -| TriAcc | [1, 2, 3, 5, 8, 10, 15, 25, 32, 37, 52, 89, 116, 136, 151, 203, 354, 468, 555, 622, 674, 877, 1551, 2074, 2483, 2805, 3060, 3263, 4140, 7403, 9992, 12058, 13715, 15050, 16130, 17007, 21147, 38154, 51898, 63053, 72142, 79574, 85671, 90688, 94828] | -| TriRevAcc | [1, 3, 2, 10, 8, 5, 37, 32, 25, 15, 151, 136, 116, 89, 52, 674, 622, 555, 468, 354, 203, 3263, 3060, 2805, 2483, 2074, 1551, 877, 17007, 16130, 15050, 13715, 12058, 9992, 7403, 4140, 94828, 90688, 85671, 79574, 72142, 63053, 51898, 38154, 21147] | -| TriAccRev | [1, 1, 3, 2, 5, 10, 5, 12, 22, 37, 15, 35, 62, 99, 151, 52, 119, 206, 320, 471, 674, 203, 458, 780, 1189, 1712, 2386, 3263, 877, 1957, 3292, 4949, 7015, 9604, 12867, 17007, 4140, 9157, 15254, 22686, 31775, 42930, 56674, 73681, 94828] | -| TriDiag | [1, 2, 5, 1, 15, 3, 52, 10, 2, 203, 37, 7, 877, 151, 27, 5, 4140, 674, 114, 20] | - -BellRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 3, 10, 37, 151, 674, 3263, 17007, 94828] | -| EvenSum | [1, 2, 7, 22, 94, 384, 1925, 9466, 54217] | -| OddSum | [0, 1, 3, 15, 57, 290, 1338, 7541, 40611] | -| AltSum | [1, 1, 4, 7, 37, 94, 587, 1925, 13606] | -| AccSum | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | -| AccRevSum | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | -| RevAccSum | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | -| DiagSum | [1, 2, 6, 18, 64, 247, 1060, 4948] | - -BellRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 30, 210, 259740, 6087317964, 150538831664272170, 9345270808867942826280, 2034479281802115592718204963760] | -| RowGcd | [1, 2, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| Middle | [1, 2, 3, 10, 27, 114, 409, 2066, 9089] | -| Central | [1, 3, 27, 409, 9089] | -| LeftSide | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| RightSide | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | -| PosHalf | [1, 5, 28, 179, 1291, 10358, 91337, 876289, 9070546] | -| NegHalf | [1, -3, 16, -89, 619, -4726, 40637, -380079, 3863854] | -| Bin | [1, 3, 13, 71, 457, 3355, 27509, 248127, 2434129] | -| InvBin | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| TransSqrs | [0, 1, 11, 83, 565, 3762, 25282, 173953, 1233887] | -| TransNat0 | [0, 1, 7, 39, 211, 1168, 6728, 40561, 256297] | -| TransNat1 | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | - -BellRev Diagonals as rows +Bell:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 1], [5, 3, 2], [15, 10, 7, 5], [52, 37, 27, 20, 15], [203, 151, 114, 87, 67, 52], [877, 674, 523, 409, 322, 255, 203], [4140, 3263, 2589, 2066, 1657, 1335, 1080, 877], [21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140]] | +| RevTabl | [[1], [1, 2], [2, 3, 5], [5, 7, 10, 15], [15, 20, 27, 37, 52], [52, 67, 87, 114, 151, 203], [203, 255, 322, 409, 523, 674, 877], [877, 1080, 1335, 1657, 2066, 2589, 3263, 4140], [4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]] | +| AntiDiag | [[1], [2], [5, 1], [15, 3], [52, 10, 2], [203, 37, 7], [877, 151, 27, 5], [4140, 674, 114, 20], [21147, 3263, 523, 87, 15]] | +| AccTabl | [[1], [2, 3], [5, 8, 10], [15, 25, 32, 37], [52, 89, 116, 136, 151], [203, 354, 468, 555, 622, 674], [877, 1551, 2074, 2483, 2805, 3060, 3263], [4140, 7403, 9992, 12058, 13715, 15050, 16130, 17007], [21147, 38154, 51898, 63053, 72142, 79574, 85671, 90688, 94828]] | +| RevAccTabl | [[1], [3, 2], [10, 8, 5], [37, 32, 25, 15], [151, 136, 116, 89, 52], [674, 622, 555, 468, 354, 203], [3263, 3060, 2805, 2483, 2074, 1551, 877], [17007, 16130, 15050, 13715, 12058, 9992, 7403, 4140], [94828, 90688, 85671, 79574, 72142, 63053, 51898, 38154, 21147]] | +| AccRevTabl | [[1], [1, 3], [2, 5, 10], [5, 12, 22, 37], [15, 35, 62, 99, 151], [52, 119, 206, 320, 471, 674], [203, 458, 780, 1189, 1712, 2386, 3263], [877, 1957, 3292, 4949, 7015, 9604, 12867, 17007], [4140, 9157, 15254, 22686, 31775, 42930, 56674, 73681, 94828]] | + +Bell:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 3, 10, 37, 151, 674, 3263, 17007, 94828] | +| EvenSum | [1, 2, 7, 22, 94, 384, 1925, 9466, 54217] | +| OddSum | [0, 1, 3, 15, 57, 290, 1338, 7541, 40611] | +| AltSum | [1, 1, 4, 7, 37, 94, 587, 1925, 13606] | +| AccSum | [1, 5, 23, 109, 544, 2876, 16113, 95495, 597155] | +| AccRevSum | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | +| AntiDiagSum | [1, 2, 6, 18, 64, 247, 1060, 4948, 25035] | + +Bell:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 30, 210, 259740, 6087317964, 150538831664272170, 9345270808867942826280, 2034479281802115592718204963760] | +| RowGcd | [1, 2, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| ColMiddle | [1, 2, 3, 10, 27, 114, 409, 2066, 9089] | +| ColECenter | [1, 3, 27, 409, 9089] | +| ColOCenter | [2, 10, 114, 2066] | +| ColLeft | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| ColRight | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | +| TransSqrs | [0, 1, 11, 83, 565, 3762, 25282, 173953, 1233887] | +| TransNat0 | [0, 1, 7, 39, 211, 1168, 6728, 40561, 256297] | +| TransNat1 | [1, 4, 17, 76, 362, 1842, 9991, 57568, 351125] | + +Bell:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 5, 15, 52, 203, 877, 4140]| @@ -175,7 +167,7 @@ BellRev Diagonals as rows | DiagRow7 | [4140, 17007, 77821, 389946, 2114719, 12303997, 76282138, 501178937, 3473600465]| | DiagRow8 | [21147, 94828, 467767, 2504665, 14418716, 88586135, 577461075, 3974779402, 28773452321]| -BellRev Diagonals as columns +Bell:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | @@ -188,7 +180,7 @@ BellRev Diagonals as columns | DiagCol7 | [877, 5017, 30304, 192713, 1286789, 8999244, 65766991, 501178937, 3974779402] | | DiagCol8 | [4140, 25287, 162409, 1094076, 7712455, 56767747, 435411946, 3473600465, 28773452321] | -BellRev Polynomial values as rows +Bell:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ BellRev Polynomial values as rows | PolyRow7 | [4140, 17007, 288158, 3256953, 20774520, 90941555, 308983962, 876586413, 2174103908] | | PolyRow8 | [21147, 94828, 2674849, 45618666, 389451031, 2135156032, 8715108813, 28866669454, 81864886291] | -BellRev Polynomial values as columns +Bell:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | diff --git a/data/md/Bessel.md b/data/md/Bessel.md index 47030ba..ad371aa 100644 --- a/data/md/Bessel.md +++ b/data/md/Bessel.md @@ -1,5 +1,5 @@ # Bessel -['A001497', 'A001498', 'A122850', 'A132062'] +['A132062', 'A001497', 'A001498', 'A122850'] Bessel Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Bessel Triangle view | Row7 | [0, 10395, 10395, 4725, 1260, 210, 21, 1] | | Row8 | [0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1] | -Bessel Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 3, 6, 7, 0, 15, 30, 36, 37, 0, 105, 210, 255, 265, 266, 0, 945, 1890, 2310, 2415, 2430, 2431, 0, 10395, 20790, 25515, 26775, 26985, 27006, 27007, 0, 135135, 270270, 332640, 349965, 353115, 353493, 353521, 353522] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 7, 6, 3, 0, 37, 36, 30, 15, 0, 266, 265, 255, 210, 105, 0, 2431, 2430, 2415, 2310, 1890, 945, 0, 27007, 27006, 26985, 26775, 25515, 20790, 10395, 0, 353522, 353521, 353493, 353115, 349965, 332640, 270270, 135135, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 4, 7, 7, 1, 7, 22, 37, 37, 1, 11, 56, 161, 266, 266, 1, 16, 121, 541, 1486, 2431, 2431, 1, 22, 232, 1492, 6217, 16612, 27007, 27007, 1, 29, 407, 3557, 20882, 83252, 218387, 353522, 353522] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 3, 1, 0, 15, 3, 0, 105, 15, 1, 0, 945, 105, 6] | +Bessel Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 3, 3, 1], [0, 15, 15, 6, 1], [0, 105, 105, 45, 10, 1], [0, 945, 945, 420, 105, 15, 1], [0, 10395, 10395, 4725, 1260, 210, 21, 1], [0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 3, 3, 0], [1, 6, 15, 15, 0], [1, 10, 45, 105, 105, 0], [1, 15, 105, 420, 945, 945, 0], [1, 21, 210, 1260, 4725, 10395, 10395, 0], [1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 3, 1], [0, 15, 3], [0, 105, 15, 1], [0, 945, 105, 6], [0, 10395, 945, 45, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 3, 6, 7], [0, 15, 30, 36, 37], [0, 105, 210, 255, 265, 266], [0, 945, 1890, 2310, 2415, 2430, 2431], [0, 10395, 20790, 25515, 26775, 26985, 27006, 27007], [0, 135135, 270270, 332640, 349965, 353115, 353493, 353521, 353522]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [7, 6, 3, 0], [37, 36, 30, 15, 0], [266, 265, 255, 210, 105, 0], [2431, 2430, 2415, 2310, 1890, 945, 0], [27007, 27006, 26985, 26775, 25515, 20790, 10395, 0], [353522, 353521, 353493, 353115, 349965, 332640, 270270, 135135, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 7, 7], [1, 7, 22, 37, 37], [1, 11, 56, 161, 266, 266], [1, 16, 121, 541, 1486, 2431, 2431], [1, 22, 232, 1492, 6217, 16612, 27007, 27007], [1, 29, 407, 3557, 20882, 83252, 218387, 353522, 353522]] | Bessel Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 7, 37, 266, 2431, 27007, 353522] | -| EvenSum | [1, 0, 1, 3, 16, 115, 1051, 11676, 152839] | -| OddSum | [0, 1, 1, 4, 21, 151, 1380, 15331, 200683] | -| AltSum | [1, -1, 0, -1, -5, -36, -329, -3655, -47844] | -| AccSum | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | -| AccRevSum | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | -| RevAccSum | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | -| DiagSum | [1, 0, 1, 1, 4, 18, 121, 1056] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 7, 37, 266, 2431, 27007, 353522] | +| EvenSum | [1, 0, 1, 3, 16, 115, 1051, 11676, 152839] | +| OddSum | [0, 1, 1, 4, 21, 151, 1380, 15331, 200683] | +| AltSum | [1, -1, 0, -1, -5, -36, -329, -3655, -47844] | +| AccSum | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | +| AccRevSum | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | +| AntiDiagSum | [1, 0, 1, 1, 4, 18, 121, 1056, 11386] | Bessel Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 30, 630, 3780, 207900, 8108100] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 15, 105, 945, 10395, 135135] | -| Middle | [1, 0, 1, 3, 15, 105, 420, 4725, 17325] | -| Central | [1, 1, 15, 420, 17325] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729] | -| NegHalf | [1, 1, -1, 7, -71, 1001, -18089, 398959, -10391023] | -| Bin | [1, 1, 3, 19, 175, 2076, 29911, 505093, 9757539] | -| InvBin | [1, 1, -1, 1, 7, -124, 1591, -19991, 256691] | -| TransSqrs | [0, 1, 5, 24, 145, 1115, 10596, 120715, 1607999] | -| TransNat0 | [0, 1, 3, 12, 67, 495, 4596, 51583, 680037] | -| TransNat1 | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 30, 630, 3780, 207900, 8108100] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 15, 105, 945, 10395, 135135] | +| ColMiddle | [1, 0, 1, 3, 15, 105, 420, 4725, 17325] | +| ColECenter | [1, 1, 15, 420, 17325] | +| ColOCenter | [0, 3, 105, 4725] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 24, 145, 1115, 10596, 120715, 1607999] | +| TransNat0 | [0, 1, 3, 12, 67, 495, 4596, 51583, 680037] | +| TransNat1 | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | Bessel Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Bessel Polynomial values as columns | PolyCol7 | [1, 7, 56, 511, 5299, 62132, 818839, 12051697, 196795172] | | PolyCol8 | [1, 8, 72, 728, 8248, 104328, 1466824, 22812056, 390433464] | -# BesselInv -True +# Bessel:Inv +[] -BesselInv Triangle view +Bessel:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ BesselInv Triangle view | Row7 | [0, 0, 0, 0, -105, 105, -21, 1] | | Row8 | [0, 0, 0, 0, 105, -420, 210, -28, 1] | -BesselInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 0, -3, 1, 0, 0, 3, -6, 1, 0, 0, 0, 15, -10, 1, 0, 0, 0, -15, 45, -15, 1, 0, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 0, 105, -420, 210, -28, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -3, 0, 0, 1, -6, 3, 0, 0, 1, -10, 15, 0, 0, 0, 1, -15, 45, -15, 0, 0, 0, 1, -21, 105, -105, 0, 0, 0, 0, 1, -28, 210, -420, 105, 0, 0, 0, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 0, -3, -2, 0, 0, 3, -3, -2, 0, 0, 0, 15, 5, 6, 0, 0, 0, -15, 30, 15, 16, 0, 0, 0, 0, -105, 0, -21, -20, 0, 0, 0, 0, 105, -315, -105, -133, -132] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, -2, -3, 0, 0, -2, -3, 3, 0, 0, 6, 5, 15, 0, 0, 0, 16, 15, 30, -15, 0, 0, 0, -20, -21, 0, -105, 0, 0, 0, 0, -132, -133, -105, -315, 105, 0, 0, 0, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -2, -2, -2, 1, -5, -2, -2, -2, 1, -9, 6, 6, 6, 6, 1, -14, 31, 16, 16, 16, 16, 1, -20, 85, -20, -20, -20, -20, -20, 1, -27, 183, -237, -132, -132, -132, -132, -132] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -3, 0, 0, 3, 1, 0, 0, 0, -6] | - -BesselInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| EvenSum | [1, 0, 1, -3, 4, -10, 46, -126, 316] | -| OddSum | [0, 1, -1, 1, -6, 16, -30, 106, -448] | -| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | -| AccSum | [1, 1, -1, -5, -2, 26, 46, -146, -580] | -| AccRevSum | [1, 1, -1, -5, -2, 26, 46, -146, -580] | -| RevAccSum | [1, 2, 1, -5, -10, 16, 82, -34, -740] | -| DiagSum | [1, 0, 1, -1, 1, -3, 4, -6] | - -BesselInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, -3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 0, -1, 0, 3, 0, -15, 0, 105] | -| Central | [1, -1, 3, -15, 105] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, -5, 1, 41, 31, -461, -895] | -| NegHalf | [1, 1, 3, 7, 25, 81, 331, 1303, 5937] | -| Bin | [1, 1, -1, -8, -5, 101, 286, -1616, -10513] | -| InvBin | [1, 1, 3, 10, 43, 201, 1066, 6028, 36975] | -| TransSqrs | [0, 1, 3, -3, -26, 0, 246, 238, -2568] | -| TransNat0 | [0, 1, 1, -3, -8, 10, 66, -14, -608] | -| TransNat1 | [1, 2, 1, -5, -10, 16, 82, -34, -740] | - -BesselInv Diagonals as rows +Bessel:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 0, -3, 1], [0, 0, 3, -6, 1], [0, 0, 0, 15, -10, 1], [0, 0, 0, -15, 45, -15, 1], [0, 0, 0, 0, -105, 105, -21, 1], [0, 0, 0, 0, 105, -420, 210, -28, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -3, 0, 0], [1, -6, 3, 0, 0], [1, -10, 15, 0, 0, 0], [1, -15, 45, -15, 0, 0, 0], [1, -21, 105, -105, 0, 0, 0, 0], [1, -28, 210, -420, 105, 0, 0, 0, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 0, 1], [0, 0, -3], [0, 0, 3, 1], [0, 0, 0, -6], [0, 0, 0, 15, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 0, -3, -2], [0, 0, 3, -3, -2], [0, 0, 0, 15, 5, 6], [0, 0, 0, -15, 30, 15, 16], [0, 0, 0, 0, -105, 0, -21, -20], [0, 0, 0, 0, 105, -315, -105, -133, -132]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [-2, -3, 0, 0], [-2, -3, 3, 0, 0], [6, 5, 15, 0, 0, 0], [16, 15, 30, -15, 0, 0, 0], [-20, -21, 0, -105, 0, 0, 0, 0], [-132, -133, -105, -315, 105, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -2, -2], [1, -5, -2, -2, -2], [1, -9, 6, 6, 6, 6], [1, -14, 31, 16, 16, 16, 16], [1, -20, 85, -20, -20, -20, -20, -20], [1, -27, 183, -237, -132, -132, -132, -132, -132]] | + +Bessel:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| EvenSum | [1, 0, 1, -3, 4, -10, 46, -126, 316] | +| OddSum | [0, 1, -1, 1, -6, 16, -30, 106, -448] | +| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | +| AccSum | [1, 1, -1, -5, -2, 26, 46, -146, -580] | +| AccRevSum | [1, 2, 1, -5, -10, 16, 82, -34, -740] | +| AntiDiagSum | [1, 0, 1, -1, 1, -3, 4, -6, 16] | + +Bessel:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| ColECenter | [1, -1, 3, -15, 105] | +| ColOCenter | [0, 0, 0, 0] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -3, -26, 0, 246, 238, -2568] | +| TransNat0 | [0, 1, 1, -3, -8, 10, 66, -14, -608] | +| TransNat1 | [1, 2, 1, -5, -10, 16, 82, -34, -740] | + +Bessel:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ BesselInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, -135135, -2027025]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 2027025]| -BesselInv Diagonals as columns +Bessel:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ BesselInv Diagonals as columns | DiagCol7 | [1, -28, 378, -3150, 17325, -62370, 135135, -135135, 0] | | DiagCol8 | [1, -36, 630, -6930, 51975, -270270, 945945, -2027025, 2027025] | -BesselInv Polynomial values as rows +Bessel:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ BesselInv Polynomial values as rows | PolyRow7 | [0, -20, 464, 3888, 11008, 12500, -19440, -134456, -397312] | | PolyRow8 | [0, -132, -1648, 4860, 63744, 237500, 509328, 605052, -225280] | -BesselInv Polynomial values as columns +Bessel:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ BesselInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 196, 490, -2058, -31556, -134456, 605052] | | PolyCol8 | [1, 8, 56, 320, 1216, -512, -52736, -397312, -225280] | -# BesselRev -True +# Bessel:Rev +[] -BesselRev Triangle view +Bessel:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ BesselRev Triangle view | Row7 | [1, 21, 210, 1260, 4725, 10395, 10395, 0] | | Row8 | [1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0] | -BesselRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 4, 7, 7, 1, 7, 22, 37, 37, 1, 11, 56, 161, 266, 266, 1, 16, 121, 541, 1486, 2431, 2431, 1, 22, 232, 1492, 6217, 16612, 27007, 27007, 1, 29, 407, 3557, 20882, 83252, 218387, 353522, 353522] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 7, 7, 4, 1, 37, 37, 22, 7, 1, 266, 266, 161, 56, 11, 1, 2431, 2431, 1486, 541, 121, 16, 1, 27007, 27007, 16612, 6217, 1492, 232, 22, 1, 353522, 353522, 218387, 83252, 20882, 3557, 407, 29, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 3, 6, 7, 0, 15, 30, 36, 37, 0, 105, 210, 255, 265, 266, 0, 945, 1890, 2310, 2415, 2430, 2431, 0, 10395, 20790, 25515, 26775, 26985, 27006, 27007, 0, 135135, 270270, 332640, 349965, 353115, 353493, 353521, 353522] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 6, 3, 1, 10, 15, 0, 1, 15, 45, 15] | - -BesselRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 7, 37, 266, 2431, 27007, 353522] | -| EvenSum | [1, 1, 1, 4, 16, 151, 1051, 15331, 152839] | -| OddSum | [0, 0, 1, 3, 21, 115, 1380, 11676, 200683] | -| AltSum | [1, 1, 0, 1, -5, 36, -329, 3655, -47844] | -| AccSum | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | -| AccRevSum | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | -| RevAccSum | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | -| DiagSum | [1, 1, 1, 2, 4, 10, 26, 76] | - -BesselRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 30, 630, 3780, 207900, 8108100] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 15, 105, 945, 10395, 135135] | -| Middle | [1, 1, 1, 3, 15, 45, 420, 1260, 17325] | -| Central | [1, 1, 15, 420, 17325] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 26, 154, 1182, 11254, 128522, 1715802] | -| NegHalf | [1, -2, 2, -2, -2, -22, -206, -2354, -31426] | -| Bin | [1, 1, 3, 19, 175, 2076, 29911, 505093, 9757539] | -| InvBin | [1, -1, -1, -1, 7, 124, 1591, 19991, 256691] | -| TransSqrs | [0, 0, 1, 15, 201, 2815, 42960, 721896, 13352815] | -| TransNat0 | [0, 0, 1, 9, 81, 835, 9990, 137466, 2148139] | -| TransNat1 | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | - -BesselRev Diagonals as rows +Bessel:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 3, 3, 0], [1, 6, 15, 15, 0], [1, 10, 45, 105, 105, 0], [1, 15, 105, 420, 945, 945, 0], [1, 21, 210, 1260, 4725, 10395, 10395, 0], [1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 3, 3, 1], [0, 15, 15, 6, 1], [0, 105, 105, 45, 10, 1], [0, 945, 945, 420, 105, 15, 1], [0, 10395, 10395, 4725, 1260, 210, 21, 1], [0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 3, 0], [1, 6, 3], [1, 10, 15, 0], [1, 15, 45, 15], [1, 21, 105, 105, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 7, 7], [1, 7, 22, 37, 37], [1, 11, 56, 161, 266, 266], [1, 16, 121, 541, 1486, 2431, 2431], [1, 22, 232, 1492, 6217, 16612, 27007, 27007], [1, 29, 407, 3557, 20882, 83252, 218387, 353522, 353522]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [7, 7, 4, 1], [37, 37, 22, 7, 1], [266, 266, 161, 56, 11, 1], [2431, 2431, 1486, 541, 121, 16, 1], [27007, 27007, 16612, 6217, 1492, 232, 22, 1], [353522, 353522, 218387, 83252, 20882, 3557, 407, 29, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 3, 6, 7], [0, 15, 30, 36, 37], [0, 105, 210, 255, 265, 266], [0, 945, 1890, 2310, 2415, 2430, 2431], [0, 10395, 20790, 25515, 26775, 26985, 27006, 27007], [0, 135135, 270270, 332640, 349965, 353115, 353493, 353521, 353522]] | + +Bessel:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 7, 37, 266, 2431, 27007, 353522] | +| EvenSum | [1, 1, 1, 4, 16, 151, 1051, 15331, 152839] | +| OddSum | [0, 0, 1, 3, 21, 115, 1380, 11676, 200683] | +| AltSum | [1, 1, 0, 1, -5, 36, -329, 3655, -47844] | +| AccSum | [1, 2, 5, 19, 104, 761, 7027, 78590, 1033559] | +| AccRevSum | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | +| AntiDiagSum | [1, 1, 1, 2, 4, 10, 26, 76, 232] | + +Bessel:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 30, 630, 3780, 207900, 8108100] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 15, 105, 945, 10395, 135135] | +| ColMiddle | [1, 1, 1, 3, 15, 45, 420, 1260, 17325] | +| ColECenter | [1, 1, 15, 420, 17325] | +| ColOCenter | [1, 3, 45, 1260] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 15, 201, 2815, 42960, 721896, 13352815] | +| TransNat0 | [0, 0, 1, 9, 81, 835, 9990, 137466, 2148139] | +| TransNat1 | [1, 1, 3, 16, 118, 1101, 12421, 164473, 2501661] | + +Bessel:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ BesselRev Diagonals as rows | DiagRow7 | [1, 28, 630, 13860, 315315, 7567560, 192972780, 5237832600, 151242416325]| | DiagRow8 | [1, 36, 990, 25740, 675675, 18378360, 523783260, 15713497800, 496939367925]| -BesselRev Diagonals as columns +Bessel:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ BesselRev Diagonals as columns | DiagCol7 | [0, 135135, 2027025, 16216200, 91891800, 413513100, 1571349780, 5237832600, 15713497800] | | DiagCol8 | [0, 2027025, 34459425, 310134825, 1964187225, 9820936125, 41247931725, 151242416325, 496939367925] | -BesselRev Polynomial values as rows +Bessel:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ BesselRev Polynomial values as rows | PolyRow7 | [1, 27007, 1084483, 10522639, 54516085, 198022231, 572224087, 1409457463, 3085622569] | | PolyRow8 | [1, 353522, 28245729, 410701432, 2836074641, 12875043966, 44642146417, 128278930004, 320939805537] | -BesselRev Polynomial values as columns +Bessel:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ BesselRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 169, 5923, 290396, 18300871, 1409457463, 128278930004] | | PolyCol8 | [1, 1, 9, 217, 8689, 486801, 35058361, 3085622569, 320939805537] | -# BesselInvRev -True +# Bessel:Inv:Rev +[] -BesselInvRev Triangle view +Bessel:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ BesselInvRev Triangle view | Row7 | [1, -21, 105, -105, 0, 0, 0, 0] | | Row8 | [1, -28, 210, -420, 105, 0, 0, 0, 0] | -BesselInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -3, 0, 0, 1, -6, 3, 0, 0, 1, -10, 15, 0, 0, 0, 1, -15, 45, -15, 0, 0, 0, 1, -21, 105, -105, 0, 0, 0, 0, 1, -28, 210, -420, 105, 0, 0, 0, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 0, -3, 1, 0, 0, 3, -6, 1, 0, 0, 0, 15, -10, 1, 0, 0, 0, -15, 45, -15, 1, 0, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 0, 105, -420, 210, -28, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -2, -2, -2, 1, -5, -2, -2, -2, 1, -9, 6, 6, 6, 6, 1, -14, 31, 16, 16, 16, 16, 1, -20, 85, -20, -20, -20, -20, -20, 1, -27, 183, -237, -132, -132, -132, -132, -132] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, -2, -2, -2, 1, -2, -2, -2, -5, 1, 6, 6, 6, 6, -9, 1, 16, 16, 16, 16, 31, -14, 1, -20, -20, -20, -20, -20, 85, -20, 1, -132, -132, -132, -132, -132, -237, 183, -27, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 0, -3, -2, 0, 0, 3, -3, -2, 0, 0, 0, 15, 5, 6, 0, 0, 0, -15, 30, 15, 16, 0, 0, 0, 0, -105, 0, -21, -20, 0, 0, 0, 0, 105, -315, -105, -133, -132] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -3, 0, 1, -6, 0, 1, -10, 3, 0, 1, -15, 15, 0] | - -BesselInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| EvenSum | [1, 1, 1, 1, 4, 16, 46, 106, 316] | -| OddSum | [0, 0, -1, -3, -6, -10, -30, -126, -448] | -| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| AccSum | [1, 2, 1, -5, -10, 16, 82, -34, -740] | -| AccRevSum | [1, 2, 1, -5, -10, 16, 82, -34, -740] | -| RevAccSum | [1, 1, -1, -5, -2, 26, 46, -146, -580] | -| DiagSum | [1, 1, 1, 0, -2, -5, -6, 1] | - -BesselInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, -3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 1, -1, -3, 3, 15, -15, -105, 105] | -| Central | [1, -1, 3, -15, 105] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, -4, -20, -8, 184, 464, -1648] | -| NegHalf | [1, -2, 6, -20, 76, -312, 1384, -6512, 32400] | -| Bin | [1, 1, -1, -8, -5, 101, 286, -1616, -10513] | -| InvBin | [1, -1, 3, -10, 43, -201, 1066, -6028, 36975] | -| TransSqrs | [0, 0, -1, -3, 6, 50, 30, -546, -1288] | -| TransNat0 | [0, 0, -1, -3, 0, 20, 30, -126, -448] | -| TransNat1 | [1, 1, -1, -5, -2, 26, 46, -146, -580] | - -BesselInvRev Diagonals as rows +Bessel:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -3, 0, 0], [1, -6, 3, 0, 0], [1, -10, 15, 0, 0, 0], [1, -15, 45, -15, 0, 0, 0], [1, -21, 105, -105, 0, 0, 0, 0], [1, -28, 210, -420, 105, 0, 0, 0, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 0, -3, 1], [0, 0, 3, -6, 1], [0, 0, 0, 15, -10, 1], [0, 0, 0, -15, 45, -15, 1], [0, 0, 0, 0, -105, 105, -21, 1], [0, 0, 0, 0, 105, -420, 210, -28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -3, 0], [1, -6, 0], [1, -10, 3, 0], [1, -15, 15, 0], [1, -21, 45, 0, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -2, -2], [1, -5, -2, -2, -2], [1, -9, 6, 6, 6, 6], [1, -14, 31, 16, 16, 16, 16], [1, -20, 85, -20, -20, -20, -20, -20], [1, -27, 183, -237, -132, -132, -132, -132, -132]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [-2, -2, -2, 1], [-2, -2, -2, -5, 1], [6, 6, 6, 6, -9, 1], [16, 16, 16, 16, 31, -14, 1], [-20, -20, -20, -20, -20, 85, -20, 1], [-132, -132, -132, -132, -132, -237, 183, -27, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 0, -3, -2], [0, 0, 3, -3, -2], [0, 0, 0, 15, 5, 6], [0, 0, 0, -15, 30, 15, 16], [0, 0, 0, 0, -105, 0, -21, -20], [0, 0, 0, 0, 105, -315, -105, -133, -132]] | + +Bessel:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| EvenSum | [1, 1, 1, 1, 4, 16, 46, 106, 316] | +| OddSum | [0, 0, -1, -3, -6, -10, -30, -126, -448] | +| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| AccSum | [1, 2, 1, -5, -10, 16, 82, -34, -740] | +| AccRevSum | [1, 1, -1, -5, -2, 26, 46, -146, -580] | +| AntiDiagSum | [1, 1, 1, 0, -2, -5, -6, 1, 25] | + +Bessel:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 1, -1, -3, 3, 15, -15, -105, 105] | +| ColECenter | [1, -1, 3, -15, 105] | +| ColOCenter | [1, -3, 15, -105] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, -3, 6, 50, 30, -546, -1288] | +| TransNat0 | [0, 0, -1, -3, 0, 20, 30, -126, -448] | +| TransNat1 | [1, 1, -1, -5, -2, 26, 46, -146, -580] | + +Bessel:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ BesselInvRev Diagonals as rows | DiagRow7 | [1, -28, 378, -3150, 17325, -62370, 135135, -135135, 0]| | DiagRow8 | [1, -36, 630, -6930, 51975, -270270, 945945, -2027025, 2027025]| -BesselInvRev Diagonals as columns +Bessel:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ BesselInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, -135135, -2027025] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 2027025] | -BesselInvRev Polynomial values as rows +Bessel:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ BesselInvRev Polynomial values as rows | PolyRow7 | [1, -20, -461, -1952, -5123, -10604, -19025, -31016, -47207] | | PolyRow8 | [1, -132, -895, -1028, 3249, 18236, 52753, 118140, 228257] | -BesselInvRev Polynomial values as columns +Bessel:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Bessel2.md b/data/md/Bessel2.md index 4bbce80..d11cbcd 100644 --- a/data/md/Bessel2.md +++ b/data/md/Bessel2.md @@ -14,45 +14,41 @@ Bessel2 Triangle view | Row7 | [1, 0, 21, 0, 105, 0, 105, 0] | | Row8 | [1, 0, 28, 0, 210, 0, 420, 0, 105] | -Bessel2 Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 10, 1, 1, 11, 11, 26, 26, 1, 1, 16, 16, 61, 61, 76, 1, 1, 22, 22, 127, 127, 232, 232, 1, 1, 29, 29, 239, 239, 659, 659, 764] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 7, 7, 1, 1, 26, 26, 11, 11, 1, 1, 76, 61, 61, 16, 16, 1, 1, 232, 232, 127, 127, 22, 22, 1, 1, 764, 659, 659, 239, 239, 29, 29, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 3, 9, 9, 10, 0, 15, 15, 25, 25, 26, 15, 15, 60, 60, 75, 75, 76, 0, 105, 105, 210, 210, 231, 231, 232, 105, 105, 525, 525, 735, 735, 763, 763, 764] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 6, 0, 1, 0, 10, 0] | +Bessel2 Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 3], [1, 0, 10, 0, 15, 0], [1, 0, 15, 0, 45, 0, 15], [1, 0, 21, 0, 105, 0, 105, 0], [1, 0, 28, 0, 210, 0, 420, 0, 105]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [3, 0, 6, 0, 1], [0, 15, 0, 10, 0, 1], [15, 0, 45, 0, 15, 0, 1], [0, 105, 0, 105, 0, 21, 0, 1], [105, 0, 420, 0, 210, 0, 28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 3], [1, 0, 6, 0], [1, 0, 10, 0], [1, 0, 15, 0, 3]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 10], [1, 1, 11, 11, 26, 26], [1, 1, 16, 16, 61, 61, 76], [1, 1, 22, 22, 127, 127, 232, 232], [1, 1, 29, 29, 239, 239, 659, 659, 764]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [4, 4, 1, 1], [10, 7, 7, 1, 1], [26, 26, 11, 11, 1, 1], [76, 61, 61, 16, 16, 1, 1], [232, 232, 127, 127, 22, 22, 1, 1], [764, 659, 659, 239, 239, 29, 29, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [3, 3, 9, 9, 10], [0, 15, 15, 25, 25, 26], [15, 15, 60, 60, 75, 75, 76], [0, 105, 105, 210, 210, 231, 231, 232], [105, 105, 525, 525, 735, 735, 763, 763, 764]] | Bessel2 Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| EvenSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| AccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| AccRevSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| RevAccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| DiagSum | [1, 1, 1, 1, 2, 4, 7, 11] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| EvenSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| AccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | +| AccRevSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | +| AntiDiagSum | [1, 1, 1, 1, 2, 4, 7, 11, 19] | Bessel2 Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 1, 0, 0, 6, 10, 0, 0, 210] | -| Central | [1, 0, 6, 0, 210] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| PosHalf | [1, 2, 5, 14, 43, 142, 499, 1850, 7193] | -| NegHalf | [1, -2, 5, -14, 43, -142, 499, -1850, 7193] | -| Bin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| InvBin | [1, -1, 2, -10, 40, -176, 916, -4852, 27350] | -| TransSqrs | [0, 0, 4, 12, 72, 280, 1320, 5544, 25312] | -| TransNat0 | [0, 0, 2, 6, 24, 80, 300, 1092, 4256] | -| TransNat1 | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 1, 0, 0, 6, 10, 0, 0, 210] | +| ColECenter | [1, 0, 6, 0, 210] | +| ColOCenter | [1, 0, 10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| TransSqrs | [0, 0, 4, 12, 72, 280, 1320, 5544, 25312] | +| TransNat0 | [0, 0, 2, 6, 24, 80, 300, 1092, 4256] | +| TransNat1 | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | Bessel2 Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Bessel2 Polynomial values as columns | PolyCol7 | [1, 1, 50, 148, 7498, 36506, 1873516, 12606280, 655222268] | | PolyCol8 | [1, 1, 65, 193, 12673, 62081, 4117441, 27956545, 1872570113] | -# Bessel2Rev -True +# Bessel2:Rev +[] -Bessel2Rev Triangle view +Bessel2:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ Bessel2Rev Triangle view | Row7 | [0, 105, 0, 105, 0, 21, 0, 1] | | Row8 | [105, 0, 420, 0, 210, 0, 28, 0, 1] | -Bessel2Rev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 3, 9, 9, 10, 0, 15, 15, 25, 25, 26, 15, 15, 60, 60, 75, 75, 76, 0, 105, 105, 210, 210, 231, 231, 232, 105, 105, 525, 525, 735, 735, 763, 763, 764] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 4, 3, 3, 0, 10, 9, 9, 3, 3, 26, 25, 25, 15, 15, 0, 76, 75, 75, 60, 60, 15, 15, 232, 231, 231, 210, 210, 105, 105, 0, 764, 763, 763, 735, 735, 525, 525, 105, 105] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 10, 1, 1, 11, 11, 26, 26, 1, 1, 16, 16, 61, 61, 76, 1, 1, 22, 22, 127, 127, 232, 232, 1, 1, 29, 29, 239, 239, 659, 659, 764] | -| TriDiag | [1, 0, 1, 1, 0, 0, 3, 3, 1, 0, 0, 0, 15, 15, 6, 1, 0, 0, 0, 0] | - -Bessel2Rev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| EvenSum | [1, 0, 2, 0, 10, 0, 76, 0, 764] | -| OddSum | [0, 1, 0, 4, 0, 26, 0, 232, 0] | -| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | -| AccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| AccRevSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| RevAccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| DiagSum | [1, 0, 2, 0, 7, 0, 37, 0] | - -Bessel2Rev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 0, 0, 3, 6, 0, 0, 105, 210] | -| Central | [1, 0, 6, 0, 210] | -| LeftSide | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 13, 73, 281, 1741, 8485, 57233] | -| NegHalf | [1, 1, 5, 13, 73, 281, 1741, 8485, 57233] | -| Bin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| InvBin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| TransSqrs | [0, 1, 4, 12, 40, 130, 456, 1624, 6112] | -| TransNat0 | [0, 1, 2, 6, 16, 50, 156, 532, 1856] | -| TransNat1 | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | - -Bessel2Rev Diagonals as rows +Bessel2:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [3, 0, 6, 0, 1], [0, 15, 0, 10, 0, 1], [15, 0, 45, 0, 15, 0, 1], [0, 105, 0, 105, 0, 21, 0, 1], [105, 0, 420, 0, 210, 0, 28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 3], [1, 0, 10, 0, 15, 0], [1, 0, 15, 0, 45, 0, 15], [1, 0, 21, 0, 105, 0, 105, 0], [1, 0, 28, 0, 210, 0, 420, 0, 105]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [3, 3, 1], [0, 0, 0], [15, 15, 6, 1], [0, 0, 0, 0], [105, 105, 45, 10, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [3, 3, 9, 9, 10], [0, 15, 15, 25, 25, 26], [15, 15, 60, 60, 75, 75, 76], [0, 105, 105, 210, 210, 231, 231, 232], [105, 105, 525, 525, 735, 735, 763, 763, 764]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [4, 3, 3, 0], [10, 9, 9, 3, 3], [26, 25, 25, 15, 15, 0], [76, 75, 75, 60, 60, 15, 15], [232, 231, 231, 210, 210, 105, 105, 0], [764, 763, 763, 735, 735, 525, 525, 105, 105]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 10], [1, 1, 11, 11, 26, 26], [1, 1, 16, 16, 61, 61, 76], [1, 1, 22, 22, 127, 127, 232, 232], [1, 1, 29, 29, 239, 239, 659, 659, 764]] | + +Bessel2:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| EvenSum | [1, 0, 2, 0, 10, 0, 76, 0, 764] | +| OddSum | [0, 1, 0, 4, 0, 26, 0, 232, 0] | +| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | +| AccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | +| AccRevSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | +| AntiDiagSum | [1, 0, 2, 0, 7, 0, 37, 0, 266] | + +Bessel2:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 0, 0, 3, 6, 0, 0, 105, 210] | +| ColECenter | [1, 0, 6, 0, 210] | +| ColOCenter | [0, 3, 0, 105] | +| ColLeft | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 12, 40, 130, 456, 1624, 6112] | +| TransNat0 | [0, 1, 2, 6, 16, 50, 156, 532, 1856] | +| TransNat1 | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | + +Bessel2:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ Bessel2Rev Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [105, 945, 4725, 17325, 51975, 135135, 315315, 675675, 1351350]| -Bessel2Rev Diagonals as columns +Bessel2:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 1, 0, 3, 0, 15, 0, 105] | @@ -188,7 +180,7 @@ Bessel2Rev Diagonals as columns | DiagCol7 | [1, 0, 36, 0, 990, 0, 25740, 0, 675675] | | DiagCol8 | [1, 0, 45, 0, 1485, 0, 45045, 0, 1351350] | -Bessel2Rev Polynomial values as rows +Bessel2:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ Bessel2Rev Polynomial values as rows | PolyRow7 | [0, 232, 1850, 10440, 45028, 157400, 466542, 1213240, 2839880] | | PolyRow8 | [105, 764, 7193, 47868, 240809, 969980, 3273369, 9583868, 25004393] | -Bessel2Rev Polynomial values as columns +Bessel2:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 1, 0, 3, 0, 15, 0, 105] | @@ -214,3 +206,107 @@ Bessel2Rev Polynomial values as columns | PolyCol7 | [1, 7, 50, 364, 2698, 20342, 155884, 1213240, 9583868] | | PolyCol8 | [1, 8, 65, 536, 4483, 38008, 326479, 2839880, 25004393] | +# Bessel2:Rev:Inv +[] + +Bessel2:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [-1, 0, 1] | +| Row3 | [0, -3, 0, 1] | +| Row4 | [3, 0, -6, 0, 1] | +| Row5 | [0, 15, 0, -10, 0, 1] | +| Row6 | [-15, 0, 45, 0, -15, 0, 1] | +| Row7 | [0, -105, 0, 105, 0, -21, 0, 1] | +| Row8 | [105, 0, -420, 0, 210, 0, -28, 0, 1] | + +Bessel2:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [3, 0, -6, 0, 1], [0, 15, 0, -10, 0, 1], [-15, 0, 45, 0, -15, 0, 1], [0, -105, 0, 105, 0, -21, 0, 1], [105, 0, -420, 0, 210, 0, -28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 3], [1, 0, -10, 0, 15, 0], [1, 0, -15, 0, 45, 0, -15], [1, 0, -21, 0, 105, 0, -105, 0], [1, 0, -28, 0, 210, 0, -420, 0, 105]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [3, -3, 1], [0, 0, 0], [-15, 15, -6, 1], [0, 0, 0, 0], [105, -105, 45, -10, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [3, 3, -3, -3, -2], [0, 15, 15, 5, 5, 6], [-15, -15, 30, 30, 15, 15, 16], [0, -105, -105, 0, 0, -21, -21, -20], [105, 105, -315, -315, -105, -105, -133, -133, -132]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-2, -3, -3, 0], [-2, -3, -3, 3, 3], [6, 5, 5, 15, 15, 0], [16, 15, 15, 30, 30, -15, -15], [-20, -21, -21, 0, 0, -105, -105, 0], [-132, -133, -133, -105, -105, -315, -315, 105, 105]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, -2], [1, 1, -9, -9, 6, 6], [1, 1, -14, -14, 31, 31, 16], [1, 1, -20, -20, 85, 85, -20, -20], [1, 1, -27, -27, 183, 183, -237, -237, -132]] | + +Bessel2:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| EvenSum | [1, 0, 0, 0, -2, 0, 16, 0, -132] | +| OddSum | [0, 1, 0, -2, 0, 6, 0, -20, 0] | +| AltSum | [1, -1, 0, 2, -2, -6, 16, 20, -132] | +| AccSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | +| AccRevSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | +| AntiDiagSum | [1, 0, 0, 0, 1, 0, -5, 0, 36] | + +Bessel2:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 0, 0, -3, -6, 0, 0, 105, 210] | +| ColECenter | [1, 0, -6, 0, 210] | +| ColOCenter | [0, -3, 0, 105] | +| ColLeft | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 6, -8, -50, -24, 364, 736] | +| TransNat0 | [0, 1, 2, 0, -8, -10, 36, 112, -160] | +| TransNat1 | [1, 2, 2, -2, -10, -4, 52, 92, -292] | + +Bessel2:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow2 | [-1, -3, -6, -10, -15, -21, -28, -36, -45]| +| DiagRow3 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow4 | [3, 15, 45, 105, 210, 378, 630, 990, 1485]| +| DiagRow5 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow6 | [-15, -105, -420, -1260, -3150, -6930, -13860, -25740, -45045]| +| DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow8 | [105, 945, 4725, 17325, 51975, 135135, 315315, 675675, 1351350]| + +Bessel2:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| DiagCol1 | [1, 0, -3, 0, 15, 0, -105, 0, 945] | +| DiagCol2 | [1, 0, -6, 0, 45, 0, -420, 0, 4725] | +| DiagCol3 | [1, 0, -10, 0, 105, 0, -1260, 0, 17325] | +| DiagCol4 | [1, 0, -15, 0, 210, 0, -3150, 0, 51975] | +| DiagCol5 | [1, 0, -21, 0, 378, 0, -6930, 0, 135135] | +| DiagCol6 | [1, 0, -28, 0, 630, 0, -13860, 0, 315315] | +| DiagCol7 | [1, 0, -36, 0, 990, 0, -25740, 0, 675675] | +| DiagCol8 | [1, 0, -45, 0, 1485, 0, -45045, 0, 1351350] | + +Bessel2:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [-1, 0, 3, 8, 15, 24, 35, 48, 63] | +| PolyRow3 | [0, -2, 2, 18, 52, 110, 198, 322, 488] | +| PolyRow4 | [3, -2, -5, 30, 163, 478, 1083, 2110, 3715] | +| PolyRow5 | [0, 6, -18, 18, 444, 1950, 5706, 13482, 27768] | +| PolyRow6 | [-15, 16, -11, -96, 961, 7360, 28821, 83824, 203569] | +| PolyRow7 | [0, -20, 86, -396, 1180, 25100, 138690, 505876, 1461944] | +| PolyRow8 | [105, -132, 249, -516, -2007, 73980, 630393, 2954364, 10270569] | + +Bessel2:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| PolyCol1 | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| PolyCol2 | [1, 2, 3, 2, -5, -18, -11, 86, 249] | +| PolyCol3 | [1, 3, 8, 18, 30, 18, -96, -396, -516] | +| PolyCol4 | [1, 4, 15, 52, 163, 444, 961, 1180, -2007] | +| PolyCol5 | [1, 5, 24, 110, 478, 1950, 7360, 25100, 73980] | +| PolyCol6 | [1, 6, 35, 198, 1083, 5706, 28821, 138690, 630393] | +| PolyCol7 | [1, 7, 48, 322, 2110, 13482, 83824, 505876, 2954364] | +| PolyCol8 | [1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569] | + diff --git a/data/md/Binomial.md b/data/md/Binomial.md index acfc13b..efeda08 100644 --- a/data/md/Binomial.md +++ b/data/md/Binomial.md @@ -14,45 +14,41 @@ Binomial Triangle view | Row7 | [1, 7, 21, 35, 35, 21, 7, 1] | | Row8 | [1, 8, 28, 56, 70, 56, 28, 8, 1] | -Binomial Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256] | -| TriRevAcc | [1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1] | -| TriAccRev | [1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 3, 1, 5, 6, 1, 1, 6, 10, 4] | +Binomial Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1], [1, 6, 15, 20, 15, 6, 1], [1, 7, 21, 35, 35, 21, 7, 1], [1, 8, 28, 56, 70, 56, 28, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1], [1, 6, 15, 20, 15, 6, 1], [1, 7, 21, 35, 35, 21, 7, 1], [1, 8, 28, 56, 70, 56, 28, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 1], [1, 4, 3], [1, 5, 6, 1], [1, 6, 10, 4], [1, 7, 15, 10, 1]] | +| AccTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 8], [1, 5, 11, 15, 16], [1, 6, 16, 26, 31, 32], [1, 7, 22, 42, 57, 63, 64], [1, 8, 29, 64, 99, 120, 127, 128], [1, 9, 37, 93, 163, 219, 247, 255, 256]] | +| RevAccTabl | [[1], [2, 1], [4, 3, 1], [8, 7, 4, 1], [16, 15, 11, 5, 1], [32, 31, 26, 16, 6, 1], [64, 63, 57, 42, 22, 7, 1], [128, 127, 120, 99, 64, 29, 8, 1], [256, 255, 247, 219, 163, 93, 37, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 8], [1, 5, 11, 15, 16], [1, 6, 16, 26, 31, 32], [1, 7, 22, 42, 57, 63, 64], [1, 8, 29, 64, 99, 120, 127, 128], [1, 9, 37, 93, 163, 219, 247, 255, 256]] | Binomial Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| OddSum | [0, 1, 2, 4, 8, 16, 32, 64, 128] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| AccRevSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| RevAccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| DiagSum | [1, 1, 2, 3, 5, 8, 13, 21] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| OddSum | [0, 1, 2, 4, 8, 16, 32, 64, 128] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | +| AccRevSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | +| AntiDiagSum | [1, 1, 2, 3, 5, 8, 13, 21, 34] | Binomial Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | -| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | -| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Middle | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Central | [1, 2, 6, 20, 70] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| NegHalf | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| Bin | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| InvBin | [1, 0, -2, 0, 6, 0, -20, 0, 70] | -| TransSqrs | [0, 1, 6, 24, 80, 240, 672, 1792, 4608] | -| TransNat0 | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | -| TransNat1 | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | +| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | +| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColMiddle | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColECenter | [1, 2, 6, 20, 70] | +| ColOCenter | [1, 3, 10, 35] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 24, 80, 240, 672, 1792, 4608] | +| TransNat0 | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | +| TransNat1 | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | Binomial Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Binomial Polynomial values as columns | PolyCol7 | [1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216] | | PolyCol8 | [1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721] | -# BinomialInv -True +# Binomial:Inv +[] -BinomialInv Triangle view +Binomial:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ BinomialInv Triangle view | Row7 | [-1, 7, -21, 35, -35, 21, -7, 1] | | Row8 | [1, -8, 28, -56, 70, -56, 28, -8, 1] | -BinomialInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 1, -4, 6, -4, 1, -1, 5, -10, 10, -5, 1, 1, -6, 15, -20, 15, -6, 1, -1, 7, -21, 35, -35, 21, -7, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1] | -| TriRev | [1, 1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -5, 10, -10, 5, -1, 1, -6, 15, -20, 15, -6, 1, 1, -7, 21, -35, 35, -21, 7, -1, 1, -8, 28, -56, 70, -56, 28, -8, 1] | -| TriAcc | [1, -1, 0, 1, -1, 0, -1, 2, -1, 0, 1, -3, 3, -1, 0, -1, 4, -6, 4, -1, 0, 1, -5, 10, -10, 5, -1, 0, -1, 6, -15, 20, -15, 6, -1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, 1, 0, -1, 2, -1, 0, -1, 3, -3, 1, 0, -1, 4, -6, 4, -1, 0, -1, 5, -10, 10, -5, 1, 0, -1, 6, -15, 20, -15, 6, -1, 0, -1, 7, -21, 35, -35, 21, -7, 1] | -| TriAccRev | [1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0] | -| TriDiag | [1, -1, 1, 1, -1, -2, 1, 3, 1, -1, -4, -3, 1, 5, 6, 1, -1, -6, -10, -4] | - -BinomialInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | -| OddSum | [0, 1, -2, 4, -8, 16, -32, 64, -128] | -| AltSum | [1, -2, 4, -8, 16, -32, 64, -128, 256] | -| AccSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| AccRevSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| RevAccSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| DiagSum | [1, -1, 2, -3, 5, -8, 13, -21] | - -BinomialInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 3, 12, 10, 60, 105, 280] | -| RowGcd | [1, 1, -2, 3, 2, 5, 1, 7, 2] | -| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Middle | [1, -1, -2, 3, 6, -10, -20, 35, 70] | -| Central | [1, -2, 6, -20, 70] | -| LeftSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| NegHalf | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| Bin | [1, 0, -2, 0, 6, 0, -20, 0, 70] | -| InvBin | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| TransSqrs | [0, 1, 2, 0, 0, 0, 0, 0, 0] | -| TransNat0 | [0, 1, 0, 0, 0, 0, 0, 0, 0] | -| TransNat1 | [1, 1, 0, 0, 0, 0, 0, 0, 0] | - -BinomialInv Diagonals as rows +Binomial:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -2, 1], [-1, 3, -3, 1], [1, -4, 6, -4, 1], [-1, 5, -10, 10, -5, 1], [1, -6, 15, -20, 15, -6, 1], [-1, 7, -21, 35, -35, 21, -7, 1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 1], [1, -3, 3, -1], [1, -4, 6, -4, 1], [1, -5, 10, -10, 5, -1], [1, -6, 15, -20, 15, -6, 1], [1, -7, 21, -35, 35, -21, 7, -1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -2], [1, 3, 1], [-1, -4, -3], [1, 5, 6, 1], [-1, -6, -10, -4], [1, 7, 15, 10, 1]] | +| AccTabl | [[1], [-1, 0], [1, -1, 0], [-1, 2, -1, 0], [1, -3, 3, -1, 0], [-1, 4, -6, 4, -1, 0], [1, -5, 10, -10, 5, -1, 0], [-1, 6, -15, 20, -15, 6, -1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 1], [0, -1, 2, -1], [0, -1, 3, -3, 1], [0, -1, 4, -6, 4, -1], [0, -1, 5, -10, 10, -5, 1], [0, -1, 6, -15, 20, -15, 6, -1], [0, -1, 7, -21, 35, -35, 21, -7, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, 0], [1, -3, 3, -1, 0], [1, -4, 6, -4, 1, 0], [1, -5, 10, -10, 5, -1, 0], [1, -6, 15, -20, 15, -6, 1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | + +Binomial:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | +| OddSum | [0, 1, -2, 4, -8, 16, -32, 64, -128] | +| AltSum | [1, -2, 4, -8, 16, -32, 64, -128, 256] | +| AccSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| AccRevSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| AntiDiagSum | [1, -1, 2, -3, 5, -8, 13, -21, 34] | + +Binomial:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | +| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | +| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColMiddle | [1, -1, -2, 3, 6, -10, -20, 35, 70] | +| ColECenter | [1, -2, 6, -20, 70] | +| ColOCenter | [-1, 3, -10, 35] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, 0, 0, 0, 0, 0, 0] | +| TransNat0 | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| TransNat1 | [1, 1, 0, 0, 0, 0, 0, 0, 0] | + +Binomial:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ BinomialInv Diagonals as rows | DiagRow7 | [-1, -8, -36, -120, -330, -792, -1716, -3432, -6435]| | DiagRow8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870]| -BinomialInv Diagonals as columns +Binomial:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -188,7 +180,7 @@ BinomialInv Diagonals as columns | DiagCol7 | [1, -8, 36, -120, 330, -792, 1716, -3432, 6435] | | DiagCol8 | [1, -9, 45, -165, 495, -1287, 3003, -6435, 12870] | -BinomialInv Polynomial values as rows +Binomial:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ BinomialInv Polynomial values as rows | PolyRow7 | [-1, 0, 1, 128, 2187, 16384, 78125, 279936, 823543] | | PolyRow8 | [1, 0, 1, 256, 6561, 65536, 390625, 1679616, 5764801] | -BinomialInv Polynomial values as columns +Binomial:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -214,10 +206,10 @@ BinomialInv Polynomial values as columns | PolyCol7 | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | | PolyCol8 | [1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801] | -# BinomialRev -True +# Binomial:Rev +[] -BinomialRev Triangle view +Binomial:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ BinomialRev Triangle view | Row7 | [1, 7, 21, 35, 35, 21, 7, 1] | | Row8 | [1, 8, 28, 56, 70, 56, 28, 8, 1] | -BinomialRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256] | -| TriRevAcc | [1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1] | -| TriAccRev | [1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 3, 1, 5, 6, 1, 1, 6, 10, 4] | - -BinomialRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| OddSum | [0, 1, 2, 4, 8, 16, 32, 64, 128] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| AccRevSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| RevAccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | -| DiagSum | [1, 1, 2, 3, 5, 8, 13, 21] | - -BinomialRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | -| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | -| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Middle | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Central | [1, 2, 6, 20, 70] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| NegHalf | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| Bin | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| InvBin | [1, 0, -2, 0, 6, 0, -20, 0, 70] | -| TransSqrs | [0, 1, 6, 24, 80, 240, 672, 1792, 4608] | -| TransNat0 | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | -| TransNat1 | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | - -BinomialRev Diagonals as rows +Binomial:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1], [1, 6, 15, 20, 15, 6, 1], [1, 7, 21, 35, 35, 21, 7, 1], [1, 8, 28, 56, 70, 56, 28, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1], [1, 6, 15, 20, 15, 6, 1], [1, 7, 21, 35, 35, 21, 7, 1], [1, 8, 28, 56, 70, 56, 28, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 1], [1, 4, 3], [1, 5, 6, 1], [1, 6, 10, 4], [1, 7, 15, 10, 1]] | +| AccTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 8], [1, 5, 11, 15, 16], [1, 6, 16, 26, 31, 32], [1, 7, 22, 42, 57, 63, 64], [1, 8, 29, 64, 99, 120, 127, 128], [1, 9, 37, 93, 163, 219, 247, 255, 256]] | +| RevAccTabl | [[1], [2, 1], [4, 3, 1], [8, 7, 4, 1], [16, 15, 11, 5, 1], [32, 31, 26, 16, 6, 1], [64, 63, 57, 42, 22, 7, 1], [128, 127, 120, 99, 64, 29, 8, 1], [256, 255, 247, 219, 163, 93, 37, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 8], [1, 5, 11, 15, 16], [1, 6, 16, 26, 31, 32], [1, 7, 22, 42, 57, 63, 64], [1, 8, 29, 64, 99, 120, 127, 128], [1, 9, 37, 93, 163, 219, 247, 255, 256]] | + +Binomial:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| OddSum | [0, 1, 2, 4, 8, 16, 32, 64, 128] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | +| AccRevSum | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | +| AntiDiagSum | [1, 1, 2, 3, 5, 8, 13, 21, 34] | + +Binomial:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | +| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | +| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColMiddle | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColECenter | [1, 2, 6, 20, 70] | +| ColOCenter | [1, 3, 10, 35] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 24, 80, 240, 672, 1792, 4608] | +| TransNat0 | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | +| TransNat1 | [1, 3, 8, 20, 48, 112, 256, 576, 1280] | + +Binomial:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ BinomialRev Diagonals as rows | DiagRow7 | [1, 8, 36, 120, 330, 792, 1716, 3432, 6435]| | DiagRow8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870]| -BinomialRev Diagonals as columns +Binomial:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ BinomialRev Diagonals as columns | DiagCol7 | [1, 8, 36, 120, 330, 792, 1716, 3432, 6435] | | DiagCol8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870] | -BinomialRev Polynomial values as rows +Binomial:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ BinomialRev Polynomial values as rows | PolyRow7 | [1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969] | | PolyRow8 | [1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721] | -BinomialRev Polynomial values as columns +Binomial:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ BinomialRev Polynomial values as columns | PolyCol7 | [1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216] | | PolyCol8 | [1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721] | -# BinomialInvRev -True +# Binomial:Inv:Rev +[] -BinomialInvRev Triangle view +Binomial:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ BinomialInvRev Triangle view | Row7 | [1, -7, 21, -35, 35, -21, 7, -1] | | Row8 | [1, -8, 28, -56, 70, -56, 28, -8, 1] | -BinomialInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -5, 10, -10, 5, -1, 1, -6, 15, -20, 15, -6, 1, 1, -7, 21, -35, 35, -21, 7, -1, 1, -8, 28, -56, 70, -56, 28, -8, 1] | -| TriRev | [1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 1, -4, 6, -4, 1, -1, 5, -10, 10, -5, 1, 1, -6, 15, -20, 15, -6, 1, -1, 7, -21, 35, -35, 21, -7, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1] | -| TriAcc | [1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0] | -| TriRevAcc | [1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 3, -3, 1, 0, 1, -4, 6, -4, 1, 0, -1, 5, -10, 10, -5, 1, 0, 1, -6, 15, -20, 15, -6, 1, 0, -1, 7, -21, 35, -35, 21, -7, 1] | -| TriAccRev | [1, -1, 0, 1, -1, 0, -1, 2, -1, 0, 1, -3, 3, -1, 0, -1, 4, -6, 4, -1, 0, 1, -5, 10, -10, 5, -1, 0, -1, 6, -15, 20, -15, 6, -1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, -2, 1, -3, 1, 1, -4, 3, 1, -5, 6, -1, 1, -6, 10, -4] | - -BinomialInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| OddSum | [0, -1, -2, -4, -8, -16, -32, -64, -128] | -| AltSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| AccSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| AccRevSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| RevAccSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| DiagSum | [1, 1, 0, -1, -1, 0, 1, 1] | - -BinomialInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 3, 12, 10, 60, 105, 280] | -| RowGcd | [1, 1, -2, 3, 2, 5, 1, 7, 2] | -| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| Middle | [1, 1, -2, -3, 6, 10, -20, -35, 70] | -| Central | [1, -2, 6, -20, 70] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| PosHalf | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| NegHalf | [1, -3, 9, -27, 81, -243, 729, -2187, 6561] | -| Bin | [1, 0, -2, 0, 6, 0, -20, 0, 70] | -| InvBin | [1, -2, 6, -20, 70, -252, 924, -3432, 12870] | -| TransSqrs | [0, -1, 2, 0, 0, 0, 0, 0, 0] | -| TransNat0 | [0, -1, 0, 0, 0, 0, 0, 0, 0] | -| TransNat1 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | - -BinomialInvRev Diagonals as rows +Binomial:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -2, 1], [1, -3, 3, -1], [1, -4, 6, -4, 1], [1, -5, 10, -10, 5, -1], [1, -6, 15, -20, 15, -6, 1], [1, -7, 21, -35, 35, -21, 7, -1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| RevTabl | [[1], [-1, 1], [1, -2, 1], [-1, 3, -3, 1], [1, -4, 6, -4, 1], [-1, 5, -10, 10, -5, 1], [1, -6, 15, -20, 15, -6, 1], [-1, 7, -21, 35, -35, 21, -7, 1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -2], [1, -3, 1], [1, -4, 3], [1, -5, 6, -1], [1, -6, 10, -4], [1, -7, 15, -10, 1]] | +| AccTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, 0], [1, -3, 3, -1, 0], [1, -4, 6, -4, 1, 0], [1, -5, 10, -10, 5, -1, 0], [1, -6, 15, -20, 15, -6, 1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | +| RevAccTabl | [[1], [0, 1], [0, -1, 1], [0, 1, -2, 1], [0, -1, 3, -3, 1], [0, 1, -4, 6, -4, 1], [0, -1, 5, -10, 10, -5, 1], [0, 1, -6, 15, -20, 15, -6, 1], [0, -1, 7, -21, 35, -35, 21, -7, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -1, 0], [-1, 2, -1, 0], [1, -3, 3, -1, 0], [-1, 4, -6, 4, -1, 0], [1, -5, 10, -10, 5, -1, 0], [-1, 6, -15, 20, -15, 6, -1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | + +Binomial:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| OddSum | [0, -1, -2, -4, -8, -16, -32, -64, -128] | +| AltSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| AccSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| AccRevSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| AntiDiagSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | + +Binomial:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | +| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | +| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColMiddle | [1, 1, -2, -3, 6, 10, -20, -35, 70] | +| ColECenter | [1, -2, 6, -20, 70] | +| ColOCenter | [1, -3, 10, -35] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| TransSqrs | [0, -1, 2, 0, 0, 0, 0, 0, 0] | +| TransNat0 | [0, -1, 0, 0, 0, 0, 0, 0, 0] | +| TransNat1 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | + +Binomial:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 1, -1, 1, -1, 1, -1, 1]| @@ -391,7 +375,7 @@ BinomialInvRev Diagonals as rows | DiagRow7 | [1, -8, 36, -120, 330, -792, 1716, -3432, 6435]| | DiagRow8 | [1, -9, 45, -165, 495, -1287, 3003, -6435, 12870]| -BinomialInvRev Diagonals as columns +Binomial:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ BinomialInvRev Diagonals as columns | DiagCol7 | [-1, -8, -36, -120, -330, -792, -1716, -3432, -6435] | | DiagCol8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870] | -BinomialInvRev Polynomial values as rows +Binomial:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ BinomialInvRev Polynomial values as rows | PolyRow7 | [1, 0, -1, -128, -2187, -16384, -78125, -279936, -823543] | | PolyRow8 | [1, 0, 1, 256, 6561, 65536, 390625, 1679616, 5764801] | -BinomialInvRev Polynomial values as columns +Binomial:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ BinomialInvRev Polynomial values as columns | PolyCol7 | [1, -6, 36, -216, 1296, -7776, 46656, -279936, 1679616] | | PolyCol8 | [1, -7, 49, -343, 2401, -16807, 117649, -823543, 5764801] | +# Binomial:Rev:Inv +[] + +Binomial:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [1, -2, 1] | +| Row3 | [-1, 3, -3, 1] | +| Row4 | [1, -4, 6, -4, 1] | +| Row5 | [-1, 5, -10, 10, -5, 1] | +| Row6 | [1, -6, 15, -20, 15, -6, 1] | +| Row7 | [-1, 7, -21, 35, -35, 21, -7, 1] | +| Row8 | [1, -8, 28, -56, 70, -56, 28, -8, 1] | + +Binomial:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -2, 1], [-1, 3, -3, 1], [1, -4, 6, -4, 1], [-1, 5, -10, 10, -5, 1], [1, -6, 15, -20, 15, -6, 1], [-1, 7, -21, 35, -35, 21, -7, 1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 1], [1, -3, 3, -1], [1, -4, 6, -4, 1], [1, -5, 10, -10, 5, -1], [1, -6, 15, -20, 15, -6, 1], [1, -7, 21, -35, 35, -21, 7, -1], [1, -8, 28, -56, 70, -56, 28, -8, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -2], [1, 3, 1], [-1, -4, -3], [1, 5, 6, 1], [-1, -6, -10, -4], [1, 7, 15, 10, 1]] | +| AccTabl | [[1], [-1, 0], [1, -1, 0], [-1, 2, -1, 0], [1, -3, 3, -1, 0], [-1, 4, -6, 4, -1, 0], [1, -5, 10, -10, 5, -1, 0], [-1, 6, -15, 20, -15, 6, -1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 1], [0, -1, 2, -1], [0, -1, 3, -3, 1], [0, -1, 4, -6, 4, -1], [0, -1, 5, -10, 10, -5, 1], [0, -1, 6, -15, 20, -15, 6, -1], [0, -1, 7, -21, 35, -35, 21, -7, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, 0], [1, -3, 3, -1, 0], [1, -4, 6, -4, 1, 0], [1, -5, 10, -10, 5, -1, 0], [1, -6, 15, -20, 15, -6, 1, 0], [1, -7, 21, -35, 35, -21, 7, -1, 0]] | + +Binomial:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | +| OddSum | [0, 1, -2, 4, -8, 16, -32, 64, -128] | +| AltSum | [1, -2, 4, -8, 16, -32, 64, -128, 256] | +| AccSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| AccRevSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| AntiDiagSum | [1, -1, 2, -3, 5, -8, 13, -21, 34] | + +Binomial:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 10, 60, 105, 280] | +| RowGcd | [1, 1, 2, 3, 2, 5, 1, 7, 2] | +| RowMax | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| ColMiddle | [1, -1, -2, 3, 6, -10, -20, 35, 70] | +| ColECenter | [1, -2, 6, -20, 70] | +| ColOCenter | [-1, 3, -10, 35] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, 0, 0, 0, 0, 0, 0] | +| TransNat0 | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| TransNat1 | [1, 1, 0, 0, 0, 0, 0, 0, 0] | + +Binomial:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -2, -3, -4, -5, -6, -7, -8, -9]| +| DiagRow2 | [1, 3, 6, 10, 15, 21, 28, 36, 45]| +| DiagRow3 | [-1, -4, -10, -20, -35, -56, -84, -120, -165]| +| DiagRow4 | [1, 5, 15, 35, 70, 126, 210, 330, 495]| +| DiagRow5 | [-1, -6, -21, -56, -126, -252, -462, -792, -1287]| +| DiagRow6 | [1, 7, 28, 84, 210, 462, 924, 1716, 3003]| +| DiagRow7 | [-1, -8, -36, -120, -330, -792, -1716, -3432, -6435]| +| DiagRow8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870]| + +Binomial:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| DiagCol1 | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| DiagCol2 | [1, -3, 6, -10, 15, -21, 28, -36, 45] | +| DiagCol3 | [1, -4, 10, -20, 35, -56, 84, -120, 165] | +| DiagCol4 | [1, -5, 15, -35, 70, -126, 210, -330, 495] | +| DiagCol5 | [1, -6, 21, -56, 126, -252, 462, -792, 1287] | +| DiagCol6 | [1, -7, 28, -84, 210, -462, 924, -1716, 3003] | +| DiagCol7 | [1, -8, 36, -120, 330, -792, 1716, -3432, 6435] | +| DiagCol8 | [1, -9, 45, -165, 495, -1287, 3003, -6435, 12870] | + +Binomial:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [1, 0, 1, 4, 9, 16, 25, 36, 49] | +| PolyRow3 | [-1, 0, 1, 8, 27, 64, 125, 216, 343] | +| PolyRow4 | [1, 0, 1, 16, 81, 256, 625, 1296, 2401] | +| PolyRow5 | [-1, 0, 1, 32, 243, 1024, 3125, 7776, 16807] | +| PolyRow6 | [1, 0, 1, 64, 729, 4096, 15625, 46656, 117649] | +| PolyRow7 | [-1, 0, 1, 128, 2187, 16384, 78125, 279936, 823543] | +| PolyRow8 | [1, 0, 1, 256, 6561, 65536, 390625, 1679616, 5764801] | + +Binomial:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol3 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| PolyCol4 | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| PolyCol5 | [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536] | +| PolyCol6 | [1, 5, 25, 125, 625, 3125, 15625, 78125, 390625] | +| PolyCol7 | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | +| PolyCol8 | [1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801] | + diff --git a/data/md/Catalan.md b/data/md/Catalan.md index e196829..b6058fc 100644 --- a/data/md/Catalan.md +++ b/data/md/Catalan.md @@ -14,45 +14,41 @@ Catalan Triangle view | Row7 | [0, 429, 572, 429, 208, 65, 12, 1] | | Row8 | [0, 1430, 2002, 1638, 910, 350, 90, 14, 1] | -Catalan Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 14, 6, 1, 0, 42, 48, 27, 8, 1, 0, 132, 165, 110, 44, 10, 1, 0, 429, 572, 429, 208, 65, 12, 1, 0, 1430, 2002, 1638, 910, 350, 90, 14, 1] | -| TriRev | [1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 14, 14, 0, 1, 8, 27, 48, 42, 0, 1, 10, 44, 110, 165, 132, 0, 1, 12, 65, 208, 429, 572, 429, 0, 1, 14, 90, 350, 910, 1638, 2002, 1430, 0] | -| TriAcc | [1, 0, 1, 0, 2, 3, 0, 5, 9, 10, 0, 14, 28, 34, 35, 0, 42, 90, 117, 125, 126, 0, 132, 297, 407, 451, 461, 462, 0, 429, 1001, 1430, 1638, 1703, 1715, 1716, 0, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435] | -| TriRevAcc | [1, 1, 0, 3, 2, 0, 10, 9, 5, 0, 35, 34, 28, 14, 0, 126, 125, 117, 90, 42, 0, 462, 461, 451, 407, 297, 132, 0, 1716, 1715, 1703, 1638, 1430, 1001, 429, 0, 6435, 6434, 6420, 6330, 5980, 5070, 3432, 1430, 0] | -| TriAccRev | [1, 1, 1, 1, 3, 3, 1, 5, 10, 10, 1, 7, 21, 35, 35, 1, 9, 36, 84, 126, 126, 1, 11, 55, 165, 330, 462, 462, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435] | -| TriDiag | [1, 0, 0, 1, 0, 2, 0, 5, 1, 0, 14, 4, 0, 42, 14, 1, 0, 132, 48, 6] | +Catalan Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 1], [0, 5, 4, 1], [0, 14, 14, 6, 1], [0, 42, 48, 27, 8, 1], [0, 132, 165, 110, 44, 10, 1], [0, 429, 572, 429, 208, 65, 12, 1], [0, 1430, 2002, 1638, 910, 350, 90, 14, 1]] | +| RevTabl | [[1], [1, 0], [1, 2, 0], [1, 4, 5, 0], [1, 6, 14, 14, 0], [1, 8, 27, 48, 42, 0], [1, 10, 44, 110, 165, 132, 0], [1, 12, 65, 208, 429, 572, 429, 0], [1, 14, 90, 350, 910, 1638, 2002, 1430, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 5, 1], [0, 14, 4], [0, 42, 14, 1], [0, 132, 48, 6], [0, 429, 165, 27, 1]] | +| AccTabl | [[1], [0, 1], [0, 2, 3], [0, 5, 9, 10], [0, 14, 28, 34, 35], [0, 42, 90, 117, 125, 126], [0, 132, 297, 407, 451, 461, 462], [0, 429, 1001, 1430, 1638, 1703, 1715, 1716], [0, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 0], [10, 9, 5, 0], [35, 34, 28, 14, 0], [126, 125, 117, 90, 42, 0], [462, 461, 451, 407, 297, 132, 0], [1716, 1715, 1703, 1638, 1430, 1001, 429, 0], [6435, 6434, 6420, 6330, 5980, 5070, 3432, 1430, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 10, 10], [1, 7, 21, 35, 35], [1, 9, 36, 84, 126, 126], [1, 11, 55, 165, 330, 462, 462], [1, 13, 78, 286, 715, 1287, 1716, 1716], [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435]] | Catalan Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| EvenSum | [1, 0, 1, 4, 15, 56, 210, 792, 3003] | -| OddSum | [0, 1, 2, 6, 20, 70, 252, 924, 3432] | -| AltSum | [1, -1, -1, -2, -5, -14, -42, -132, -429] | -| AccSum | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | -| AccRevSum | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | -| RevAccSum | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | -| DiagSum | [1, 0, 1, 2, 6, 18, 57, 186] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| EvenSum | [1, 0, 1, 4, 15, 56, 210, 792, 3003] | +| OddSum | [0, 1, 2, 6, 20, 70, 252, 924, 3432] | +| AltSum | [1, -1, -1, -2, -5, -14, -42, -132, -429] | +| AccSum | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | +| AccRevSum | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | +| AntiDiagSum | [1, 0, 1, 2, 6, 18, 57, 186, 622] | Catalan Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 20, 42, 3024, 660, 34320, 450450] | -| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 2, 5, 14, 48, 165, 572, 2002] | -| Middle | [1, 0, 2, 5, 14, 48, 110, 429, 910] | -| Central | [1, 2, 14, 110, 910] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 29, 181, 1181, 7941, 54573, 381333] | -| NegHalf | [1, 1, -3, 13, -67, 381, -2307, 14589, -95235] | -| Bin | [1, 1, 5, 28, 165, 1001, 6188, 38760, 245157] | -| InvBin | [1, 1, -3, 4, 5, -39, 84, 8, -603] | -| TransSqrs | [0, 1, 6, 30, 140, 630, 2772, 12012, 51480] | -| TransNat0 | [0, 1, 4, 16, 64, 256, 1024, 4096, 16384] | -| TransNat1 | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 20, 42, 3024, 660, 34320, 450450] | +| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 2, 5, 14, 48, 165, 572, 2002] | +| ColMiddle | [1, 0, 2, 5, 14, 48, 110, 429, 910] | +| ColECenter | [1, 2, 14, 110, 910] | +| ColOCenter | [0, 5, 48, 429] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 30, 140, 630, 2772, 12012, 51480] | +| TransNat0 | [0, 1, 4, 16, 64, 256, 1024, 4096, 16384] | +| TransNat1 | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | Catalan Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,114 @@ Catalan Polynomial values as columns | PolyCol7 | [1, 7, 63, 574, 5243, 47922, 438102, 4005372, 36620115] | | PolyCol8 | [1, 8, 80, 808, 8176, 82768, 837984, 8484456, 85904688] | -# CatalanRev -True +# Catalan:Inv +[] -CatalanRev Triangle view +Catalan:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [0, -2, 1] | +| Row3 | [0, 3, -4, 1] | +| Row4 | [0, -4, 10, -6, 1] | +| Row5 | [0, 5, -20, 21, -8, 1] | +| Row6 | [0, -6, 35, -56, 36, -10, 1] | +| Row7 | [0, 7, -56, 126, -120, 55, -12, 1] | +| Row8 | [0, -8, 84, -252, 330, -220, 78, -14, 1] | + +Catalan:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -2, 1], [0, 3, -4, 1], [0, -4, 10, -6, 1], [0, 5, -20, 21, -8, 1], [0, -6, 35, -56, 36, -10, 1], [0, 7, -56, 126, -120, 55, -12, 1], [0, -8, 84, -252, 330, -220, 78, -14, 1]] | +| RevTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 3, 0], [1, -6, 10, -4, 0], [1, -8, 21, -20, 5, 0], [1, -10, 36, -56, 35, -6, 0], [1, -12, 55, -120, 126, -56, 7, 0], [1, -14, 78, -220, 330, -252, 84, -8, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -2], [0, 3, 1], [0, -4, -4], [0, 5, 10, 1], [0, -6, -20, -6], [0, 7, 35, 21, 1]] | +| AccTabl | [[1], [0, 1], [0, -2, -1], [0, 3, -1, 0], [0, -4, 6, 0, 1], [0, 5, -15, 6, -2, -1], [0, -6, 29, -27, 9, -1, 0], [0, 7, -49, 77, -43, 12, 0, 1], [0, -8, 76, -176, 154, -66, 12, -2, -1]] | +| RevAccTabl | [[1], [1, 0], [-1, -2, 0], [0, -1, 3, 0], [1, 0, 6, -4, 0], [-1, -2, 6, -15, 5, 0], [0, -1, 9, -27, 29, -6, 0], [1, 0, 12, -43, 77, -49, 7, 0], [-1, -2, 12, -66, 154, -176, 76, -8, 0]] | +| AccRevTabl | [[1], [1, 1], [1, -1, -1], [1, -3, 0, 0], [1, -5, 5, 1, 1], [1, -7, 14, -6, -1, -1], [1, -9, 27, -29, 6, 0, 0], [1, -11, 44, -76, 50, -6, 1, 1], [1, -13, 65, -155, 175, -77, 7, -1, -1]] | + +Catalan:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, 0, 1, -1, 0, 1, -1] | +| EvenSum | [1, 0, 1, -4, 11, -28, 72, -188, 493] | +| OddSum | [0, 1, -2, 4, -10, 27, -72, 189, -494] | +| AltSum | [1, -1, 3, -8, 21, -55, 144, -377, 987] | +| AccSum | [1, 1, -3, 2, 3, -7, 4, 5, -11] | +| AccRevSum | [1, 2, -1, -2, 3, 0, -4, 4, 1] | +| AntiDiagSum | [1, 0, 1, -2, 4, -8, 16, -32, 64] | + +Catalan:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 60, 840, 2520, 27720, 360360] | +| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 2, 4, 10, 21, 56, 126, 330] | +| ColMiddle | [1, 0, -2, 3, 10, -20, -56, 126, 330] | +| ColECenter | [1, -2, 10, -56, 330] | +| ColOCenter | [0, 3, -20, 126] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -4, -2, 11, -8, -11, 26] | +| TransNat0 | [0, 1, 0, -2, 2, 1, -4, 3, 2] | +| TransNat1 | [1, 2, -1, -2, 3, 0, -4, 4, 1] | + +Catalan:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [0, -2, -4, -6, -8, -10, -12, -14, -16]| +| DiagRow2 | [0, 3, 10, 21, 36, 55, 78, 105, 136]| +| DiagRow3 | [0, -4, -20, -56, -120, -220, -364, -560, -816]| +| DiagRow4 | [0, 5, 35, 126, 330, 715, 1365, 2380, 3876]| +| DiagRow5 | [0, -6, -56, -252, -792, -2002, -4368, -8568, -15504]| +| DiagRow6 | [0, 7, 84, 462, 1716, 5005, 12376, 27132, 54264]| +| DiagRow7 | [0, -8, -120, -792, -3432, -11440, -31824, -77520, -170544]| +| DiagRow8 | [0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314]| + +Catalan:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| DiagCol2 | [1, -4, 10, -20, 35, -56, 84, -120, 165] | +| DiagCol3 | [1, -6, 21, -56, 126, -252, 462, -792, 1287] | +| DiagCol4 | [1, -8, 36, -120, 330, -792, 1716, -3432, 6435] | +| DiagCol5 | [1, -10, 55, -220, 715, -2002, 5005, -11440, 24310] | +| DiagCol6 | [1, -12, 78, -364, 1365, -4368, 12376, -31824, 75582] | +| DiagCol7 | [1, -14, 105, -560, 2380, -8568, 27132, -77520, 203490] | +| DiagCol8 | [1, -16, 136, -816, 3876, -15504, 54264, -170544, 490314] | + +Catalan:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [0, -1, 0, 3, 8, 15, 24, 35, 48] | +| PolyRow3 | [0, 0, -2, 0, 12, 40, 90, 168, 280] | +| PolyRow4 | [0, 1, 0, -3, 16, 105, 336, 805, 1632] | +| PolyRow5 | [0, -1, 2, -3, 20, 275, 1254, 3857, 9512] | +| PolyRow6 | [0, 0, 0, 0, 24, 720, 4680, 18480, 55440] | +| PolyRow7 | [0, 1, -2, 3, 28, 1885, 17466, 88543, 323128] | +| PolyRow8 | [0, -1, 0, 3, 32, 4935, 65184, 424235, 1883328] | + +Catalan:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 1, -1, 0, 1, -1, 0, 1, -1] | +| PolyCol2 | [1, 2, 0, -2, 0, 2, 0, -2, 0] | +| PolyCol3 | [1, 3, 3, 0, -3, -3, 0, 3, 3] | +| PolyCol4 | [1, 4, 8, 12, 16, 20, 24, 28, 32] | +| PolyCol5 | [1, 5, 15, 40, 105, 275, 720, 1885, 4935] | +| PolyCol6 | [1, 6, 24, 90, 336, 1254, 4680, 17466, 65184] | +| PolyCol7 | [1, 7, 35, 168, 805, 3857, 18480, 88543, 424235] | +| PolyCol8 | [1, 8, 48, 280, 1632, 9512, 55440, 323128, 1883328] | + +# Catalan:Rev +[] + +Catalan:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +222,43 @@ CatalanRev Triangle view | Row7 | [1, 12, 65, 208, 429, 572, 429, 0] | | Row8 | [1, 14, 90, 350, 910, 1638, 2002, 1430, 0] | -CatalanRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 14, 14, 0, 1, 8, 27, 48, 42, 0, 1, 10, 44, 110, 165, 132, 0, 1, 12, 65, 208, 429, 572, 429, 0, 1, 14, 90, 350, 910, 1638, 2002, 1430, 0] | -| TriRev | [1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 14, 6, 1, 0, 42, 48, 27, 8, 1, 0, 132, 165, 110, 44, 10, 1, 0, 429, 572, 429, 208, 65, 12, 1, 0, 1430, 2002, 1638, 910, 350, 90, 14, 1] | -| TriAcc | [1, 1, 1, 1, 3, 3, 1, 5, 10, 10, 1, 7, 21, 35, 35, 1, 9, 36, 84, 126, 126, 1, 11, 55, 165, 330, 462, 462, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435] | -| TriRevAcc | [1, 1, 1, 3, 3, 1, 10, 10, 5, 1, 35, 35, 21, 7, 1, 126, 126, 84, 36, 9, 1, 462, 462, 330, 165, 55, 11, 1, 1716, 1716, 1287, 715, 286, 78, 13, 1, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1] | -| TriAccRev | [1, 0, 1, 0, 2, 3, 0, 5, 9, 10, 0, 14, 28, 34, 35, 0, 42, 90, 117, 125, 126, 0, 132, 297, 407, 451, 461, 462, 0, 429, 1001, 1430, 1638, 1703, 1715, 1716, 0, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435] | -| TriDiag | [1, 1, 1, 0, 1, 2, 1, 4, 0, 1, 6, 5, 1, 8, 14, 0, 1, 10, 27, 14] | - -CatalanRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| EvenSum | [1, 1, 1, 6, 15, 70, 210, 924, 3003] | -| OddSum | [0, 0, 2, 4, 20, 56, 252, 792, 3432] | -| AltSum | [1, 1, -1, 2, -5, 14, -42, 132, -429] | -| AccSum | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | -| AccRevSum | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | -| RevAccSum | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | -| DiagSum | [1, 1, 1, 3, 5, 12, 23, 52] | - -CatalanRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 20, 42, 3024, 660, 34320, 450450] | -| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 2, 5, 14, 48, 165, 572, 2002] | -| Middle | [1, 1, 2, 4, 14, 27, 110, 208, 910] | -| Central | [1, 2, 14, 110, 910] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 8, 34, 148, 652, 2892, 12882, 57540] | -| NegHalf | [1, -2, 0, -2, -4, -12, -36, -114, -372] | -| Bin | [1, 1, 5, 28, 165, 1001, 6188, 38760, 245157] | -| InvBin | [1, -1, -3, -4, 5, 39, 84, -8, -603] | -| TransSqrs | [0, 0, 2, 24, 188, 1220, 7116, 38752, 201176] | -| TransNat0 | [0, 0, 2, 14, 76, 374, 1748, 7916, 35096] | -| TransNat1 | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | - -CatalanRev Diagonals as rows +Catalan:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 2, 0], [1, 4, 5, 0], [1, 6, 14, 14, 0], [1, 8, 27, 48, 42, 0], [1, 10, 44, 110, 165, 132, 0], [1, 12, 65, 208, 429, 572, 429, 0], [1, 14, 90, 350, 910, 1638, 2002, 1430, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 1], [0, 5, 4, 1], [0, 14, 14, 6, 1], [0, 42, 48, 27, 8, 1], [0, 132, 165, 110, 44, 10, 1], [0, 429, 572, 429, 208, 65, 12, 1], [0, 1430, 2002, 1638, 910, 350, 90, 14, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 2], [1, 4, 0], [1, 6, 5], [1, 8, 14, 0], [1, 10, 27, 14], [1, 12, 44, 48, 0]] | +| AccTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 10, 10], [1, 7, 21, 35, 35], [1, 9, 36, 84, 126, 126], [1, 11, 55, 165, 330, 462, 462], [1, 13, 78, 286, 715, 1287, 1716, 1716], [1, 15, 105, 455, 1365, 3003, 5005, 6435, 6435]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 1], [10, 10, 5, 1], [35, 35, 21, 7, 1], [126, 126, 84, 36, 9, 1], [462, 462, 330, 165, 55, 11, 1], [1716, 1716, 1287, 715, 286, 78, 13, 1], [6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 3], [0, 5, 9, 10], [0, 14, 28, 34, 35], [0, 42, 90, 117, 125, 126], [0, 132, 297, 407, 451, 461, 462], [0, 429, 1001, 1430, 1638, 1703, 1715, 1716], [0, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435]] | + +Catalan:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| EvenSum | [1, 1, 1, 6, 15, 70, 210, 924, 3003] | +| OddSum | [0, 0, 2, 4, 20, 56, 252, 792, 3432] | +| AltSum | [1, 1, -1, 2, -5, 14, -42, 132, -429] | +| AccSum | [1, 2, 7, 26, 99, 382, 1486, 5812, 22819] | +| AccRevSum | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | +| AntiDiagSum | [1, 1, 1, 3, 5, 12, 23, 52, 105] | + +Catalan:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 20, 42, 3024, 660, 34320, 450450] | +| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 2, 5, 14, 48, 165, 572, 2002] | +| ColMiddle | [1, 1, 2, 4, 14, 27, 110, 208, 910] | +| ColECenter | [1, 2, 14, 110, 910] | +| ColOCenter | [1, 4, 27, 208] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 24, 188, 1220, 7116, 38752, 201176] | +| TransNat0 | [0, 0, 2, 14, 76, 374, 1748, 7916, 35096] | +| TransNat1 | [1, 1, 5, 24, 111, 500, 2210, 9632, 41531] | + +Catalan:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +271,7 @@ CatalanRev Diagonals as rows | DiagRow7 | [1, 14, 119, 798, 4655, 24794, 123970, 592020, 2731365]| | DiagRow8 | [1, 16, 152, 1120, 7084, 40480, 215280, 1085760, 5259150]| -CatalanRev Diagonals as columns +Catalan:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -188,7 +284,7 @@ CatalanRev Diagonals as columns | DiagCol7 | [0, 1430, 7072, 23256, 62016, 144210, 303600, 592020, 1085760] | | DiagCol8 | [0, 4862, 25194, 87210, 245157, 600875, 1332045, 2731365, 5259150] | -CatalanRev Polynomial values as rows +Catalan:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +297,7 @@ CatalanRev Polynomial values as rows | PolyRow7 | [1, 1716, 54573, 492724, 2467137, 8786436, 25066621, 61189668, 133071009] | | PolyRow8 | [1, 6435, 381333, 5068915, 33563481, 148733571, 507709165, 1443039123, 3581326065] | -CatalanRev Polynomial values as columns +Catalan:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -214,3 +310,107 @@ CatalanRev Polynomial values as columns | PolyCol7 | [1, 1, 15, 274, 5531, 118686, 2654646, 61189668, 1443039123] | | PolyCol8 | [1, 1, 17, 353, 8113, 198401, 5060433, 133071009, 3581326065] | +# Catalan:Inv:Rev +[] + +Catalan:Inv:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 0] | +| Row2 | [1, -2, 0] | +| Row3 | [1, -4, 3, 0] | +| Row4 | [1, -6, 10, -4, 0] | +| Row5 | [1, -8, 21, -20, 5, 0] | +| Row6 | [1, -10, 36, -56, 35, -6, 0] | +| Row7 | [1, -12, 55, -120, 126, -56, 7, 0] | +| Row8 | [1, -14, 78, -220, 330, -252, 84, -8, 0] | + +Catalan:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -2, 0], [1, -4, 3, 0], [1, -6, 10, -4, 0], [1, -8, 21, -20, 5, 0], [1, -10, 36, -56, 35, -6, 0], [1, -12, 55, -120, 126, -56, 7, 0], [1, -14, 78, -220, 330, -252, 84, -8, 0]] | +| RevTabl | [[1], [0, 1], [0, -2, 1], [0, 3, -4, 1], [0, -4, 10, -6, 1], [0, 5, -20, 21, -8, 1], [0, -6, 35, -56, 36, -10, 1], [0, 7, -56, 126, -120, 55, -12, 1], [0, -8, 84, -252, 330, -220, 78, -14, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -2], [1, -4, 0], [1, -6, 3], [1, -8, 10, 0], [1, -10, 21, -4], [1, -12, 36, -20, 0]] | +| AccTabl | [[1], [1, 1], [1, -1, -1], [1, -3, 0, 0], [1, -5, 5, 1, 1], [1, -7, 14, -6, -1, -1], [1, -9, 27, -29, 6, 0, 0], [1, -11, 44, -76, 50, -6, 1, 1], [1, -13, 65, -155, 175, -77, 7, -1, -1]] | +| RevAccTabl | [[1], [1, 1], [-1, -1, 1], [0, 0, -3, 1], [1, 1, 5, -5, 1], [-1, -1, -6, 14, -7, 1], [0, 0, 6, -29, 27, -9, 1], [1, 1, -6, 50, -76, 44, -11, 1], [-1, -1, 7, -77, 175, -155, 65, -13, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -2, -1], [0, 3, -1, 0], [0, -4, 6, 0, 1], [0, 5, -15, 6, -2, -1], [0, -6, 29, -27, 9, -1, 0], [0, 7, -49, 77, -43, 12, 0, 1], [0, -8, 76, -176, 154, -66, 12, -2, -1]] | + +Catalan:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, 0, 1, -1, 0, 1, -1] | +| EvenSum | [1, 1, 1, 4, 11, 27, 72, 189, 493] | +| OddSum | [0, 0, -2, -4, -10, -28, -72, -188, -494] | +| AltSum | [1, 1, 3, 8, 21, 55, 144, 377, 987] | +| AccSum | [1, 2, -1, -2, 3, 0, -4, 4, 1] | +| AccRevSum | [1, 1, -3, 2, 3, -7, 4, 5, -11] | +| AntiDiagSum | [1, 1, 1, -1, -3, -2, 3, 8, 5] | + +Catalan:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 60, 840, 2520, 27720, 360360] | +| RowGcd | [1, 1, 2, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 2, 4, 10, 21, 56, 126, 330] | +| ColMiddle | [1, 1, -2, -4, 10, 21, -56, -120, 330] | +| ColECenter | [1, -2, 10, -56, 330] | +| ColOCenter | [1, -4, 21, -120] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -2, 8, -2, -24, 40, -4, -70] | +| TransNat0 | [0, 0, -2, 2, 2, -6, 4, 4, -10] | +| TransNat1 | [1, 1, -3, 2, 3, -7, 4, 5, -11] | + +Catalan:Inv:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow1 | [1, -2, 3, -4, 5, -6, 7, -8, 9]| +| DiagRow2 | [1, -4, 10, -20, 35, -56, 84, -120, 165]| +| DiagRow3 | [1, -6, 21, -56, 126, -252, 462, -792, 1287]| +| DiagRow4 | [1, -8, 36, -120, 330, -792, 1716, -3432, 6435]| +| DiagRow5 | [1, -10, 55, -220, 715, -2002, 5005, -11440, 24310]| +| DiagRow6 | [1, -12, 78, -364, 1365, -4368, 12376, -31824, 75582]| +| DiagRow7 | [1, -14, 105, -560, 2380, -8568, 27132, -77520, 203490]| +| DiagRow8 | [1, -16, 136, -816, 3876, -15504, 54264, -170544, 490314]| + +Catalan:Inv:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [0, -2, -4, -6, -8, -10, -12, -14, -16] | +| DiagCol2 | [0, 3, 10, 21, 36, 55, 78, 105, 136] | +| DiagCol3 | [0, -4, -20, -56, -120, -220, -364, -560, -816] | +| DiagCol4 | [0, 5, 35, 126, 330, 715, 1365, 2380, 3876] | +| DiagCol5 | [0, -6, -56, -252, -792, -2002, -4368, -8568, -15504] | +| DiagCol6 | [0, 7, 84, 462, 1716, 5005, 12376, 27132, 54264] | +| DiagCol7 | [0, -8, -120, -792, -3432, -11440, -31824, -77520, -170544] | +| DiagCol8 | [0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314] | + +Catalan:Inv:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow2 | [1, -1, -3, -5, -7, -9, -11, -13, -15] | +| PolyRow3 | [1, 0, 5, 16, 33, 56, 85, 120, 161] | +| PolyRow4 | [1, 1, -3, -35, -119, -279, -539, -923, -1455] | +| PolyRow5 | [1, -1, -11, 31, 305, 1111, 2869, 6119, 11521] | +| PolyRow6 | [1, 0, 45, 160, -231, -3024, -12155, -34320, -79695] | +| PolyRow7 | [1, 1, -91, -1079, -3263, -559, 30421, 146329, 458081] | +| PolyRow8 | [1, -1, 93, 3955, 26537, 80631, 102949, -220597, -1770735] | + +Catalan:Inv:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 1, -1, 0, 1, -1, 0, 1, -1] | +| PolyCol2 | [1, 1, -3, 5, -3, -11, 45, -91, 93] | +| PolyCol3 | [1, 1, -5, 16, -35, 31, 160, -1079, 3955] | +| PolyCol4 | [1, 1, -7, 33, -119, 305, -231, -3263, 26537] | +| PolyCol5 | [1, 1, -9, 56, -279, 1111, -3024, -559, 80631] | +| PolyCol6 | [1, 1, -11, 85, -539, 2869, -12155, 30421, 102949] | +| PolyCol7 | [1, 1, -13, 120, -923, 6119, -34320, 146329, -220597] | +| PolyCol8 | [1, 1, -15, 161, -1455, 11521, -79695, 458081, -1770735] | + diff --git a/data/md/CatalanAer.md b/data/md/CatalanAer.md index 7c1a190..216d9ab 100644 --- a/data/md/CatalanAer.md +++ b/data/md/CatalanAer.md @@ -1,5 +1,5 @@ # CatalanAer -['A052173', 'A053121', 'A112554', 'A322378'] +['A053121', 'A052173', 'A112554', 'A322378'] CatalanAer Triangle view | Row | Seq | @@ -14,45 +14,41 @@ CatalanAer Triangle view | Row7 | [0, 14, 0, 14, 0, 6, 0, 1] | | Row8 | [14, 0, 28, 0, 20, 0, 7, 0, 1] | -CatalanAer Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 1, 0, 4, 0, 5, 0, 1, 0, 5, 0, 9, 0, 5, 1, 0, 6, 0, 14, 0, 14, 0, 1, 0, 7, 0, 20, 0, 28, 0, 14] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 2, 2, 3, 2, 2, 5, 5, 6, 0, 5, 5, 9, 9, 10, 5, 5, 14, 14, 19, 19, 20, 0, 14, 14, 28, 28, 34, 34, 35, 14, 14, 42, 42, 62, 62, 69, 69, 70] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 3, 2, 2, 0, 6, 5, 5, 2, 2, 10, 9, 9, 5, 5, 0, 20, 19, 19, 14, 14, 5, 5, 35, 34, 34, 28, 28, 14, 14, 0, 70, 69, 69, 62, 62, 42, 42, 14, 14] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70] | -| TriDiag | [1, 0, 1, 1, 0, 0, 2, 2, 1, 0, 0, 0, 5, 5, 3, 1, 0, 0, 0, 0] | +CatalanAer Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 2, 0, 1], [2, 0, 3, 0, 1], [0, 5, 0, 4, 0, 1], [5, 0, 9, 0, 5, 0, 1], [0, 14, 0, 14, 0, 6, 0, 1], [14, 0, 28, 0, 20, 0, 7, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 2, 0], [1, 0, 3, 0, 2], [1, 0, 4, 0, 5, 0], [1, 0, 5, 0, 9, 0, 5], [1, 0, 6, 0, 14, 0, 14, 0], [1, 0, 7, 0, 20, 0, 28, 0, 14]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [2, 2, 1], [0, 0, 0], [5, 5, 3, 1], [0, 0, 0, 0], [14, 14, 9, 4, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 2, 2, 3], [2, 2, 5, 5, 6], [0, 5, 5, 9, 9, 10], [5, 5, 14, 14, 19, 19, 20], [0, 14, 14, 28, 28, 34, 34, 35], [14, 14, 42, 42, 62, 62, 69, 69, 70]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [3, 2, 2, 0], [6, 5, 5, 2, 2], [10, 9, 9, 5, 5, 0], [20, 19, 19, 14, 14, 5, 5], [35, 34, 34, 28, 28, 14, 14, 0], [70, 69, 69, 62, 62, 42, 42, 14, 14]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 3, 3], [1, 1, 4, 4, 6], [1, 1, 5, 5, 10, 10], [1, 1, 6, 6, 15, 15, 20], [1, 1, 7, 7, 21, 21, 35, 35], [1, 1, 8, 8, 28, 28, 56, 56, 70]] | CatalanAer Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| EvenSum | [1, 0, 2, 0, 6, 0, 20, 0, 70] | -| OddSum | [0, 1, 0, 3, 0, 10, 0, 35, 0] | -| AltSum | [1, -1, 2, -3, 6, -10, 20, -35, 70] | -| AccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| AccRevSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| RevAccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| DiagSum | [1, 0, 2, 0, 5, 0, 14, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| EvenSum | [1, 0, 2, 0, 6, 0, 20, 0, 70] | +| OddSum | [0, 1, 0, 3, 0, 10, 0, 35, 0] | +| AltSum | [1, -1, 2, -3, 6, -10, 20, -35, 70] | +| AccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | +| AccRevSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| AntiDiagSum | [1, 0, 2, 0, 5, 0, 14, 0, 42] | CatalanAer Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | -| Middle | [1, 0, 0, 2, 3, 0, 0, 14, 20] | -| Central | [1, 0, 3, 0, 20] | -| LeftSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 9, 45, 97, 485, 1145, 5725] | -| NegHalf | [1, 1, 5, 9, 45, 97, 485, 1145, 5725] | -| Bin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| InvBin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| TransSqrs | [0, 1, 4, 11, 28, 66, 152, 339, 748] | -| TransNat0 | [0, 1, 2, 5, 10, 22, 44, 93, 186] | -| TransNat1 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | +| ColMiddle | [1, 0, 0, 2, 3, 0, 0, 14, 20] | +| ColECenter | [1, 0, 3, 0, 20] | +| ColOCenter | [0, 2, 0, 14] | +| ColLeft | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 11, 28, 66, 152, 339, 748] | +| TransNat0 | [0, 1, 2, 5, 10, 22, 44, 93, 186] | +| TransNat1 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | CatalanAer Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ CatalanAer Polynomial values as columns | PolyCol7 | [1, 7, 50, 357, 2550, 18214, 130100, 929285, 6637750] | | PolyCol8 | [1, 8, 65, 528, 4290, 34856, 283205, 2301040, 18695950] | -# CatalanAerInv -True +# CatalanAer:Inv +[] -CatalanAerInv Triangle view +CatalanAer:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ CatalanAerInv Triangle view | Row7 | [0, -4, 0, 10, 0, -6, 0, 1] | | Row8 | [1, 0, -10, 0, 15, 0, -7, 0, 1] | -CatalanAerInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 0, 3, 0, -4, 0, 1, -1, 0, 6, 0, -5, 0, 1, 0, -4, 0, 10, 0, -6, 0, 1, 1, 0, -10, 0, 15, 0, -7, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1] | -| TriAcc | [1, 0, 1, -1, -1, 0, 0, -2, -2, -1, 1, 1, -2, -2, -1, 0, 3, 3, -1, -1, 0, -1, -1, 5, 5, 0, 0, 1, 0, -4, -4, 6, 6, 0, 0, 1, 1, 1, -9, -9, 6, 6, -1, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, -1, -1, -2, -2, 0, -1, -2, -2, 1, 1, 0, -1, -1, 3, 3, 0, 1, 0, 0, 5, 5, -1, -1, 1, 0, 0, 6, 6, -4, -4, 0, 0, -1, -1, 6, 6, -9, -9, 1, 1] | -| TriAccRev | [1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -1, 1, 1, -3, -3, 0, 0, 1, 1, -4, -4, 2, 2, 1, 1, 1, -5, -5, 5, 5, 1, 1, 1, 1, -6, -6, 9, 9, -1, -1, 0] | -| TriDiag | [1, 0, -1, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 3, -3, 1, 0, 0, 0, 0] | - -CatalanAerInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| EvenSum | [1, 0, 0, 0, -1, 0, 1, 0, 0] | -| OddSum | [0, 1, 0, -1, 0, 0, 0, 1, 0] | -| AltSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | -| AccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| AccRevSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| RevAccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| DiagSum | [1, 0, 0, 0, 0, 0, 0, 0] | - -CatalanAerInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -3, 12, 30, 60, 210] | -| RowGcd | [1, 1, 1, -2, -3, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | -| Middle | [1, 0, 0, -2, -3, 0, 0, 10, 15] | -| Central | [1, 0, -3, 0, 15] | -| LeftSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, -7, 5, 33, 13, -119, -171] | -| NegHalf | [1, 1, -3, -7, 5, 33, 13, -119, -171] | -| Bin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| InvBin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| TransSqrs | [0, 1, 4, 7, 4, -8, -20, -15, 12] | -| TransNat0 | [0, 1, 2, 1, -2, -4, -2, 3, 6] | -| TransNat1 | [1, 2, 2, 0, -3, -4, -1, 4, 6] | - -CatalanAerInv Diagonals as rows +CatalanAer:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -2, 0, 1], [1, 0, -3, 0, 1], [0, 3, 0, -4, 0, 1], [-1, 0, 6, 0, -5, 0, 1], [0, -4, 0, 10, 0, -6, 0, 1], [1, 0, -10, 0, 15, 0, -7, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -2, 0], [1, 0, -3, 0, 1], [1, 0, -4, 0, 3, 0], [1, 0, -5, 0, 6, 0, -1], [1, 0, -6, 0, 10, 0, -4, 0], [1, 0, -7, 0, 15, 0, -10, 0, 1]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [1, -2, 1], [0, 0, 0], [-1, 3, -3, 1], [0, 0, 0, 0], [1, -4, 6, -4, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -2, -2, -1], [1, 1, -2, -2, -1], [0, 3, 3, -1, -1, 0], [-1, -1, 5, 5, 0, 0, 1], [0, -4, -4, 6, 6, 0, 0, 1], [1, 1, -9, -9, 6, 6, -1, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-1, -2, -2, 0], [-1, -2, -2, 1, 1], [0, -1, -1, 3, 3, 0], [1, 0, 0, 5, 5, -1, -1], [1, 0, 0, 6, 6, -4, -4, 0], [0, -1, -1, 6, 6, -9, -9, 1, 1]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -1, -1], [1, 1, -2, -2, -1], [1, 1, -3, -3, 0, 0], [1, 1, -4, -4, 2, 2, 1], [1, 1, -5, -5, 5, 5, 1, 1], [1, 1, -6, -6, 9, 9, -1, -1, 0]] | + +CatalanAer:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| EvenSum | [1, 0, 0, 0, -1, 0, 1, 0, 0] | +| OddSum | [0, 1, 0, -1, 0, 0, 0, 1, 0] | +| AltSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | +| AccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | +| AccRevSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | +| AntiDiagSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | + +CatalanAer:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 3, 12, 30, 60, 210] | +| RowGcd | [1, 1, 1, 2, 3, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | +| ColMiddle | [1, 0, 0, -2, -3, 0, 0, 10, 15] | +| ColECenter | [1, 0, -3, 0, 15] | +| ColOCenter | [0, -2, 0, 10] | +| ColLeft | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 7, 4, -8, -20, -15, 12] | +| TransNat0 | [0, 1, 2, 1, -2, -4, -2, 3, 6] | +| TransNat1 | [1, 2, 2, 0, -3, -4, -1, 4, 6] | + +CatalanAer:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ CatalanAerInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, 5, 15, 35, 70, 126, 210, 330, 495]| -CatalanAerInv Diagonals as columns +CatalanAer:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, -1, 0, 1, 0, -1, 0, 1] | @@ -188,7 +180,7 @@ CatalanAerInv Diagonals as columns | DiagCol7 | [1, 0, -8, 0, 36, 0, -120, 0, 330] | | DiagCol8 | [1, 0, -9, 0, 45, 0, -165, 0, 495] | -CatalanAerInv Polynomial values as rows +CatalanAer:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ CatalanAerInv Polynomial values as rows | PolyRow7 | [0, 1, 8, 987, 10864, 60605, 235416, 726103, 1905632] | | PolyRow8 | [1, 0, 9, 2584, 40545, 290376, 1372105, 4976784, 15003009] | -CatalanAerInv Polynomial values as columns +CatalanAer:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, -1, 0, 1, 0, -1, 0, 1] | @@ -214,10 +206,10 @@ CatalanAerInv Polynomial values as columns | PolyCol7 | [1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784] | | PolyCol8 | [1, 8, 63, 496, 3905, 30744, 242047, 1905632, 15003009] | -# CatalanAerRev -True +# CatalanAer:Rev +[] -CatalanAerRev Triangle view +CatalanAer:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ CatalanAerRev Triangle view | Row7 | [1, 0, 6, 0, 14, 0, 14, 0] | | Row8 | [1, 0, 7, 0, 20, 0, 28, 0, 14] | -CatalanAerRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 1, 0, 4, 0, 5, 0, 1, 0, 5, 0, 9, 0, 5, 1, 0, 6, 0, 14, 0, 14, 0, 1, 0, 7, 0, 20, 0, 28, 0, 14] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 4, 4, 1, 1, 10, 10, 5, 5, 1, 1, 20, 15, 15, 6, 6, 1, 1, 35, 35, 21, 21, 7, 7, 1, 1, 70, 56, 56, 28, 28, 8, 8, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 2, 2, 3, 2, 2, 5, 5, 6, 0, 5, 5, 9, 9, 10, 5, 5, 14, 14, 19, 19, 20, 0, 14, 14, 28, 28, 34, 34, 35, 14, 14, 42, 42, 62, 62, 69, 69, 70] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0] | - -CatalanAerRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| EvenSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| AccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| AccRevSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RevAccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| DiagSum | [1, 1, 1, 1, 2, 3, 4, 5] | - -CatalanAerRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | -| Middle | [1, 1, 0, 0, 3, 4, 0, 0, 20] | -| Central | [1, 0, 3, 0, 20] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| PosHalf | [1, 2, 5, 12, 30, 74, 185, 460, 1150] | -| NegHalf | [1, -2, 5, -12, 30, -74, 185, -460, 1150] | -| Bin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| InvBin | [1, -1, 2, -7, 21, -66, 216, -715, 2395] | -| TransSqrs | [0, 0, 4, 8, 44, 96, 344, 752, 2252] | -| TransNat0 | [0, 0, 2, 4, 14, 28, 76, 152, 374] | -| TransNat1 | [1, 1, 4, 7, 20, 38, 96, 187, 444] | - -CatalanAerRev Diagonals as rows +CatalanAer:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 2, 0], [1, 0, 3, 0, 2], [1, 0, 4, 0, 5, 0], [1, 0, 5, 0, 9, 0, 5], [1, 0, 6, 0, 14, 0, 14, 0], [1, 0, 7, 0, 20, 0, 28, 0, 14]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 2, 0, 1], [2, 0, 3, 0, 1], [0, 5, 0, 4, 0, 1], [5, 0, 9, 0, 5, 0, 1], [0, 14, 0, 14, 0, 6, 0, 1], [14, 0, 28, 0, 20, 0, 7, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 2], [1, 0, 3, 0], [1, 0, 4, 0], [1, 0, 5, 0, 2]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 3, 3], [1, 1, 4, 4, 6], [1, 1, 5, 5, 10, 10], [1, 1, 6, 6, 15, 15, 20], [1, 1, 7, 7, 21, 21, 35, 35], [1, 1, 8, 8, 28, 28, 56, 56, 70]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [3, 3, 1, 1], [6, 4, 4, 1, 1], [10, 10, 5, 5, 1, 1], [20, 15, 15, 6, 6, 1, 1], [35, 35, 21, 21, 7, 7, 1, 1], [70, 56, 56, 28, 28, 8, 8, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 2, 2, 3], [2, 2, 5, 5, 6], [0, 5, 5, 9, 9, 10], [5, 5, 14, 14, 19, 19, 20], [0, 14, 14, 28, 28, 34, 34, 35], [14, 14, 42, 42, 62, 62, 69, 69, 70]] | + +CatalanAer:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| EvenSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| AccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| AccRevSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | +| AntiDiagSum | [1, 1, 1, 1, 2, 3, 4, 5, 8] | + +CatalanAer:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | +| ColMiddle | [1, 1, 0, 0, 3, 4, 0, 0, 20] | +| ColECenter | [1, 0, 3, 0, 20] | +| ColOCenter | [1, 0, 4, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| TransSqrs | [0, 0, 4, 8, 44, 96, 344, 752, 2252] | +| TransNat0 | [0, 0, 2, 4, 14, 28, 76, 152, 374] | +| TransNat1 | [1, 1, 4, 7, 20, 38, 96, 187, 444] | + +CatalanAer:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 0, 2, 0, 5, 0, 14]| @@ -283,7 +271,7 @@ CatalanAerRev Diagonals as rows | DiagRow7 | [1, 0, 8, 0, 44, 0, 208, 0, 910]| | DiagRow8 | [1, 0, 9, 0, 54, 0, 273, 0, 1260]| -CatalanAerRev Diagonals as columns +CatalanAer:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ CatalanAerRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [14, 42, 90, 165, 275, 429, 637, 910, 1260] | -CatalanAerRev Polynomial values as rows +CatalanAer:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ CatalanAerRev Polynomial values as rows | PolyRow7 | [1, 35, 1145, 11395, 61025, 227651, 671545, 1680995, 3727745] | | PolyRow8 | [1, 70, 5725, 113950, 1037425, 5918926, 24847165, 84049750, 242303425] | -CatalanAerRev Polynomial values as columns +CatalanAer:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ CatalanAerRev Polynomial values as columns | PolyCol7 | [1, 1, 50, 99, 4950, 12202, 610100, 1680995, 84049750] | | PolyCol8 | [1, 1, 65, 129, 8385, 20737, 1347905, 3727745, 242303425] | -# CatalanAerInvRev -True +# CatalanAer:Inv:Rev +[] -CatalanAerInvRev Triangle view +CatalanAer:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ CatalanAerInvRev Triangle view | Row7 | [1, 0, -6, 0, 10, 0, -4, 0] | | Row8 | [1, 0, -7, 0, 15, 0, -10, 0, 1] | -CatalanAerInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1] | -| TriRev | [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 0, 3, 0, -4, 0, 1, -1, 0, 6, 0, -5, 0, 1, 0, -4, 0, 10, 0, -6, 0, 1, 1, 0, -10, 0, 15, 0, -7, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -1, 1, 1, -3, -3, 0, 0, 1, 1, -4, -4, 2, 2, 1, 1, 1, -5, -5, 5, 5, 1, 1, 1, 1, -6, -6, 9, 9, -1, -1, 0] | -| TriRevAcc | [1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -2, -2, 1, 1, 0, 0, -3, -3, 1, 1, 1, 2, 2, -4, -4, 1, 1, 1, 1, 5, 5, -5, -5, 1, 1, 0, -1, -1, 9, 9, -6, -6, 1, 1] | -| TriAccRev | [1, 0, 1, -1, -1, 0, 0, -2, -2, -1, 1, 1, -2, -2, -1, 0, 3, 3, -1, -1, 0, -1, -1, 5, 5, 0, 0, 1, 0, -4, -4, 6, 6, 0, 0, 1, 1, 1, -9, -9, 6, 6, -1, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -2, 1, 0, -3, 0, 1, 0, -4, 0] | - -CatalanAerInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| EvenSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| AccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| AccRevSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| RevAccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| DiagSum | [1, 1, 1, 1, 0, -1, -2, -3] | - -CatalanAerInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -3, 12, 30, 60, 210] | -| RowGcd | [1, 1, 1, -2, -3, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | -| Middle | [1, 1, 0, 0, -3, -4, 0, 0, 15] | -| Central | [1, 0, -3, 0, 15] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| PosHalf | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| NegHalf | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| Bin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| InvBin | [1, -1, 0, 5, -16, 24, 15, -197, 576] | -| TransSqrs | [0, 0, -4, -8, 4, 32, 40, -8, -84] | -| TransNat0 | [0, 0, -2, -4, -2, 4, 8, 4, -6] | -| TransNat1 | [1, 1, -2, -5, -3, 4, 9, 5, -6] | - -CatalanAerInvRev Diagonals as rows +CatalanAer:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, -1], [1, 0, -2, 0], [1, 0, -3, 0, 1], [1, 0, -4, 0, 3, 0], [1, 0, -5, 0, 6, 0, -1], [1, 0, -6, 0, 10, 0, -4, 0], [1, 0, -7, 0, 15, 0, -10, 0, 1]] | +| RevTabl | [[1], [0, 1], [-1, 0, 1], [0, -2, 0, 1], [1, 0, -3, 0, 1], [0, 3, 0, -4, 0, 1], [-1, 0, 6, 0, -5, 0, 1], [0, -4, 0, 10, 0, -6, 0, 1], [1, 0, -10, 0, 15, 0, -7, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, -1], [1, 0, -2], [1, 0, -3, 0], [1, 0, -4, 0], [1, 0, -5, 0, 1]] | +| AccTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -1, -1], [1, 1, -2, -2, -1], [1, 1, -3, -3, 0, 0], [1, 1, -4, -4, 2, 2, 1], [1, 1, -5, -5, 5, 5, 1, 1], [1, 1, -6, -6, 9, 9, -1, -1, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 1, 1], [-1, -1, 1, 1], [-1, -2, -2, 1, 1], [0, 0, -3, -3, 1, 1], [1, 2, 2, -4, -4, 1, 1], [1, 1, 5, 5, -5, -5, 1, 1], [0, -1, -1, 9, 9, -6, -6, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 0], [0, -2, -2, -1], [1, 1, -2, -2, -1], [0, 3, 3, -1, -1, 0], [-1, -1, 5, 5, 0, 0, 1], [0, -4, -4, 6, 6, 0, 0, 1], [1, 1, -9, -9, 6, 6, -1, -1, 0]] | + +CatalanAer:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| EvenSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| AccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | +| AccRevSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | +| AntiDiagSum | [1, 1, 1, 1, 0, -1, -2, -3, -3] | + +CatalanAer:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 3, 12, 30, 60, 210] | +| RowGcd | [1, 1, 1, 2, 3, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | +| ColMiddle | [1, 1, 0, 0, -3, -4, 0, 0, 15] | +| ColECenter | [1, 0, -3, 0, 15] | +| ColOCenter | [1, 0, -4, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| TransSqrs | [0, 0, -4, -8, 4, 32, 40, -8, -84] | +| TransNat0 | [0, 0, -2, -4, -2, 4, 8, 4, -6] | +| TransNat1 | [1, 1, -2, -5, -3, 4, 9, 5, -6] | + +CatalanAer:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 1, 0, -1, 0, 1]| @@ -391,7 +375,7 @@ CatalanAerInvRev Diagonals as rows | DiagRow7 | [1, 0, -8, 0, 36, 0, -120, 0, 330]| | DiagRow8 | [1, 0, -9, 0, 45, 0, -165, 0, 495]| -CatalanAerInvRev Diagonals as columns +CatalanAer:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ CatalanAerInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, 5, 15, 35, 70, 126, 210, 330, 495] | -CatalanAerInvRev Polynomial values as rows +CatalanAer:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ CatalanAerInvRev Polynomial values as rows | PolyRow7 | [1, 1, -119, -2159, -13919, -56399, -173879, -446879, -1007999] | | PolyRow8 | [1, 0, -171, 424, 28305, 243576, 1232245, 4623984, 14216769] | -CatalanAerInvRev Polynomial values as columns +CatalanAer:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/CatalanSqr.md b/data/md/CatalanSqr.md new file mode 100644 index 0000000..46b1b49 --- /dev/null +++ b/data/md/CatalanSqr.md @@ -0,0 +1,416 @@ +# CatalanSqr +['A039599', 'A050155'] + +CatalanSqr Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 1] | +| Row2 | [2, 3, 1] | +| Row3 | [5, 9, 5, 1] | +| Row4 | [14, 28, 20, 7, 1] | +| Row5 | [42, 90, 75, 35, 9, 1] | +| Row6 | [132, 297, 275, 154, 54, 11, 1] | +| Row7 | [429, 1001, 1001, 637, 273, 77, 13, 1] | +| Row8 | [1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1] | + +CatalanSqr Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 3, 1], [5, 9, 5, 1], [14, 28, 20, 7, 1], [42, 90, 75, 35, 9, 1], [132, 297, 275, 154, 54, 11, 1], [429, 1001, 1001, 637, 273, 77, 13, 1], [1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 2], [1, 5, 9, 5], [1, 7, 20, 28, 14], [1, 9, 35, 75, 90, 42], [1, 11, 54, 154, 275, 297, 132], [1, 13, 77, 273, 637, 1001, 1001, 429], [1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430]] | +| AntiDiag | [[1], [1], [2, 1], [5, 3], [14, 9, 1], [42, 28, 5], [132, 90, 20, 1], [429, 297, 75, 7], [1430, 1001, 275, 35, 1]] | +| AccTabl | [[1], [1, 2], [2, 5, 6], [5, 14, 19, 20], [14, 42, 62, 69, 70], [42, 132, 207, 242, 251, 252], [132, 429, 704, 858, 912, 923, 924], [429, 1430, 2431, 3068, 3341, 3418, 3431, 3432], [1430, 4862, 8502, 11050, 12310, 12750, 12854, 12869, 12870]] | +| RevAccTabl | [[1], [2, 1], [6, 5, 2], [20, 19, 14, 5], [70, 69, 62, 42, 14], [252, 251, 242, 207, 132, 42], [924, 923, 912, 858, 704, 429, 132], [3432, 3431, 3418, 3341, 3068, 2431, 1430, 429], [12870, 12869, 12854, 12750, 12310, 11050, 8502, 4862, 1430]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 6], [1, 6, 15, 20], [1, 8, 28, 56, 70], [1, 10, 45, 120, 210, 252], [1, 12, 66, 220, 495, 792, 924], [1, 14, 91, 364, 1001, 2002, 3003, 3432], [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870]] | + +CatalanSqr Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| EvenSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| OddSum | [0, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 13, 58, 257, 1126, 4882, 20980, 89497] | +| AccRevSum | [1, 3, 11, 42, 163, 638, 2510, 9908, 39203] | +| AntiDiagSum | [1, 1, 3, 8, 24, 75, 243, 808, 2742] | + +CatalanSqr Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 45, 140, 3150, 207900, 21021, 2522520] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 9, 28, 90, 297, 1001, 3640] | +| ColMiddle | [1, 1, 3, 9, 20, 75, 154, 637, 1260] | +| ColECenter | [1, 3, 20, 154, 1260] | +| ColOCenter | [1, 9, 75, 637] | +| ColLeft | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 38, 187, 874, 3958, 17548, 76627] | +| TransNat0 | [0, 1, 5, 22, 93, 386, 1586, 6476, 26333] | +| TransNat1 | [1, 3, 11, 42, 163, 638, 2510, 9908, 39203] | + +CatalanSqr Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [1, 3, 5, 7, 9, 11, 13, 15, 17]| +| DiagRow2 | [2, 9, 20, 35, 54, 77, 104, 135, 170]| +| DiagRow3 | [5, 28, 75, 154, 273, 440, 663, 950, 1309]| +| DiagRow4 | [14, 90, 275, 637, 1260, 2244, 3705, 5775, 8602]| +| DiagRow5 | [42, 297, 1001, 2548, 5508, 10659, 19019, 31878, 50830]| +| DiagRow6 | [132, 1001, 3640, 9996, 23256, 48279, 92092, 164450, 278460]| +| DiagRow7 | [429, 3432, 13260, 38760, 95931, 211508, 427570, 807300, 1442025]| +| DiagRow8 | [1430, 11934, 48450, 149226, 389367, 904475, 1924065, 3817125, 7152444]| + +CatalanSqr Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| DiagCol1 | [1, 3, 9, 28, 90, 297, 1001, 3432, 11934] | +| DiagCol2 | [1, 5, 20, 75, 275, 1001, 3640, 13260, 48450] | +| DiagCol3 | [1, 7, 35, 154, 637, 2548, 9996, 38760, 149226] | +| DiagCol4 | [1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367] | +| DiagCol5 | [1, 11, 77, 440, 2244, 10659, 48279, 211508, 904475] | +| DiagCol6 | [1, 13, 104, 663, 3705, 19019, 92092, 427570, 1924065] | +| DiagCol7 | [1, 15, 135, 950, 5775, 31878, 164450, 807300, 3817125] | +| DiagCol8 | [1, 17, 170, 1309, 8602, 50830, 278460, 1442025, 7152444] | + +CatalanSqr Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| PolyRow2 | [2, 6, 12, 20, 30, 42, 56, 72, 90] | +| PolyRow3 | [5, 20, 51, 104, 185, 300, 455, 656, 909] | +| PolyRow4 | [14, 70, 222, 548, 1150, 2154, 3710, 5992, 9198] | +| PolyRow5 | [42, 252, 978, 2904, 7170, 15492, 30282, 54768, 93114] | +| PolyRow6 | [132, 924, 4338, 15432, 44760, 111492, 247254, 500688, 942732] | +| PolyRow7 | [429, 3432, 19323, 82128, 279585, 802584, 2019087, 4577568, 9545013] | +| PolyRow8 | [1430, 12870, 86310, 437444, 1746870, 5778090, 16488710, 41851560, 96642774] | + +CatalanSqr Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| PolyCol1 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| PolyCol2 | [1, 3, 12, 51, 222, 978, 4338, 19323, 86310] | +| PolyCol3 | [1, 4, 20, 104, 548, 2904, 15432, 82128, 437444] | +| PolyCol4 | [1, 5, 30, 185, 1150, 7170, 44760, 279585, 1746870] | +| PolyCol5 | [1, 6, 42, 300, 2154, 15492, 111492, 802584, 5778090] | +| PolyCol6 | [1, 7, 56, 455, 3710, 30282, 247254, 2019087, 16488710] | +| PolyCol7 | [1, 8, 72, 656, 5992, 54768, 500688, 4577568, 41851560] | +| PolyCol8 | [1, 9, 90, 909, 9198, 93114, 942732, 9545013, 96642774] | + +# CatalanSqr:Inv +[] + +CatalanSqr:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [1, -3, 1] | +| Row3 | [-1, 6, -5, 1] | +| Row4 | [1, -10, 15, -7, 1] | +| Row5 | [-1, 15, -35, 28, -9, 1] | +| Row6 | [1, -21, 70, -84, 45, -11, 1] | +| Row7 | [-1, 28, -126, 210, -165, 66, -13, 1] | +| Row8 | [1, -36, 210, -462, 495, -286, 91, -15, 1] | + +CatalanSqr:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -3, 1], [-1, 6, -5, 1], [1, -10, 15, -7, 1], [-1, 15, -35, 28, -9, 1], [1, -21, 70, -84, 45, -11, 1], [-1, 28, -126, 210, -165, 66, -13, 1], [1, -36, 210, -462, 495, -286, 91, -15, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 1], [1, -5, 6, -1], [1, -7, 15, -10, 1], [1, -9, 28, -35, 15, -1], [1, -11, 45, -84, 70, -21, 1], [1, -13, 66, -165, 210, -126, 28, -1], [1, -15, 91, -286, 495, -462, 210, -36, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -3], [1, 6, 1], [-1, -10, -5], [1, 15, 15, 1], [-1, -21, -35, -7], [1, 28, 70, 28, 1]] | +| AccTabl | [[1], [-1, 0], [1, -2, -1], [-1, 5, 0, 1], [1, -9, 6, -1, 0], [-1, 14, -21, 7, -2, -1], [1, -20, 50, -34, 11, 0, 1], [-1, 27, -99, 111, -54, 12, -1, 0], [1, -35, 175, -287, 208, -78, 13, -2, -1]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 1], [1, 0, 5, -1], [0, -1, 6, -9, 1], [-1, -2, 7, -21, 14, -1], [1, 0, 11, -34, 50, -20, 1], [0, -1, 12, -54, 111, -99, 27, -1], [-1, -2, 13, -78, 208, -287, 175, -35, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -2, -1], [1, -4, 2, 1], [1, -6, 9, -1, 0], [1, -8, 20, -15, 0, -1], [1, -10, 35, -49, 21, 0, 1], [1, -12, 54, -111, 99, -27, 1, 0], [1, -14, 77, -209, 286, -176, 34, -2, -1]] | + +CatalanSqr:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 1, 0, -1, 1, 0, -1] | +| EvenSum | [1, -1, 2, -6, 17, -45, 117, -305, 798] | +| OddSum | [0, 1, -3, 7, -17, 44, -116, 305, -799] | +| AltSum | [1, -2, 5, -13, 34, -89, 233, -610, 1597] | +| AccSum | [1, -1, -2, 5, -3, -4, 9, -5, -6] | +| AccRevSum | [1, 1, -2, 0, 3, -3, -1, 5, -4] | +| AntiDiagSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | + +CatalanSqr:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 30, 210, 1260, 13860, 180180, 180180] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 6, 15, 35, 84, 210, 495] | +| ColMiddle | [1, -1, -3, 6, 15, -35, -84, 210, 495] | +| ColECenter | [1, -3, 15, -84, 495] | +| ColOCenter | [-1, 6, -35, 210] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -5, 3, 8, -16, 5, 21] | +| TransNat0 | [0, 1, -1, -1, 3, -2, -2, 5, -3] | +| TransNat1 | [1, 1, -2, 0, 3, -3, -1, 5, -4] | + +CatalanSqr:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -3, -5, -7, -9, -11, -13, -15, -17]| +| DiagRow2 | [1, 6, 15, 28, 45, 66, 91, 120, 153]| +| DiagRow3 | [-1, -10, -35, -84, -165, -286, -455, -680, -969]| +| DiagRow4 | [1, 15, 70, 210, 495, 1001, 1820, 3060, 4845]| +| DiagRow5 | [-1, -21, -126, -462, -1287, -3003, -6188, -11628, -20349]| +| DiagRow6 | [1, 28, 210, 924, 3003, 8008, 18564, 38760, 74613]| +| DiagRow7 | [-1, -36, -330, -1716, -6435, -19448, -50388, -116280, -245157]| +| DiagRow8 | [1, 45, 495, 3003, 12870, 43758, 125970, 319770, 735471]| + +CatalanSqr:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| DiagCol1 | [1, -3, 6, -10, 15, -21, 28, -36, 45] | +| DiagCol2 | [1, -5, 15, -35, 70, -126, 210, -330, 495] | +| DiagCol3 | [1, -7, 28, -84, 210, -462, 924, -1716, 3003] | +| DiagCol4 | [1, -9, 45, -165, 495, -1287, 3003, -6435, 12870] | +| DiagCol5 | [1, -11, 66, -286, 1001, -3003, 8008, -19448, 43758] | +| DiagCol6 | [1, -13, 91, -455, 1820, -6188, 18564, -50388, 125970] | +| DiagCol7 | [1, -15, 120, -680, 3060, -11628, 38760, -116280, 319770] | +| DiagCol8 | [1, -17, 153, -969, 4845, -20349, 74613, -245157, 735471] | + +CatalanSqr:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [1, -1, -1, 1, 5, 11, 19, 29, 41] | +| PolyRow3 | [-1, 1, -1, -1, 7, 29, 71, 139, 239] | +| PolyRow4 | [1, 0, 1, -2, 9, 76, 265, 666, 1393] | +| PolyRow5 | [-1, -1, 1, -1, 11, 199, 989, 3191, 8119] | +| PolyRow6 | [1, 1, -1, 1, 13, 521, 3691, 15289, 47321] | +| PolyRow7 | [-1, 0, -1, 2, 15, 1364, 13775, 73254, 275807] | +| PolyRow8 | [1, -1, 1, 1, 17, 3571, 51409, 350981, 1607521] | + +CatalanSqr:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| PolyCol1 | [1, 0, -1, 1, 0, -1, 1, 0, -1] | +| PolyCol2 | [1, 1, -1, -1, 1, 1, -1, -1, 1] | +| PolyCol3 | [1, 2, 1, -1, -2, -1, 1, 2, 1] | +| PolyCol4 | [1, 3, 5, 7, 9, 11, 13, 15, 17] | +| PolyCol5 | [1, 4, 11, 29, 76, 199, 521, 1364, 3571] | +| PolyCol6 | [1, 5, 19, 71, 265, 989, 3691, 13775, 51409] | +| PolyCol7 | [1, 6, 29, 139, 666, 3191, 15289, 73254, 350981] | +| PolyCol8 | [1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521] | + +# CatalanSqr:Rev +[] + +CatalanSqr:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 1] | +| Row2 | [1, 3, 2] | +| Row3 | [1, 5, 9, 5] | +| Row4 | [1, 7, 20, 28, 14] | +| Row5 | [1, 9, 35, 75, 90, 42] | +| Row6 | [1, 11, 54, 154, 275, 297, 132] | +| Row7 | [1, 13, 77, 273, 637, 1001, 1001, 429] | +| Row8 | [1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430] | + +CatalanSqr:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 2], [1, 5, 9, 5], [1, 7, 20, 28, 14], [1, 9, 35, 75, 90, 42], [1, 11, 54, 154, 275, 297, 132], [1, 13, 77, 273, 637, 1001, 1001, 429], [1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430]] | +| RevTabl | [[1], [1, 1], [2, 3, 1], [5, 9, 5, 1], [14, 28, 20, 7, 1], [42, 90, 75, 35, 9, 1], [132, 297, 275, 154, 54, 11, 1], [429, 1001, 1001, 637, 273, 77, 13, 1], [1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 5, 2], [1, 7, 9], [1, 9, 20, 5], [1, 11, 35, 28], [1, 13, 54, 75, 14]] | +| AccTabl | [[1], [1, 2], [1, 4, 6], [1, 6, 15, 20], [1, 8, 28, 56, 70], [1, 10, 45, 120, 210, 252], [1, 12, 66, 220, 495, 792, 924], [1, 14, 91, 364, 1001, 2002, 3003, 3432], [1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870]] | +| RevAccTabl | [[1], [2, 1], [6, 4, 1], [20, 15, 6, 1], [70, 56, 28, 8, 1], [252, 210, 120, 45, 10, 1], [924, 792, 495, 220, 66, 12, 1], [3432, 3003, 2002, 1001, 364, 91, 14, 1], [12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 5, 6], [5, 14, 19, 20], [14, 42, 62, 69, 70], [42, 132, 207, 242, 251, 252], [132, 429, 704, 858, 912, 923, 924], [429, 1430, 2431, 3068, 3341, 3418, 3431, 3432], [1430, 4862, 8502, 11050, 12310, 12750, 12854, 12869, 12870]] | + +CatalanSqr:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| EvenSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| OddSum | [0, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 11, 42, 163, 638, 2510, 9908, 39203] | +| AccRevSum | [1, 3, 13, 58, 257, 1126, 4882, 20980, 89497] | +| AntiDiagSum | [1, 1, 2, 4, 8, 17, 35, 75, 157] | + +CatalanSqr:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 45, 140, 3150, 207900, 21021, 2522520] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 9, 28, 90, 297, 1001, 3640] | +| ColMiddle | [1, 1, 3, 5, 20, 35, 154, 273, 1260] | +| ColECenter | [1, 3, 20, 154, 1260] | +| ColOCenter | [1, 5, 35, 273] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| TransSqrs | [0, 1, 11, 86, 563, 3314, 18190, 95052, 478979] | +| TransNat0 | [0, 1, 7, 38, 187, 874, 3958, 17548, 76627] | +| TransNat1 | [1, 3, 13, 58, 257, 1126, 4882, 20980, 89497] | + +CatalanSqr:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 2, 5, 14, 42, 132, 429, 1430]| +| DiagRow1 | [1, 3, 9, 28, 90, 297, 1001, 3432, 11934]| +| DiagRow2 | [1, 5, 20, 75, 275, 1001, 3640, 13260, 48450]| +| DiagRow3 | [1, 7, 35, 154, 637, 2548, 9996, 38760, 149226]| +| DiagRow4 | [1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367]| +| DiagRow5 | [1, 11, 77, 440, 2244, 10659, 48279, 211508, 904475]| +| DiagRow6 | [1, 13, 104, 663, 3705, 19019, 92092, 427570, 1924065]| +| DiagRow7 | [1, 15, 135, 950, 5775, 31878, 164450, 807300, 3817125]| +| DiagRow8 | [1, 17, 170, 1309, 8602, 50830, 278460, 1442025, 7152444]| + +CatalanSqr:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [1, 3, 5, 7, 9, 11, 13, 15, 17] | +| DiagCol2 | [2, 9, 20, 35, 54, 77, 104, 135, 170] | +| DiagCol3 | [5, 28, 75, 154, 273, 440, 663, 950, 1309] | +| DiagCol4 | [14, 90, 275, 637, 1260, 2244, 3705, 5775, 8602] | +| DiagCol5 | [42, 297, 1001, 2548, 5508, 10659, 19019, 31878, 50830] | +| DiagCol6 | [132, 1001, 3640, 9996, 23256, 48279, 92092, 164450, 278460] | +| DiagCol7 | [429, 3432, 13260, 38760, 95931, 211508, 427570, 807300, 1442025] | +| DiagCol8 | [1430, 11934, 48450, 149226, 389367, 904475, 1924065, 3817125, 7152444] | + +CatalanSqr:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| PolyRow2 | [1, 6, 15, 28, 45, 66, 91, 120, 153] | +| PolyRow3 | [1, 20, 87, 232, 485, 876, 1435, 2192, 3177] | +| PolyRow4 | [1, 70, 543, 2092, 5725, 12786, 24955, 44248, 73017] | +| PolyRow5 | [1, 252, 3543, 19864, 71445, 197796, 460747, 949488, 1785609] | +| PolyRow6 | [1, 924, 23823, 195352, 925965, 3183156, 8859739, 21237168, 45543897] | +| PolyRow7 | [1, 3432, 163719, 1970896, 12335685, 52718616, 175466347, 489517344, 1197639081] | +| PolyRow8 | [1, 12870, 1143999, 20275660, 167817405, 892401426, 3553964155, 11544312984, 32231934585] | + +CatalanSqr:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| PolyCol2 | [1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999] | +| PolyCol3 | [1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660] | +| PolyCol4 | [1, 5, 45, 485, 5725, 71445, 925965, 12335685, 167817405] | +| PolyCol5 | [1, 6, 66, 876, 12786, 197796, 3183156, 52718616, 892401426] | +| PolyCol6 | [1, 7, 91, 1435, 24955, 460747, 8859739, 175466347, 3553964155] | +| PolyCol7 | [1, 8, 120, 2192, 44248, 949488, 21237168, 489517344, 11544312984] | +| PolyCol8 | [1, 9, 153, 3177, 73017, 1785609, 45543897, 1197639081, 32231934585] | + +# CatalanSqr:Inv:Rev +[] + +CatalanSqr:Inv:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, -1] | +| Row2 | [1, -3, 1] | +| Row3 | [1, -5, 6, -1] | +| Row4 | [1, -7, 15, -10, 1] | +| Row5 | [1, -9, 28, -35, 15, -1] | +| Row6 | [1, -11, 45, -84, 70, -21, 1] | +| Row7 | [1, -13, 66, -165, 210, -126, 28, -1] | +| Row8 | [1, -15, 91, -286, 495, -462, 210, -36, 1] | + +CatalanSqr:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -3, 1], [1, -5, 6, -1], [1, -7, 15, -10, 1], [1, -9, 28, -35, 15, -1], [1, -11, 45, -84, 70, -21, 1], [1, -13, 66, -165, 210, -126, 28, -1], [1, -15, 91, -286, 495, -462, 210, -36, 1]] | +| RevTabl | [[1], [-1, 1], [1, -3, 1], [-1, 6, -5, 1], [1, -10, 15, -7, 1], [-1, 15, -35, 28, -9, 1], [1, -21, 70, -84, 45, -11, 1], [-1, 28, -126, 210, -165, 66, -13, 1], [1, -36, 210, -462, 495, -286, 91, -15, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -3], [1, -5, 1], [1, -7, 6], [1, -9, 15, -1], [1, -11, 28, -10], [1, -13, 45, -35, 1]] | +| AccTabl | [[1], [1, 0], [1, -2, -1], [1, -4, 2, 1], [1, -6, 9, -1, 0], [1, -8, 20, -15, 0, -1], [1, -10, 35, -49, 21, 0, 1], [1, -12, 54, -111, 99, -27, 1, 0], [1, -14, 77, -209, 286, -176, 34, -2, -1]] | +| RevAccTabl | [[1], [0, 1], [-1, -2, 1], [1, 2, -4, 1], [0, -1, 9, -6, 1], [-1, 0, -15, 20, -8, 1], [1, 0, 21, -49, 35, -10, 1], [0, 1, -27, 99, -111, 54, -12, 1], [-1, -2, 34, -176, 286, -209, 77, -14, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -2, -1], [-1, 5, 0, 1], [1, -9, 6, -1, 0], [-1, 14, -21, 7, -2, -1], [1, -20, 50, -34, 11, 0, 1], [-1, 27, -99, 111, -54, 12, -1, 0], [1, -35, 175, -287, 208, -78, 13, -2, -1]] | + +CatalanSqr:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 1, 0, -1, 1, 0, -1] | +| EvenSum | [1, 1, 2, 7, 17, 44, 117, 305, 798] | +| OddSum | [0, -1, -3, -6, -17, -45, -116, -305, -799] | +| AltSum | [1, 2, 5, 13, 34, 89, 233, 610, 1597] | +| AccSum | [1, 1, -2, 0, 3, -3, -1, 5, -4] | +| AccRevSum | [1, -1, -2, 5, -3, -4, 9, -5, -6] | +| AntiDiagSum | [1, 1, 0, -2, -3, 0, 6, 8, -1] | + +CatalanSqr:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 30, 210, 1260, 13860, 180180, 180180] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 6, 15, 35, 84, 210, 495] | +| ColMiddle | [1, 1, -3, -5, 15, 28, -84, -165, 495] | +| ColECenter | [1, -3, 15, -84, 495] | +| ColOCenter | [1, -5, 28, -165] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| TransSqrs | [0, -1, 1, 10, -21, 3, 44, -65, 5] | +| TransNat0 | [0, -1, -1, 4, -3, -3, 8, -5, -5] | +| TransNat1 | [1, -1, -2, 5, -3, -4, 9, -5, -6] | + +CatalanSqr:Inv:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, -1, 1, -1, 1, -1, 1, -1, 1]| +| DiagRow1 | [1, -3, 6, -10, 15, -21, 28, -36, 45]| +| DiagRow2 | [1, -5, 15, -35, 70, -126, 210, -330, 495]| +| DiagRow3 | [1, -7, 28, -84, 210, -462, 924, -1716, 3003]| +| DiagRow4 | [1, -9, 45, -165, 495, -1287, 3003, -6435, 12870]| +| DiagRow5 | [1, -11, 66, -286, 1001, -3003, 8008, -19448, 43758]| +| DiagRow6 | [1, -13, 91, -455, 1820, -6188, 18564, -50388, 125970]| +| DiagRow7 | [1, -15, 120, -680, 3060, -11628, 38760, -116280, 319770]| +| DiagRow8 | [1, -17, 153, -969, 4845, -20349, 74613, -245157, 735471]| + +CatalanSqr:Inv:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [-1, -3, -5, -7, -9, -11, -13, -15, -17] | +| DiagCol2 | [1, 6, 15, 28, 45, 66, 91, 120, 153] | +| DiagCol3 | [-1, -10, -35, -84, -165, -286, -455, -680, -969] | +| DiagCol4 | [1, 15, 70, 210, 495, 1001, 1820, 3060, 4845] | +| DiagCol5 | [-1, -21, -126, -462, -1287, -3003, -6188, -11628, -20349] | +| DiagCol6 | [1, 28, 210, 924, 3003, 8008, 18564, 38760, 74613] | +| DiagCol7 | [-1, -36, -330, -1716, -6435, -19448, -50388, -116280, -245157] | +| DiagCol8 | [1, 45, 495, 3003, 12870, 43758, 125970, 319770, 735471] | + +CatalanSqr:Inv:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| PolyRow2 | [1, -1, -1, 1, 5, 11, 19, 29, 41] | +| PolyRow3 | [1, 1, 7, 13, 13, 1, -29, -83, -167] | +| PolyRow4 | [1, 0, -17, -74, -171, -284, -365, -342, -119] | +| PolyRow5 | [1, -1, 23, 253, 989, 2531, 5059, 8513, 12473] | +| PolyRow6 | [1, 1, -1, -599, -4187, -15679, -42509, -93911, -179479] | +| PolyRow7 | [1, 0, -89, 718, 13485, 77836, 285475, 803706, 1893913] | +| PolyRow8 | [1, -1, 271, 1801, -27403, -308549, -1609901, -5846539, -16922039] | + +CatalanSqr:Inv:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 0, -1, 1, 0, -1, 1, 0, -1] | +| PolyCol2 | [1, -1, -1, 7, -17, 23, -1, -89, 271] | +| PolyCol3 | [1, -2, 1, 13, -74, 253, -599, 718, 1801] | +| PolyCol4 | [1, -3, 5, 13, -171, 989, -4187, 13485, -27403] | +| PolyCol5 | [1, -4, 11, 1, -284, 2531, -15679, 77836, -308549] | +| PolyCol6 | [1, -5, 19, -29, -365, 5059, -42509, 285475, -1609901] | +| PolyCol7 | [1, -6, 29, -83, -342, 8513, -93911, 803706, -5846539] | +| PolyCol8 | [1, -7, 41, -167, -119, 12473, -179479, 1893913, -16922039] | + diff --git a/data/md/CentralCycle.md b/data/md/CentralCycle.md new file mode 100644 index 0000000..d7d4648 --- /dev/null +++ b/data/md/CentralCycle.md @@ -0,0 +1,208 @@ +# CentralCycle +['A269940', 'A111999', 'A259456'] + +CentralCycle Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [0, 2, 3] | +| Row3 | [0, 6, 20, 15] | +| Row4 | [0, 24, 130, 210, 105] | +| Row5 | [0, 120, 924, 2380, 2520, 945] | +| Row6 | [0, 720, 7308, 26432, 44100, 34650, 10395] | +| Row7 | [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135] | +| Row8 | [0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025] | + +CentralCycle Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 3], [0, 6, 20, 15], [0, 24, 130, 210, 105], [0, 120, 924, 2380, 2520, 945], [0, 720, 7308, 26432, 44100, 34650, 10395], [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135], [0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025]] | +| RevTabl | [[1], [1, 0], [3, 2, 0], [15, 20, 6, 0], [105, 210, 130, 24, 0], [945, 2520, 2380, 924, 120, 0], [10395, 34650, 44100, 26432, 7308, 720, 0], [135135, 540540, 866250, 705320, 303660, 64224, 5040, 0], [2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 6, 3], [0, 24, 20], [0, 120, 130, 15], [0, 720, 924, 210], [0, 5040, 7308, 2380, 105]] | +| AccTabl | [[1], [0, 1], [0, 2, 5], [0, 6, 26, 41], [0, 24, 154, 364, 469], [0, 120, 1044, 3424, 5944, 6889], [0, 720, 8028, 34460, 78560, 113210, 123605], [0, 5040, 69264, 372924, 1078244, 1944494, 2485034, 2620169], [0, 40320, 663696, 4342536, 15441316, 34300156, 52588426, 62047876, 64074901]] | +| RevAccTabl | [[1], [1, 0], [5, 2, 0], [41, 26, 6, 0], [469, 364, 154, 24, 0], [6889, 5944, 3424, 1044, 120, 0], [123605, 113210, 78560, 34460, 8028, 720, 0], [2620169, 2485034, 1944494, 1078244, 372924, 69264, 5040, 0], [64074901, 62047876, 52588426, 34300156, 15441316, 4342536, 663696, 40320, 0]] | +| AccRevTabl | [[1], [1, 1], [3, 5, 5], [15, 35, 41, 41], [105, 315, 445, 469, 469], [945, 3465, 5845, 6769, 6889, 6889], [10395, 45045, 89145, 115577, 122885, 123605, 123605], [135135, 675675, 1541925, 2247245, 2550905, 2615129, 2620169, 2620169], [2027025, 11486475, 29774745, 48633585, 59732365, 63411205, 64034581, 64074901, 64074901]] | + +CentralCycle Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 5, 41, 469, 6889, 123605, 2620169, 64074901] | +| EvenSum | [1, 0, 3, 20, 235, 3444, 61803, 1310084, 32037451] | +| OddSum | [0, 1, 2, 21, 234, 3445, 61802, 1310085, 32037450] | +| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| AccSum | [1, 1, 7, 73, 1011, 17421, 358583, 8575169, 233499227] | +| AccRevSum | [1, 2, 13, 132, 1803, 30802, 630257, 15006352, 407249783] | +| AntiDiagSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | + +CentralCycle Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 60, 10920, 1413720, 39840292800, 6652505239380000, 2048503569859181635200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 20, 210, 2520, 44100, 866250, 18858840] | +| ColMiddle | [1, 0, 2, 6, 130, 924, 26432, 303660, 11098780] | +| ColECenter | [1, 2, 130, 26432, 11098780] | +| ColOCenter | [0, 6, 924, 303660] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| TransSqrs | [0, 1, 14, 221, 4114, 89181, 2213910, 62017301, 1936315234] | +| TransNat0 | [0, 1, 8, 91, 1334, 23913, 506652, 12386183, 343174882] | +| TransNat1 | [1, 2, 13, 132, 1803, 30802, 630257, 15006352, 407249783] | + +CentralCycle Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]| +| DiagRow1 | [0, 2, 20, 210, 2520, 34650, 540540, 9459450, 183783600]| +| DiagRow2 | [0, 6, 130, 2380, 44100, 866250, 18288270, 416215800, 10199989800]| +| DiagRow3 | [0, 24, 924, 26432, 705320, 18858840, 520059540, 14980405440, 453247114320]| +| DiagRow4 | [0, 120, 7308, 303660, 11098780, 389449060, 13642629000, 486591585480, 17856935296200]| +| DiagRow5 | [0, 720, 64224, 3678840, 177331440, 7934927000, 345240896000, 14972984666640, 656598399256800]| +| DiagRow6 | [0, 5040, 623376, 47324376, 2920525608, 162831789120, 8637235647040, 448594185959920, 23209044289551120]| +| DiagRow7 | [0, 40320, 6636960, 647536032, 49952862960, 3404554433280, 216752221445760, 13306928148113600, 803351393628623840]| +| DiagRow8 | [0, 362880, 76998240, 9418945536, 890577127440, 73017236532240, 5505619701582000, 395063504843607600, 27563542664861323120]| + +CentralCycle Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, 2, 6, 24, 120, 720, 5040, 40320, 362880] | +| DiagCol2 | [3, 20, 130, 924, 7308, 64224, 623376, 6636960, 76998240] | +| DiagCol3 | [15, 210, 2380, 26432, 303660, 3678840, 47324376, 647536032, 9418945536] | +| DiagCol4 | [105, 2520, 44100, 705320, 11098780, 177331440, 2920525608, 49952862960, 890577127440] | +| DiagCol5 | [945, 34650, 866250, 18858840, 389449060, 7934927000, 162831789120, 3404554433280, 73017236532240] | +| DiagCol6 | [10395, 540540, 18288270, 520059540, 13642629000, 345240896000, 8637235647040, 216752221445760, 5505619701582000] | +| DiagCol7 | [135135, 9459450, 416215800, 14980405440, 486591585480, 14972984666640, 448594185959920, 13306928148113600, 395063504843607600] | +| DiagCol8 | [2027025, 183783600, 10199989800, 453247114320, 17856935296200, 656598399256800, 23209044289551120, 803351393628623840, 27563542664861323120] | + +CentralCycle Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [0, 5, 16, 33, 56, 85, 120, 161, 208] | +| PolyRow3 | [0, 41, 212, 603, 1304, 2405, 3996, 6167, 9008] | +| PolyRow4 | [0, 469, 3928, 15417, 42496, 95245, 186264, 330673, 546112] | +| PolyRow5 | [0, 6889, 93536, 506691, 1780384, 4849325, 11162304, 22795591, 42566336] | +| PolyRow6 | [0, 123605, 2721808, 20351601, 91160576, 301755925, 817557840, 1920637313, 4055038336] | +| PolyRow7 | [0, 2620169, 93593216, 966015531, 5516187584, 22190803925, 70766573184, 191243348471, 456531848576] | +| PolyRow8 | [0, 64074901, 3713245504, 52906036329, 385133814016, 1882926644125, 7067769414336, 21972157620049, 59305251856384] | + +CentralCycle Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 1, 5, 41, 469, 6889, 123605, 2620169, 64074901] | +| PolyCol2 | [1, 2, 16, 212, 3928, 93536, 2721808, 93593216, 3713245504] | +| PolyCol3 | [1, 3, 33, 603, 15417, 506691, 20351601, 966015531, 52906036329] | +| PolyCol4 | [1, 4, 56, 1304, 42496, 1780384, 91160576, 5516187584, 385133814016] | +| PolyCol5 | [1, 5, 85, 2405, 95245, 4849325, 301755925, 22190803925, 1882926644125] | +| PolyCol6 | [1, 6, 120, 3996, 186264, 11162304, 817557840, 70766573184, 7067769414336] | +| PolyCol7 | [1, 7, 161, 6167, 330673, 22795591, 1920637313, 191243348471, 21972157620049] | +| PolyCol8 | [1, 8, 208, 9008, 546112, 42566336, 4055038336, 456531848576, 59305251856384] | + +# CentralCycle:Rev +[] + +CentralCycle:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 0] | +| Row2 | [3, 2, 0] | +| Row3 | [15, 20, 6, 0] | +| Row4 | [105, 210, 130, 24, 0] | +| Row5 | [945, 2520, 2380, 924, 120, 0] | +| Row6 | [10395, 34650, 44100, 26432, 7308, 720, 0] | +| Row7 | [135135, 540540, 866250, 705320, 303660, 64224, 5040, 0] | +| Row8 | [2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, 0] | + +CentralCycle:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [3, 2, 0], [15, 20, 6, 0], [105, 210, 130, 24, 0], [945, 2520, 2380, 924, 120, 0], [10395, 34650, 44100, 26432, 7308, 720, 0], [135135, 540540, 866250, 705320, 303660, 64224, 5040, 0], [2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 20, 15], [0, 24, 130, 210, 105], [0, 120, 924, 2380, 2520, 945], [0, 720, 7308, 26432, 44100, 34650, 10395], [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135], [0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025]] | +| AntiDiag | [[1], [1], [3, 0], [15, 2], [105, 20, 0], [945, 210, 6], [10395, 2520, 130, 0], [135135, 34650, 2380, 24], [2027025, 540540, 44100, 924, 0]] | +| AccTabl | [[1], [1, 1], [3, 5, 5], [15, 35, 41, 41], [105, 315, 445, 469, 469], [945, 3465, 5845, 6769, 6889, 6889], [10395, 45045, 89145, 115577, 122885, 123605, 123605], [135135, 675675, 1541925, 2247245, 2550905, 2615129, 2620169, 2620169], [2027025, 11486475, 29774745, 48633585, 59732365, 63411205, 64034581, 64074901, 64074901]] | +| RevAccTabl | [[1], [1, 1], [5, 5, 3], [41, 41, 35, 15], [469, 469, 445, 315, 105], [6889, 6889, 6769, 5845, 3465, 945], [123605, 123605, 122885, 115577, 89145, 45045, 10395], [2620169, 2620169, 2615129, 2550905, 2247245, 1541925, 675675, 135135], [64074901, 64074901, 64034581, 63411205, 59732365, 48633585, 29774745, 11486475, 2027025]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 5], [0, 6, 26, 41], [0, 24, 154, 364, 469], [0, 120, 1044, 3424, 5944, 6889], [0, 720, 8028, 34460, 78560, 113210, 123605], [0, 5040, 69264, 372924, 1078244, 1944494, 2485034, 2620169], [0, 40320, 663696, 4342536, 15441316, 34300156, 52588426, 62047876, 64074901]] | + +CentralCycle:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 5, 41, 469, 6889, 123605, 2620169, 64074901] | +| EvenSum | [1, 1, 3, 21, 235, 3445, 61803, 1310085, 32037451] | +| OddSum | [0, 0, 2, 20, 234, 3444, 61802, 1310084, 32037450] | +| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AccSum | [1, 2, 13, 132, 1803, 30802, 630257, 15006352, 407249783] | +| AccRevSum | [1, 1, 7, 73, 1011, 17421, 358583, 8575169, 233499227] | +| AntiDiagSum | [1, 1, 3, 17, 125, 1161, 13045, 172189, 2612589] | + +CentralCycle:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 60, 10920, 1413720, 39840292800, 6652505239380000, 2048503569859181635200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 20, 210, 2520, 44100, 866250, 18858840] | +| ColMiddle | [1, 1, 2, 20, 130, 2380, 26432, 705320, 11098780] | +| ColECenter | [1, 2, 130, 26432, 11098780] | +| ColOCenter | [1, 20, 2380, 705320] | +| ColLeft | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 44, 946, 22276, 583866, 16999020, 546310786] | +| TransNat0 | [0, 0, 2, 32, 542, 10532, 234978, 5955000, 169424326] | +| TransNat1 | [1, 1, 7, 73, 1011, 17421, 358583, 8575169, 233499227] | + +CentralCycle:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow1 | [1, 2, 6, 24, 120, 720, 5040, 40320, 362880]| +| DiagRow2 | [3, 20, 130, 924, 7308, 64224, 623376, 6636960, 76998240]| +| DiagRow3 | [15, 210, 2380, 26432, 303660, 3678840, 47324376, 647536032, 9418945536]| +| DiagRow4 | [105, 2520, 44100, 705320, 11098780, 177331440, 2920525608, 49952862960, 890577127440]| +| DiagRow5 | [945, 34650, 866250, 18858840, 389449060, 7934927000, 162831789120, 3404554433280, 73017236532240]| +| DiagRow6 | [10395, 540540, 18288270, 520059540, 13642629000, 345240896000, 8637235647040, 216752221445760, 5505619701582000]| +| DiagRow7 | [135135, 9459450, 416215800, 14980405440, 486591585480, 14972984666640, 448594185959920, 13306928148113600, 395063504843607600]| +| DiagRow8 | [2027025, 183783600, 10199989800, 453247114320, 17856935296200, 656598399256800, 23209044289551120, 803351393628623840, 27563542664861323120]| + +CentralCycle:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| DiagCol1 | [0, 2, 20, 210, 2520, 34650, 540540, 9459450, 183783600] | +| DiagCol2 | [0, 6, 130, 2380, 44100, 866250, 18288270, 416215800, 10199989800] | +| DiagCol3 | [0, 24, 924, 26432, 705320, 18858840, 520059540, 14980405440, 453247114320] | +| DiagCol4 | [0, 120, 7308, 303660, 11098780, 389449060, 13642629000, 486591585480, 17856935296200] | +| DiagCol5 | [0, 720, 64224, 3678840, 177331440, 7934927000, 345240896000, 14972984666640, 656598399256800] | +| DiagCol6 | [0, 5040, 623376, 47324376, 2920525608, 162831789120, 8637235647040, 448594185959920, 23209044289551120] | +| DiagCol7 | [0, 40320, 6636960, 647536032, 49952862960, 3404554433280, 216752221445760, 13306928148113600, 803351393628623840] | +| DiagCol8 | [0, 362880, 76998240, 9418945536, 890577127440, 73017236532240, 5505619701582000, 395063504843607600, 27563542664861323120] | + +CentralCycle:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow2 | [3, 5, 7, 9, 11, 13, 15, 17, 19] | +| PolyRow3 | [15, 41, 79, 129, 191, 265, 351, 449, 559] | +| PolyRow4 | [105, 469, 1237, 2553, 4561, 7405, 11229, 16177, 22393] | +| PolyRow5 | [945, 6889, 24817, 64593, 138961, 263545, 456849, 740257, 1138033] | +| PolyRow6 | [10395, 123605, 607519, 1991817, 5154371, 11407645, 22585095, 41127569, 70169707] | +| PolyRow7 | [135135, 2620169, 17560063, 72473697, 225443951, 581896585, 1315007919, 2689741313, 5090512447] | +| PolyRow8 | [2027025, 64074901, 585330109, 3039768729, 11361813721, 34187248525, 88152599781, 202456201969, 424884156769] | + +CentralCycle:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| PolyCol1 | [1, 1, 5, 41, 469, 6889, 123605, 2620169, 64074901] | +| PolyCol2 | [1, 1, 7, 79, 1237, 24817, 607519, 17560063, 585330109] | +| PolyCol3 | [1, 1, 9, 129, 2553, 64593, 1991817, 72473697, 3039768729] | +| PolyCol4 | [1, 1, 11, 191, 4561, 138961, 5154371, 225443951, 11361813721] | +| PolyCol5 | [1, 1, 13, 265, 7405, 263545, 11407645, 581896585, 34187248525] | +| PolyCol6 | [1, 1, 15, 351, 11229, 456849, 22585095, 1315007919, 88152599781] | +| PolyCol7 | [1, 1, 17, 449, 16177, 740257, 41127569, 2689741313, 202456201969] | +| PolyCol8 | [1, 1, 19, 559, 22393, 1138033, 70169707, 5090512447, 424884156769] | + diff --git a/data/md/CentralSet.md b/data/md/CentralSet.md index 9b10724..9204f0c 100644 --- a/data/md/CentralSet.md +++ b/data/md/CentralSet.md @@ -1,5 +1,5 @@ # CentralSet -['A008957', 'A036969', 'A269945'] +['A269945', 'A008957', 'A036969'] CentralSet Triangle view | Row | Seq | @@ -14,45 +14,41 @@ CentralSet Triangle view | Row7 | [0, 1, 1365, 13013, 11440, 2002, 91, 1] | | Row8 | [0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1] | -CentralSet Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 14, 21, 1, 0, 1, 30, 147, 85, 1, 0, 1, 55, 627, 1408, 341, 1, 0, 1, 91, 2002, 11440, 13013, 1365, 1, 0, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 6, 7, 0, 1, 22, 36, 37, 0, 1, 86, 233, 263, 264, 0, 1, 342, 1750, 2377, 2432, 2433, 0, 1, 1366, 14379, 25819, 27821, 27912, 27913, 0, 1, 5462, 123944, 319997, 381487, 386765, 386905, 386906] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 7, 6, 1, 0, 37, 36, 22, 1, 0, 264, 263, 233, 86, 1, 0, 2433, 2432, 2377, 1750, 342, 1, 0, 27913, 27912, 27821, 25819, 14379, 1366, 1, 0, 386906, 386905, 386765, 381487, 319997, 123944, 5462, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 6, 7, 7, 1, 15, 36, 37, 37, 1, 31, 178, 263, 264, 264, 1, 56, 683, 2091, 2432, 2433, 2433, 1, 92, 2094, 13534, 26547, 27912, 27913, 27913, 1, 141, 5419, 66909, 262962, 381444, 386905, 386906, 386906] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 5, 0, 1, 21, 1, 0, 1, 85, 14] | +CentralSet Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 5, 1], [0, 1, 21, 14, 1], [0, 1, 85, 147, 30, 1], [0, 1, 341, 1408, 627, 55, 1], [0, 1, 1365, 13013, 11440, 2002, 91, 1], [0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 5, 1, 0], [1, 14, 21, 1, 0], [1, 30, 147, 85, 1, 0], [1, 55, 627, 1408, 341, 1, 0], [1, 91, 2002, 11440, 13013, 1365, 1, 0], [1, 140, 5278, 61490, 196053, 118482, 5461, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 5], [0, 1, 21, 1], [0, 1, 85, 14], [0, 1, 341, 147, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 6, 7], [0, 1, 22, 36, 37], [0, 1, 86, 233, 263, 264], [0, 1, 342, 1750, 2377, 2432, 2433], [0, 1, 1366, 14379, 25819, 27821, 27912, 27913], [0, 1, 5462, 123944, 319997, 381487, 386765, 386905, 386906]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [7, 6, 1, 0], [37, 36, 22, 1, 0], [264, 263, 233, 86, 1, 0], [2433, 2432, 2377, 1750, 342, 1, 0], [27913, 27912, 27821, 25819, 14379, 1366, 1, 0], [386906, 386905, 386765, 381487, 319997, 123944, 5462, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 6, 7, 7], [1, 15, 36, 37, 37], [1, 31, 178, 263, 264, 264], [1, 56, 683, 2091, 2432, 2433, 2433], [1, 92, 2094, 13534, 26547, 27912, 27913, 27913], [1, 141, 5419, 66909, 262962, 381444, 386905, 386906, 386906]] | CentralSet Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 7, 37, 264, 2433, 27913, 386906] | -| EvenSum | [1, 0, 1, 5, 22, 115, 969, 12896, 206793] | -| OddSum | [0, 1, 1, 2, 15, 149, 1464, 15017, 180113] | -| AltSum | [1, -1, 0, 3, 7, -34, -495, -2121, 26680] | -| AccSum | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | -| AccRevSum | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | -| RevAccSum | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | -| DiagSum | [1, 0, 1, 1, 2, 6, 23, 100] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 7, 37, 264, 2433, 27913, 386906] | +| EvenSum | [1, 0, 1, 5, 22, 115, 969, 12896, 206793] | +| OddSum | [0, 1, 1, 2, 15, 149, 1464, 15017, 180113] | +| AltSum | [1, -1, 0, 3, 7, -34, -495, -2121, 26680] | +| AccSum | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | +| AccRevSum | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | +| AntiDiagSum | [1, 0, 1, 1, 2, 6, 23, 100, 490] | CentralSet Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 5, 42, 24990, 12439680, 3123120, 943260039381660] | -| RowGcd | [1, 1, 1, 5, 7, 1, 11, 13, 1] | -| RowMax | [1, 1, 1, 5, 21, 147, 1408, 13013, 196053] | -| Middle | [1, 0, 1, 1, 21, 85, 1408, 13013, 196053] | -| Central | [1, 1, 21, 1408, 196053] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 15, 121, 1345, 19371, 351663, 7791217] | -| NegHalf | [1, 1, -1, -5, 49, -135, -3441, 80899, -776287] | -| Bin | [1, 1, 3, 19, 187, 2476, 43017, 927207, 24103963] | -| InvBin | [1, 1, -1, -11, 67, 476, -13975, 67803, 3944843] | -| TransSqrs | [0, 1, 5, 30, 227, 2169, 25480, 358993, 5959213] | -| TransNat0 | [0, 1, 3, 14, 89, 737, 7696, 98093, 1490687] | -| TransNat1 | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 5, 42, 24990, 12439680, 3123120, 943260039381660] | +| RowGcd | [1, 1, 1, 5, 7, 1, 11, 13, 1] | +| RowMax | [1, 1, 1, 5, 21, 147, 1408, 13013, 196053] | +| ColMiddle | [1, 0, 1, 1, 21, 85, 1408, 13013, 196053] | +| ColECenter | [1, 1, 21, 1408, 196053] | +| ColOCenter | [0, 1, 85, 13013] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 30, 227, 2169, 25480, 358993, 5959213] | +| TransNat0 | [0, 1, 3, 14, 89, 737, 7696, 98093, 1490687] | +| TransNat1 | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | CentralSet Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ CentralSet Polynomial values as columns | PolyCol7 | [1, 7, 56, 595, 8239, 143430, 3047121, 77175007, 2287104848] | | PolyCol8 | [1, 8, 72, 840, 12616, 236360, 5375304, 145162056, 4572924232] | -# CentralSetInv -True +# CentralSet:Inv +[] -CentralSetInv Triangle view +CentralSet:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ CentralSetInv Triangle view | Row7 | [0, 518400, -773136, 296296, -44473, 3003, -91, 1] | | Row8 | [0, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1] | -CentralSetInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 4, -5, 1, 0, -36, 49, -14, 1, 0, 576, -820, 273, -30, 1, 0, -14400, 21076, -7645, 1023, -55, 1, 0, 518400, -773136, 296296, -44473, 3003, -91, 1, 0, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -5, 4, 0, 1, -14, 49, -36, 0, 1, -30, 273, -820, 576, 0, 1, -55, 1023, -7645, 21076, -14400, 0, 1, -91, 3003, -44473, 296296, -773136, 518400, 0, 1, -140, 7462, -191620, 2475473, -15291640, 38402064, -25401600, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 4, -1, 0, 0, -36, 13, -1, 0, 0, 576, -244, 29, -1, 0, 0, -14400, 6676, -969, 54, -1, 0, 0, 518400, -254736, 41560, -2913, 90, -1, 0, 0, -25401600, 13000464, -2291176, 184297, -7323, 139, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, 0, -1, 4, 0, 0, -1, 13, -36, 0, 0, -1, 29, -244, 576, 0, 0, -1, 54, -969, 6676, -14400, 0, 0, -1, 90, -2913, 41560, -254736, 518400, 0, 0, -1, 139, -7323, 184297, -2291176, 13000464, -25401600, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -4, 0, 0, 1, -13, 36, 0, 0, 1, -29, 244, -576, 0, 0, 1, -54, 969, -6676, 14400, 0, 0, 1, -90, 2913, -41560, 254736, -518400, 0, 0, 1, -139, 7323, -184297, 2291176, -13000464, 25401600, 0, 0] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 4, 1, 0, -36, -5, 0, 576, 49, 1, 0, -14400, -820, -14] | - -CentralSetInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 0, 1, -5, 50, -850, 22100, -817700, 40885000] | -| OddSum | [0, 1, -1, 5, -50, 850, -22100, 817700, -40885000] | -| AltSum | [1, -1, 2, -10, 100, -1700, 44200, -1635400, 81770000] | -| AccSum | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | -| AccRevSum | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | -| RevAccSum | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | -| DiagSum | [1, 0, 1, -1, 5, -41, 626, -15234] | - -CentralSetInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 20, 1764, 10745280, 326939342400, 352300447699200, 27645512323747439317996800] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 5, 49, 820, 21076, 773136, 38402064] | -| Middle | [1, 0, -1, 4, 49, -820, -7645, 296296, 2475473] | -| Central | [1, -1, 49, -7645, 2475473] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, 7, -119, 3689, -180761, 12834031, -1244901007] | -| NegHalf | [1, 1, 3, 27, 513, 16929, 863379, 63026667, 6239640033] | -| Bin | [1, 1, -1, -2, 95, -2739, 91856, -3730824, 178473359] | -| InvBin | [1, 1, 3, 28, 495, 13961, 571116, 31855272, 2319026319] | -| TransSqrs | [0, 1, 3, -7, 50, -702, 16128, -547200, 25660800] | -| TransNat0 | [0, 1, 1, -3, 24, -360, 8640, -302400, 14515200] | -| TransNat1 | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | - -CentralSetInv Diagonals as rows +CentralSet:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 4, -5, 1], [0, -36, 49, -14, 1], [0, 576, -820, 273, -30, 1], [0, -14400, 21076, -7645, 1023, -55, 1], [0, 518400, -773136, 296296, -44473, 3003, -91, 1], [0, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -5, 4, 0], [1, -14, 49, -36, 0], [1, -30, 273, -820, 576, 0], [1, -55, 1023, -7645, 21076, -14400, 0], [1, -91, 3003, -44473, 296296, -773136, 518400, 0], [1, -140, 7462, -191620, 2475473, -15291640, 38402064, -25401600, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 4, 1], [0, -36, -5], [0, 576, 49, 1], [0, -14400, -820, -14], [0, 518400, 21076, 273, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 4, -1, 0], [0, -36, 13, -1, 0], [0, 576, -244, 29, -1, 0], [0, -14400, 6676, -969, 54, -1, 0], [0, 518400, -254736, 41560, -2913, 90, -1, 0], [0, -25401600, 13000464, -2291176, 184297, -7323, 139, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 4, 0], [0, -1, 13, -36, 0], [0, -1, 29, -244, 576, 0], [0, -1, 54, -969, 6676, -14400, 0], [0, -1, 90, -2913, 41560, -254736, 518400, 0], [0, -1, 139, -7323, 184297, -2291176, 13000464, -25401600, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -4, 0, 0], [1, -13, 36, 0, 0], [1, -29, 244, -576, 0, 0], [1, -54, 969, -6676, 14400, 0, 0], [1, -90, 2913, -41560, 254736, -518400, 0, 0], [1, -139, 7323, -184297, 2291176, -13000464, 25401600, 0, 0]] | + +CentralSet:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -5, 50, -850, 22100, -817700, 40885000] | +| OddSum | [0, 1, -1, 5, -50, 850, -22100, 817700, -40885000] | +| AltSum | [1, -1, 2, -10, 100, -1700, 44200, -1635400, 81770000] | +| AccSum | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | +| AccRevSum | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | +| AntiDiagSum | [1, 0, 1, -1, 5, -41, 626, -15234, 539750] | + +CentralSet:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 20, 1764, 10745280, 326939342400, 352300447699200, 27645512323747439317996800] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 5, 49, 820, 21076, 773136, 38402064] | +| ColMiddle | [1, 0, -1, 4, 49, -820, -7645, 296296, 2475473] | +| ColECenter | [1, -1, 49, -7645, 2475473] | +| ColOCenter | [0, 4, -820, 296296] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -7, 50, -702, 16128, -547200, 25660800] | +| TransNat0 | [0, 1, 1, -3, 24, -360, 8640, -302400, 14515200] | +| TransNat1 | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | + +CentralSet:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ CentralSetInv Diagonals as rows | DiagRow7 | [0, -25401600, -2483133696, -84865562640, -1593719752240, -19967312312156, -185789298737900, -1373222414339685, -8439654758970225]| | DiagRow8 | [0, 1625702400, 202759531776, 8689315795776, 201529405816816, 3076822378767280, 34475213865472380, 303626807076050640, 2202549127844351265]| -CentralSetInv Diagonals as columns +CentralSet:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ CentralSetInv Diagonals as columns | DiagCol7 | [1, -140, 16422, -1999370, 268880381, -40796457506, 7026231453265, -1373222414339685, 303626807076050640] | | DiagCol8 | [1, -204, 32946, -5293970, 909450751, -171757365650, 36053226248115, -8439654758970225, 2202549127844351265] | -CentralSetInv Polynomial values as rows +CentralSet:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ CentralSetInv Polynomial values as rows | PolyRow7 | [0, 0, -306544, -339768, 0, 545600, 1026000, 1183896, 852992] | | PolyRow8 | [0, 0, 14407568, 15629328, 0, -24006400, -44118000, -49723632, -34972672] | -CentralSetInv Polynomial values as columns +CentralSet:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ CentralSetInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 126, -252, 2268, -40824, 1183896, -49723632] | | PolyCol8 | [1, 8, 56, 224, -224, 1792, -30464, 852992, -34972672] | -# CentralSetRev -True +# CentralSet:Rev +[] -CentralSetRev Triangle view +CentralSet:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ CentralSetRev Triangle view | Row7 | [1, 91, 2002, 11440, 13013, 1365, 1, 0] | | Row8 | [1, 140, 5278, 61490, 196053, 118482, 5461, 1, 0] | -CentralSetRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 14, 21, 1, 0, 1, 30, 147, 85, 1, 0, 1, 55, 627, 1408, 341, 1, 0, 1, 91, 2002, 11440, 13013, 1365, 1, 0, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 6, 7, 7, 1, 15, 36, 37, 37, 1, 31, 178, 263, 264, 264, 1, 56, 683, 2091, 2432, 2433, 2433, 1, 92, 2094, 13534, 26547, 27912, 27913, 27913, 1, 141, 5419, 66909, 262962, 381444, 386905, 386906, 386906] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 7, 7, 6, 1, 37, 37, 36, 15, 1, 264, 264, 263, 178, 31, 1, 2433, 2433, 2432, 2091, 683, 56, 1, 27913, 27913, 27912, 26547, 13534, 2094, 92, 1, 386906, 386906, 386905, 381444, 262962, 66909, 5419, 141, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 6, 7, 0, 1, 22, 36, 37, 0, 1, 86, 233, 263, 264, 0, 1, 342, 1750, 2377, 2432, 2433, 0, 1, 1366, 14379, 25819, 27821, 27912, 27913, 0, 1, 5462, 123944, 319997, 381487, 386765, 386905, 386906] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 5, 0, 1, 14, 1, 1, 30, 21, 0, 1, 55, 147, 1] | - -CentralSetRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 7, 37, 264, 2433, 27913, 386906] | -| EvenSum | [1, 1, 1, 2, 22, 149, 969, 15017, 206793] | -| OddSum | [0, 0, 1, 5, 15, 115, 1464, 12896, 180113] | -| AltSum | [1, 1, 0, -3, 7, 34, -495, 2121, 26680] | -| AccSum | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | -| AccRevSum | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | -| RevAccSum | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | -| DiagSum | [1, 1, 1, 2, 6, 16, 52, 204] | - -CentralSetRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 5, 42, 24990, 12439680, 3123120, 943260039381660] | -| RowGcd | [1, 1, 1, 5, 7, 1, 11, 13, 1] | -| RowMax | [1, 1, 1, 5, 21, 147, 1408, 13013, 196053] | -| Middle | [1, 1, 1, 5, 21, 147, 1408, 11440, 196053] | -| Central | [1, 1, 21, 1408, 196053] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 30, 214, 2030, 24486, 362622, 6430198] | -| NegHalf | [1, -2, 2, 10, -14, -390, -1566, 26026, 563282] | -| Bin | [1, 1, 3, 19, 187, 2476, 43017, 927207, 24103963] | -| InvBin | [1, -1, -1, 11, 67, -476, -13975, -67803, 3944843] | -| TransSqrs | [0, 0, 1, 9, 107, 1399, 20716, 353428, 6870205] | -| TransNat0 | [0, 0, 1, 7, 59, 583, 6902, 97298, 1604561] | -| TransNat1 | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | - -CentralSetRev Diagonals as rows +CentralSet:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 5, 1, 0], [1, 14, 21, 1, 0], [1, 30, 147, 85, 1, 0], [1, 55, 627, 1408, 341, 1, 0], [1, 91, 2002, 11440, 13013, 1365, 1, 0], [1, 140, 5278, 61490, 196053, 118482, 5461, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 5, 1], [0, 1, 21, 14, 1], [0, 1, 85, 147, 30, 1], [0, 1, 341, 1408, 627, 55, 1], [0, 1, 1365, 13013, 11440, 2002, 91, 1], [0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 5, 0], [1, 14, 1], [1, 30, 21, 0], [1, 55, 147, 1], [1, 91, 627, 85, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 6, 7, 7], [1, 15, 36, 37, 37], [1, 31, 178, 263, 264, 264], [1, 56, 683, 2091, 2432, 2433, 2433], [1, 92, 2094, 13534, 26547, 27912, 27913, 27913], [1, 141, 5419, 66909, 262962, 381444, 386905, 386906, 386906]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [7, 7, 6, 1], [37, 37, 36, 15, 1], [264, 264, 263, 178, 31, 1], [2433, 2433, 2432, 2091, 683, 56, 1], [27913, 27913, 27912, 26547, 13534, 2094, 92, 1], [386906, 386906, 386905, 381444, 262962, 66909, 5419, 141, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 6, 7], [0, 1, 22, 36, 37], [0, 1, 86, 233, 263, 264], [0, 1, 342, 1750, 2377, 2432, 2433], [0, 1, 1366, 14379, 25819, 27821, 27912, 27913], [0, 1, 5462, 123944, 319997, 381487, 386765, 386905, 386906]] | + +CentralSet:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 7, 37, 264, 2433, 27913, 386906] | +| EvenSum | [1, 1, 1, 2, 22, 149, 969, 15017, 206793] | +| OddSum | [0, 0, 1, 5, 15, 115, 1464, 12896, 180113] | +| AltSum | [1, 1, 0, -3, 7, 34, -495, 2121, 26680] | +| AccSum | [1, 2, 5, 21, 126, 1001, 10129, 126006, 1877593] | +| AccRevSum | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | +| AntiDiagSum | [1, 1, 1, 2, 6, 16, 52, 204, 804] | + +CentralSet:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 5, 42, 24990, 12439680, 3123120, 943260039381660] | +| RowGcd | [1, 1, 1, 5, 7, 1, 11, 13, 1] | +| RowMax | [1, 1, 1, 5, 21, 147, 1408, 13013, 196053] | +| ColMiddle | [1, 1, 1, 5, 21, 147, 1408, 11440, 196053] | +| ColECenter | [1, 1, 21, 1408, 196053] | +| ColOCenter | [1, 5, 147, 11440] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 9, 107, 1399, 20716, 353428, 6870205] | +| TransNat0 | [0, 0, 1, 7, 59, 583, 6902, 97298, 1604561] | +| TransNat1 | [1, 1, 3, 14, 96, 847, 9335, 125211, 1991467] | + +CentralSet:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ CentralSetRev Diagonals as rows | DiagRow7 | [1, 140, 12138, 846260, 52253971, 2995372800, 163648537860, 8657594647800, 448016038000965]| | DiagRow8 | [1, 204, 25194, 2458676, 209609235, 16410363840, 1213911823620, 86347951359480, 5974284925007685]| -CentralSetRev Diagonals as columns +CentralSet:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ CentralSetRev Diagonals as columns | DiagCol7 | [0, 1, 21845, 9668036, 860181300, 31306548900, 638816292660, 8657594647800, 86347951359480] | | DiagCol8 | [0, 1, 87381, 87099705, 13850000505, 796513723005, 23793900258765, 448016038000965, 5974284925007685] | -CentralSetRev Polynomial values as rows +CentralSet:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ CentralSetRev Polynomial values as rows | PolyRow7 | [1, 27913, 351663, 1713649, 5497741, 13894881, 30069403, 58326073, 104277849] | | PolyRow8 | [1, 386906, 7791217, 50362828, 197920145, 586014526, 1443941761, 3126705632, 6150938593] | -CentralSetRev Polynomial values as columns +CentralSet:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ CentralSetRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 85, 1471, 38970, 1349601, 58326073, 3126705632] | | PolyCol8 | [1, 1, 9, 105, 1969, 57265, 2190969, 104277849, 6150938593] | -# CentralSetInvRev -True +# CentralSet:Inv:Rev +[] -CentralSetInvRev Triangle view +CentralSet:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ CentralSetInvRev Triangle view | Row7 | [1, -91, 3003, -44473, 296296, -773136, 518400, 0] | | Row8 | [1, -140, 7462, -191620, 2475473, -15291640, 38402064, -25401600, 0] | -CentralSetInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -5, 4, 0, 1, -14, 49, -36, 0, 1, -30, 273, -820, 576, 0, 1, -55, 1023, -7645, 21076, -14400, 0, 1, -91, 3003, -44473, 296296, -773136, 518400, 0, 1, -140, 7462, -191620, 2475473, -15291640, 38402064, -25401600, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 4, -5, 1, 0, -36, 49, -14, 1, 0, 576, -820, 273, -30, 1, 0, -14400, 21076, -7645, 1023, -55, 1, 0, 518400, -773136, 296296, -44473, 3003, -91, 1, 0, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -4, 0, 0, 1, -13, 36, 0, 0, 1, -29, 244, -576, 0, 0, 1, -54, 969, -6676, 14400, 0, 0, 1, -90, 2913, -41560, 254736, -518400, 0, 0, 1, -139, 7323, -184297, 2291176, -13000464, 25401600, 0, 0] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, 0, 0, -4, 1, 0, 0, 36, -13, 1, 0, 0, -576, 244, -29, 1, 0, 0, 14400, -6676, 969, -54, 1, 0, 0, -518400, 254736, -41560, 2913, -90, 1, 0, 0, 25401600, -13000464, 2291176, -184297, 7323, -139, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 4, -1, 0, 0, -36, 13, -1, 0, 0, 576, -244, 29, -1, 0, 0, -14400, 6676, -969, 54, -1, 0, 0, 518400, -254736, 41560, -2913, 90, -1, 0, 0, -25401600, 13000464, -2291176, 184297, -7323, 139, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -5, 0, 1, -14, 4, 1, -30, 49, 0, 1, -55, 273, -36] | - -CentralSetInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 5, 50, 850, 22100, 817700, 40885000] | -| OddSum | [0, 0, -1, -5, -50, -850, -22100, -817700, -40885000] | -| AltSum | [1, 1, 2, 10, 100, 1700, 44200, 1635400, 81770000] | -| AccSum | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | -| AccRevSum | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | -| RevAccSum | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | -| DiagSum | [1, 1, 1, 0, -4, -9, 20, 183] | - -CentralSetInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 20, 1764, 10745280, 326939342400, 352300447699200, 27645512323747439317996800] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 5, 49, 820, 21076, 773136, 38402064] | -| Middle | [1, 1, -1, -5, 49, 273, -7645, -44473, 2475473] | -| Central | [1, -1, 49, -7645, 2475473] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, -4, 28, -392, 9016, -306544, 14407568] | -| NegHalf | [1, -2, 6, -36, 396, -7128, 192456, -7313328, 372979728] | -| Bin | [1, 1, -1, -2, 95, -2739, 91856, -3730824, 178473359] | -| InvBin | [1, -1, 3, -28, 495, -13961, 571116, -31855272, 2319026319] | -| TransSqrs | [0, 0, -1, 11, -142, 2898, -87552, 3686400, -206582400] | -| TransNat0 | [0, 0, -1, 3, -24, 360, -8640, 302400, -14515200] | -| TransNat1 | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | - -CentralSetInvRev Diagonals as rows +CentralSet:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -5, 4, 0], [1, -14, 49, -36, 0], [1, -30, 273, -820, 576, 0], [1, -55, 1023, -7645, 21076, -14400, 0], [1, -91, 3003, -44473, 296296, -773136, 518400, 0], [1, -140, 7462, -191620, 2475473, -15291640, 38402064, -25401600, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 4, -5, 1], [0, -36, 49, -14, 1], [0, 576, -820, 273, -30, 1], [0, -14400, 21076, -7645, 1023, -55, 1], [0, 518400, -773136, 296296, -44473, 3003, -91, 1], [0, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -5, 0], [1, -14, 4], [1, -30, 49, 0], [1, -55, 273, -36], [1, -91, 1023, -820, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -4, 0, 0], [1, -13, 36, 0, 0], [1, -29, 244, -576, 0, 0], [1, -54, 969, -6676, 14400, 0, 0], [1, -90, 2913, -41560, 254736, -518400, 0, 0], [1, -139, 7323, -184297, 2291176, -13000464, 25401600, 0, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [0, 0, -4, 1], [0, 0, 36, -13, 1], [0, 0, -576, 244, -29, 1], [0, 0, 14400, -6676, 969, -54, 1], [0, 0, -518400, 254736, -41560, 2913, -90, 1], [0, 0, 25401600, -13000464, 2291176, -184297, 7323, -139, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 4, -1, 0], [0, -36, 13, -1, 0], [0, 576, -244, 29, -1, 0], [0, -14400, 6676, -969, 54, -1, 0], [0, 518400, -254736, 41560, -2913, 90, -1, 0], [0, -25401600, 13000464, -2291176, 184297, -7323, 139, -1, 0]] | + +CentralSet:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 5, 50, 850, 22100, 817700, 40885000] | +| OddSum | [0, 0, -1, -5, -50, -850, -22100, -817700, -40885000] | +| AltSum | [1, 1, 2, 10, 100, 1700, 44200, 1635400, 81770000] | +| AccSum | [1, 2, 1, -3, 24, -360, 8640, -302400, 14515200] | +| AccRevSum | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | +| AntiDiagSum | [1, 1, 1, 0, -4, -9, 20, 183, 113] | + +CentralSet:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 20, 1764, 10745280, 326939342400, 352300447699200, 27645512323747439317996800] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 5, 49, 820, 21076, 773136, 38402064] | +| ColMiddle | [1, 1, -1, -5, 49, 273, -7645, -44473, 2475473] | +| ColECenter | [1, -1, 49, -7645, 2475473] | +| ColOCenter | [1, -5, 273, -44473] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, 11, -142, 2898, -87552, 3686400, -206582400] | +| TransNat0 | [0, 0, -1, 3, -24, 360, -8640, 302400, -14515200] | +| TransNat1 | [1, 1, -1, 3, -24, 360, -8640, 302400, -14515200] | + +CentralSet:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ CentralSetInvRev Diagonals as rows | DiagRow7 | [1, -140, 16422, -1999370, 268880381, -40796457506, 7026231453265, -1373222414339685, 303626807076050640]| | DiagRow8 | [1, -204, 32946, -5293970, 909450751, -171757365650, 36053226248115, -8439654758970225, 2202549127844351265]| -CentralSetInvRev Diagonals as columns +CentralSet:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ CentralSetInvRev Diagonals as columns | DiagCol7 | [0, -25401600, -2483133696, -84865562640, -1593719752240, -19967312312156, -185789298737900, -1373222414339685, -8439654758970225] | | DiagCol8 | [0, 1625702400, 202759531776, 8689315795776, 201529405816816, 3076822378767280, 34475213865472380, 303626807076050640, 2202549127844351265] | -CentralSetInvRev Polynomial values as rows +CentralSet:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ CentralSetInvRev Polynomial values as rows | PolyRow7 | [1, 0, 12834031, 212867512, 1404728325, 5863650496, 18549065875, 48691443816, 111752378857] | | PolyRow8 | [1, 0, -1244901007, -31078656752, -273922023375, -1430730721024, -5434876301375, -16652473785072, -43695180133087] | -CentralSetInvRev Polynomial values as columns +CentralSet:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/ChebyshevS.md b/data/md/ChebyshevS.md index a385b79..e06a820 100644 --- a/data/md/ChebyshevS.md +++ b/data/md/ChebyshevS.md @@ -14,45 +14,41 @@ ChebyshevS Triangle view | Row7 | [0, -4, 0, 10, 0, -6, 0, 1] | | Row8 | [1, 0, -10, 0, 15, 0, -7, 0, 1] | -ChebyshevS Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 0, 3, 0, -4, 0, 1, -1, 0, 6, 0, -5, 0, 1, 0, -4, 0, 10, 0, -6, 0, 1, 1, 0, -10, 0, 15, 0, -7, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1] | -| TriAcc | [1, 0, 1, -1, -1, 0, 0, -2, -2, -1, 1, 1, -2, -2, -1, 0, 3, 3, -1, -1, 0, -1, -1, 5, 5, 0, 0, 1, 0, -4, -4, 6, 6, 0, 0, 1, 1, 1, -9, -9, 6, 6, -1, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, -1, -1, -2, -2, 0, -1, -2, -2, 1, 1, 0, -1, -1, 3, 3, 0, 1, 0, 0, 5, 5, -1, -1, 1, 0, 0, 6, 6, -4, -4, 0, 0, -1, -1, 6, 6, -9, -9, 1, 1] | -| TriAccRev | [1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -1, 1, 1, -3, -3, 0, 0, 1, 1, -4, -4, 2, 2, 1, 1, 1, -5, -5, 5, 5, 1, 1, 1, 1, -6, -6, 9, 9, -1, -1, 0] | -| TriDiag | [1, 0, -1, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 3, -3, 1, 0, 0, 0, 0] | +ChebyshevS Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -2, 0, 1], [1, 0, -3, 0, 1], [0, 3, 0, -4, 0, 1], [-1, 0, 6, 0, -5, 0, 1], [0, -4, 0, 10, 0, -6, 0, 1], [1, 0, -10, 0, 15, 0, -7, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -2, 0], [1, 0, -3, 0, 1], [1, 0, -4, 0, 3, 0], [1, 0, -5, 0, 6, 0, -1], [1, 0, -6, 0, 10, 0, -4, 0], [1, 0, -7, 0, 15, 0, -10, 0, 1]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [1, -2, 1], [0, 0, 0], [-1, 3, -3, 1], [0, 0, 0, 0], [1, -4, 6, -4, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -2, -2, -1], [1, 1, -2, -2, -1], [0, 3, 3, -1, -1, 0], [-1, -1, 5, 5, 0, 0, 1], [0, -4, -4, 6, 6, 0, 0, 1], [1, 1, -9, -9, 6, 6, -1, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-1, -2, -2, 0], [-1, -2, -2, 1, 1], [0, -1, -1, 3, 3, 0], [1, 0, 0, 5, 5, -1, -1], [1, 0, 0, 6, 6, -4, -4, 0], [0, -1, -1, 6, 6, -9, -9, 1, 1]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -1, -1], [1, 1, -2, -2, -1], [1, 1, -3, -3, 0, 0], [1, 1, -4, -4, 2, 2, 1], [1, 1, -5, -5, 5, 5, 1, 1], [1, 1, -6, -6, 9, 9, -1, -1, 0]] | ChebyshevS Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| EvenSum | [1, 0, 0, 0, -1, 0, 1, 0, 0] | -| OddSum | [0, 1, 0, -1, 0, 0, 0, 1, 0] | -| AltSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | -| AccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| AccRevSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| RevAccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| DiagSum | [1, 0, 0, 0, 0, 0, 0, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| EvenSum | [1, 0, 0, 0, -1, 0, 1, 0, 0] | +| OddSum | [0, 1, 0, -1, 0, 0, 0, 1, 0] | +| AltSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | +| AccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | +| AccRevSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | +| AntiDiagSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | ChebyshevS Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -3, 12, 30, 60, 210] | -| RowGcd | [1, 1, 1, -2, -3, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | -| Middle | [1, 0, 0, -2, -3, 0, 0, 10, 15] | -| Central | [1, 0, -3, 0, 15] | -| LeftSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, -7, 5, 33, 13, -119, -171] | -| NegHalf | [1, 1, -3, -7, 5, 33, 13, -119, -171] | -| Bin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| InvBin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| TransSqrs | [0, 1, 4, 7, 4, -8, -20, -15, 12] | -| TransNat0 | [0, 1, 2, 1, -2, -4, -2, 3, 6] | -| TransNat1 | [1, 2, 2, 0, -3, -4, -1, 4, 6] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 3, 12, 30, 60, 210] | +| RowGcd | [1, 1, 1, 2, 3, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | +| ColMiddle | [1, 0, 0, -2, -3, 0, 0, 10, 15] | +| ColECenter | [1, 0, -3, 0, 15] | +| ColOCenter | [0, -2, 0, 10] | +| ColLeft | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 7, 4, -8, -20, -15, 12] | +| TransNat0 | [0, 1, 2, 1, -2, -4, -2, 3, 6] | +| TransNat1 | [1, 2, 2, 0, -3, -4, -1, 4, 6] | ChebyshevS Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ ChebyshevS Polynomial values as columns | PolyCol7 | [1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784] | | PolyCol8 | [1, 8, 63, 496, 3905, 30744, 242047, 1905632, 15003009] | -# ChebyshevSInv -True +# ChebyshevS:Inv +[] -ChebyshevSInv Triangle view +ChebyshevS:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ ChebyshevSInv Triangle view | Row7 | [0, 14, 0, 14, 0, 6, 0, 1] | | Row8 | [14, 0, 28, 0, 20, 0, 7, 0, 1] | -ChebyshevSInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 1, 0, 4, 0, 5, 0, 1, 0, 5, 0, 9, 0, 5, 1, 0, 6, 0, 14, 0, 14, 0, 1, 0, 7, 0, 20, 0, 28, 0, 14] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 2, 2, 3, 2, 2, 5, 5, 6, 0, 5, 5, 9, 9, 10, 5, 5, 14, 14, 19, 19, 20, 0, 14, 14, 28, 28, 34, 34, 35, 14, 14, 42, 42, 62, 62, 69, 69, 70] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 3, 2, 2, 0, 6, 5, 5, 2, 2, 10, 9, 9, 5, 5, 0, 20, 19, 19, 14, 14, 5, 5, 35, 34, 34, 28, 28, 14, 14, 0, 70, 69, 69, 62, 62, 42, 42, 14, 14] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70] | -| TriDiag | [1, 0, 1, 1, 0, 0, 2, 2, 1, 0, 0, 0, 5, 5, 3, 1, 0, 0, 0, 0] | - -ChebyshevSInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| EvenSum | [1, 0, 2, 0, 6, 0, 20, 0, 70] | -| OddSum | [0, 1, 0, 3, 0, 10, 0, 35, 0] | -| AltSum | [1, -1, 2, -3, 6, -10, 20, -35, 70] | -| AccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| AccRevSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| RevAccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| DiagSum | [1, 0, 2, 0, 5, 0, 14, 0] | - -ChebyshevSInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | -| Middle | [1, 0, 0, 2, 3, 0, 0, 14, 20] | -| Central | [1, 0, 3, 0, 20] | -| LeftSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 9, 45, 97, 485, 1145, 5725] | -| NegHalf | [1, 1, 5, 9, 45, 97, 485, 1145, 5725] | -| Bin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| InvBin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| TransSqrs | [0, 1, 4, 11, 28, 66, 152, 339, 748] | -| TransNat0 | [0, 1, 2, 5, 10, 22, 44, 93, 186] | -| TransNat1 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | - -ChebyshevSInv Diagonals as rows +ChebyshevS:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 2, 0, 1], [2, 0, 3, 0, 1], [0, 5, 0, 4, 0, 1], [5, 0, 9, 0, 5, 0, 1], [0, 14, 0, 14, 0, 6, 0, 1], [14, 0, 28, 0, 20, 0, 7, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 2, 0], [1, 0, 3, 0, 2], [1, 0, 4, 0, 5, 0], [1, 0, 5, 0, 9, 0, 5], [1, 0, 6, 0, 14, 0, 14, 0], [1, 0, 7, 0, 20, 0, 28, 0, 14]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [2, 2, 1], [0, 0, 0], [5, 5, 3, 1], [0, 0, 0, 0], [14, 14, 9, 4, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 2, 2, 3], [2, 2, 5, 5, 6], [0, 5, 5, 9, 9, 10], [5, 5, 14, 14, 19, 19, 20], [0, 14, 14, 28, 28, 34, 34, 35], [14, 14, 42, 42, 62, 62, 69, 69, 70]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [3, 2, 2, 0], [6, 5, 5, 2, 2], [10, 9, 9, 5, 5, 0], [20, 19, 19, 14, 14, 5, 5], [35, 34, 34, 28, 28, 14, 14, 0], [70, 69, 69, 62, 62, 42, 42, 14, 14]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 3, 3], [1, 1, 4, 4, 6], [1, 1, 5, 5, 10, 10], [1, 1, 6, 6, 15, 15, 20], [1, 1, 7, 7, 21, 21, 35, 35], [1, 1, 8, 8, 28, 28, 56, 56, 70]] | + +ChebyshevS:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| EvenSum | [1, 0, 2, 0, 6, 0, 20, 0, 70] | +| OddSum | [0, 1, 0, 3, 0, 10, 0, 35, 0] | +| AltSum | [1, -1, 2, -3, 6, -10, 20, -35, 70] | +| AccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | +| AccRevSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| AntiDiagSum | [1, 0, 2, 0, 5, 0, 14, 0, 42] | + +ChebyshevS:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | +| ColMiddle | [1, 0, 0, 2, 3, 0, 0, 14, 20] | +| ColECenter | [1, 0, 3, 0, 20] | +| ColOCenter | [0, 2, 0, 14] | +| ColLeft | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 11, 28, 66, 152, 339, 748] | +| TransNat0 | [0, 1, 2, 5, 10, 22, 44, 93, 186] | +| TransNat1 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | + +ChebyshevS:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ ChebyshevSInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [14, 42, 90, 165, 275, 429, 637, 910, 1260]| -ChebyshevSInv Diagonals as columns +ChebyshevS:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 1, 0, 2, 0, 5, 0, 14] | @@ -188,7 +180,7 @@ ChebyshevSInv Diagonals as columns | DiagCol7 | [1, 0, 8, 0, 44, 0, 208, 0, 910] | | DiagCol8 | [1, 0, 9, 0, 54, 0, 273, 0, 1260] | -ChebyshevSInv Polynomial values as rows +ChebyshevS:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ ChebyshevSInv Polynomial values as rows | PolyRow7 | [0, 35, 460, 4065, 23480, 98695, 329700, 929285, 2301040] | | PolyRow8 | [14, 70, 1150, 13550, 99790, 513214, 2033150, 6637750, 18695950] | -ChebyshevSInv Polynomial values as columns +ChebyshevS:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 1, 0, 2, 0, 5, 0, 14] | @@ -214,10 +206,10 @@ ChebyshevSInv Polynomial values as columns | PolyCol7 | [1, 7, 50, 357, 2550, 18214, 130100, 929285, 6637750] | | PolyCol8 | [1, 8, 65, 528, 4290, 34856, 283205, 2301040, 18695950] | -# ChebyshevSRev -True +# ChebyshevS:Rev +[] -ChebyshevSRev Triangle view +ChebyshevS:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ ChebyshevSRev Triangle view | Row7 | [1, 0, -6, 0, 10, 0, -4, 0] | | Row8 | [1, 0, -7, 0, 15, 0, -10, 0, 1] | -ChebyshevSRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1] | -| TriRev | [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 0, 3, 0, -4, 0, 1, -1, 0, 6, 0, -5, 0, 1, 0, -4, 0, 10, 0, -6, 0, 1, 1, 0, -10, 0, 15, 0, -7, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -1, 1, 1, -3, -3, 0, 0, 1, 1, -4, -4, 2, 2, 1, 1, 1, -5, -5, 5, 5, 1, 1, 1, 1, -6, -6, 9, 9, -1, -1, 0] | -| TriRevAcc | [1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -2, -2, 1, 1, 0, 0, -3, -3, 1, 1, 1, 2, 2, -4, -4, 1, 1, 1, 1, 5, 5, -5, -5, 1, 1, 0, -1, -1, 9, 9, -6, -6, 1, 1] | -| TriAccRev | [1, 0, 1, -1, -1, 0, 0, -2, -2, -1, 1, 1, -2, -2, -1, 0, 3, 3, -1, -1, 0, -1, -1, 5, 5, 0, 0, 1, 0, -4, -4, 6, 6, 0, 0, 1, 1, 1, -9, -9, 6, 6, -1, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -2, 1, 0, -3, 0, 1, 0, -4, 0] | - -ChebyshevSRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| EvenSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| AccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| AccRevSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | -| RevAccSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | -| DiagSum | [1, 1, 1, 1, 0, -1, -2, -3] | - -ChebyshevSRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -3, 12, 30, 60, 210] | -| RowGcd | [1, 1, 1, -2, -3, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | -| Middle | [1, 1, 0, 0, -3, -4, 0, 0, 15] | -| Central | [1, 0, -3, 0, 15] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| PosHalf | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| NegHalf | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| Bin | [1, 1, 0, -5, -16, -24, 15, 197, 576] | -| InvBin | [1, -1, 0, 5, -16, 24, 15, -197, 576] | -| TransSqrs | [0, 0, -4, -8, 4, 32, 40, -8, -84] | -| TransNat0 | [0, 0, -2, -4, -2, 4, 8, 4, -6] | -| TransNat1 | [1, 1, -2, -5, -3, 4, 9, 5, -6] | - -ChebyshevSRev Diagonals as rows +ChebyshevS:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, -1], [1, 0, -2, 0], [1, 0, -3, 0, 1], [1, 0, -4, 0, 3, 0], [1, 0, -5, 0, 6, 0, -1], [1, 0, -6, 0, 10, 0, -4, 0], [1, 0, -7, 0, 15, 0, -10, 0, 1]] | +| RevTabl | [[1], [0, 1], [-1, 0, 1], [0, -2, 0, 1], [1, 0, -3, 0, 1], [0, 3, 0, -4, 0, 1], [-1, 0, 6, 0, -5, 0, 1], [0, -4, 0, 10, 0, -6, 0, 1], [1, 0, -10, 0, 15, 0, -7, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, -1], [1, 0, -2], [1, 0, -3, 0], [1, 0, -4, 0], [1, 0, -5, 0, 1]] | +| AccTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -1, -1], [1, 1, -2, -2, -1], [1, 1, -3, -3, 0, 0], [1, 1, -4, -4, 2, 2, 1], [1, 1, -5, -5, 5, 5, 1, 1], [1, 1, -6, -6, 9, 9, -1, -1, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 1, 1], [-1, -1, 1, 1], [-1, -2, -2, 1, 1], [0, 0, -3, -3, 1, 1], [1, 2, 2, -4, -4, 1, 1], [1, 1, 5, 5, -5, -5, 1, 1], [0, -1, -1, 9, 9, -6, -6, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 0], [0, -2, -2, -1], [1, 1, -2, -2, -1], [0, 3, 3, -1, -1, 0], [-1, -1, 5, 5, 0, 0, 1], [0, -4, -4, 6, 6, 0, 0, 1], [1, 1, -9, -9, 6, 6, -1, -1, 0]] | + +ChebyshevS:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| EvenSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 0, -1, -1, 0, 1, 1, 0] | +| AccSum | [1, 2, 2, 0, -3, -4, -1, 4, 6] | +| AccRevSum | [1, 1, -2, -5, -3, 4, 9, 5, -6] | +| AntiDiagSum | [1, 1, 1, 1, 0, -1, -2, -3, -3] | + +ChebyshevS:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 3, 12, 30, 60, 210] | +| RowGcd | [1, 1, 1, 2, 3, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 4, 6, 10, 15] | +| ColMiddle | [1, 1, 0, 0, -3, -4, 0, 0, 15] | +| ColECenter | [1, 0, -3, 0, 15] | +| ColOCenter | [1, 0, -4, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| TransSqrs | [0, 0, -4, -8, 4, 32, 40, -8, -84] | +| TransNat0 | [0, 0, -2, -4, -2, 4, 8, 4, -6] | +| TransNat1 | [1, 1, -2, -5, -3, 4, 9, 5, -6] | + +ChebyshevS:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 1, 0, -1, 0, 1]| @@ -283,7 +271,7 @@ ChebyshevSRev Diagonals as rows | DiagRow7 | [1, 0, -8, 0, 36, 0, -120, 0, 330]| | DiagRow8 | [1, 0, -9, 0, 45, 0, -165, 0, 495]| -ChebyshevSRev Diagonals as columns +ChebyshevS:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ ChebyshevSRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, 5, 15, 35, 70, 126, 210, 330, 495] | -ChebyshevSRev Polynomial values as rows +ChebyshevS:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ ChebyshevSRev Polynomial values as rows | PolyRow7 | [1, 1, -119, -2159, -13919, -56399, -173879, -446879, -1007999] | | PolyRow8 | [1, 0, -171, 424, 28305, 243576, 1232245, 4623984, 14216769] | -ChebyshevSRev Polynomial values as columns +ChebyshevS:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ ChebyshevSRev Polynomial values as columns | PolyCol7 | [1, 1, -48, -97, 2255, 7008, -103487, -446879, 4623984] | | PolyCol8 | [1, 1, -63, -127, 3905, 12033, -237887, -1007999, 14216769] | -# ChebyshevSInvRev -True +# ChebyshevS:Inv:Rev +[] -ChebyshevSInvRev Triangle view +ChebyshevS:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ ChebyshevSInvRev Triangle view | Row7 | [1, 0, 6, 0, 14, 0, 14, 0] | | Row8 | [1, 0, 7, 0, 20, 0, 28, 0, 14] | -ChebyshevSInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 1, 0, 4, 0, 5, 0, 1, 0, 5, 0, 9, 0, 5, 1, 0, 6, 0, 14, 0, 14, 0, 1, 0, 7, 0, 20, 0, 28, 0, 14] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 14, 0, 14, 0, 6, 0, 1, 14, 0, 28, 0, 20, 0, 7, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 4, 4, 1, 1, 10, 10, 5, 5, 1, 1, 20, 15, 15, 6, 6, 1, 1, 35, 35, 21, 21, 7, 7, 1, 1, 70, 56, 56, 28, 28, 8, 8, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 2, 2, 3, 2, 2, 5, 5, 6, 0, 5, 5, 9, 9, 10, 5, 5, 14, 14, 19, 19, 20, 0, 14, 14, 28, 28, 34, 34, 35, 14, 14, 42, 42, 62, 62, 69, 69, 70] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0] | - -ChebyshevSInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| EvenSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | -| AccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| AccRevSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RevAccSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | -| DiagSum | [1, 1, 1, 1, 2, 3, 4, 5] | - -ChebyshevSInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | -| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | -| Middle | [1, 1, 0, 0, 3, 4, 0, 0, 20] | -| Central | [1, 0, 3, 0, 20] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| PosHalf | [1, 2, 5, 12, 30, 74, 185, 460, 1150] | -| NegHalf | [1, -2, 5, -12, 30, -74, 185, -460, 1150] | -| Bin | [1, 1, 2, 7, 21, 66, 216, 715, 2395] | -| InvBin | [1, -1, 2, -7, 21, -66, 216, -715, 2395] | -| TransSqrs | [0, 0, 4, 8, 44, 96, 344, 752, 2252] | -| TransNat0 | [0, 0, 2, 4, 14, 28, 76, 152, 374] | -| TransNat1 | [1, 1, 4, 7, 20, 38, 96, 187, 444] | - -ChebyshevSInvRev Diagonals as rows +ChebyshevS:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 2, 0], [1, 0, 3, 0, 2], [1, 0, 4, 0, 5, 0], [1, 0, 5, 0, 9, 0, 5], [1, 0, 6, 0, 14, 0, 14, 0], [1, 0, 7, 0, 20, 0, 28, 0, 14]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 2, 0, 1], [2, 0, 3, 0, 1], [0, 5, 0, 4, 0, 1], [5, 0, 9, 0, 5, 0, 1], [0, 14, 0, 14, 0, 6, 0, 1], [14, 0, 28, 0, 20, 0, 7, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 2], [1, 0, 3, 0], [1, 0, 4, 0], [1, 0, 5, 0, 2]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 3, 3], [1, 1, 4, 4, 6], [1, 1, 5, 5, 10, 10], [1, 1, 6, 6, 15, 15, 20], [1, 1, 7, 7, 21, 21, 35, 35], [1, 1, 8, 8, 28, 28, 56, 56, 70]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [3, 3, 1, 1], [6, 4, 4, 1, 1], [10, 10, 5, 5, 1, 1], [20, 15, 15, 6, 6, 1, 1], [35, 35, 21, 21, 7, 7, 1, 1], [70, 56, 56, 28, 28, 8, 8, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 2, 2, 3], [2, 2, 5, 5, 6], [0, 5, 5, 9, 9, 10], [5, 5, 14, 14, 19, 19, 20], [0, 14, 14, 28, 28, 34, 34, 35], [14, 14, 42, 42, 62, 62, 69, 69, 70]] | + +ChebyshevS:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| EvenSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 3, 6, 10, 20, 35, 70] | +| AccSum | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| AccRevSum | [1, 1, 4, 7, 20, 38, 96, 187, 444] | +| AntiDiagSum | [1, 1, 1, 1, 2, 3, 4, 5, 8] | + +ChebyshevS:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 20, 45, 42, 140] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 2, 1] | +| RowMax | [1, 1, 1, 2, 3, 5, 9, 14, 28] | +| ColMiddle | [1, 1, 0, 0, 3, 4, 0, 0, 20] | +| ColECenter | [1, 0, 3, 0, 20] | +| ColOCenter | [1, 0, 4, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| TransSqrs | [0, 0, 4, 8, 44, 96, 344, 752, 2252] | +| TransNat0 | [0, 0, 2, 4, 14, 28, 76, 152, 374] | +| TransNat1 | [1, 1, 4, 7, 20, 38, 96, 187, 444] | + +ChebyshevS:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 0, 2, 0, 5, 0, 14]| @@ -391,7 +375,7 @@ ChebyshevSInvRev Diagonals as rows | DiagRow7 | [1, 0, 8, 0, 44, 0, 208, 0, 910]| | DiagRow8 | [1, 0, 9, 0, 54, 0, 273, 0, 1260]| -ChebyshevSInvRev Diagonals as columns +ChebyshevS:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ ChebyshevSInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [14, 42, 90, 165, 275, 429, 637, 910, 1260] | -ChebyshevSInvRev Polynomial values as rows +ChebyshevS:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ ChebyshevSInvRev Polynomial values as rows | PolyRow7 | [1, 35, 1145, 11395, 61025, 227651, 671545, 1680995, 3727745] | | PolyRow8 | [1, 70, 5725, 113950, 1037425, 5918926, 24847165, 84049750, 242303425] | -ChebyshevSInvRev Polynomial values as columns +ChebyshevS:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/ChebyshevT.md b/data/md/ChebyshevT.md index 0aabc66..8de28d8 100644 --- a/data/md/ChebyshevT.md +++ b/data/md/ChebyshevT.md @@ -1,5 +1,5 @@ # ChebyshevT -['A039991', 'A053120', 'A081265'] +['A053120', 'A039991', 'A081265'] ChebyshevT Triangle view | Row | Seq | @@ -14,45 +14,41 @@ ChebyshevT Triangle view | Row7 | [0, -7, 0, 56, 0, -112, 0, 64] | | Row8 | [1, 0, -32, 0, 160, 0, -256, 0, 128] | -ChebyshevT Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 2, 0, -3, 0, 4, 1, 0, -8, 0, 8, 0, 5, 0, -20, 0, 16, -1, 0, 18, 0, -48, 0, 32, 0, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128] | -| TriRev | [1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1] | -| TriAcc | [1, 0, 1, -1, -1, 1, 0, -3, -3, 1, 1, 1, -7, -7, 1, 0, 5, 5, -15, -15, 1, -1, -1, 17, 17, -31, -31, 1, 0, -7, -7, 49, 49, -63, -63, 1, 1, 1, -31, -31, 129, 129, -127, -127, 1] | -| TriRevAcc | [1, 1, 0, 1, -1, -1, 1, -3, -3, 0, 1, -7, -7, 1, 1, 1, -15, -15, 5, 5, 0, 1, -31, -31, 17, 17, -1, -1, 1, -63, -63, 49, 49, -7, -7, 0, 1, -127, -127, 129, 129, -31, -31, 1, 1] | -| TriAccRev | [1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 8, 0, 0, 1, 16, 16, -4, -4, 1, 1, 32, 32, -16, -16, 2, 2, 1, 64, 64, -48, -48, 8, 8, 1, 1, 128, 128, -128, -128, 32, 32, 0, 0, 1] | -| TriDiag | [1, 0, -1, 1, 0, 0, 1, -3, 2, 0, 0, 0, -1, 5, -8, 4, 0, 0, 0, 0] | +ChebyshevT Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 2], [0, -3, 0, 4], [1, 0, -8, 0, 8], [0, 5, 0, -20, 0, 16], [-1, 0, 18, 0, -48, 0, 32], [0, -7, 0, 56, 0, -112, 0, 64], [1, 0, -32, 0, 160, 0, -256, 0, 128]] | +| RevTabl | [[1], [1, 0], [2, 0, -1], [4, 0, -3, 0], [8, 0, -8, 0, 1], [16, 0, -20, 0, 5, 0], [32, 0, -48, 0, 18, 0, -1], [64, 0, -112, 0, 56, 0, -7, 0], [128, 0, -256, 0, 160, 0, -32, 0, 1]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [1, -3, 2], [0, 0, 0], [-1, 5, -8, 4], [0, 0, 0, 0], [1, -7, 18, -20, 8]] | +| AccTabl | [[1], [0, 1], [-1, -1, 1], [0, -3, -3, 1], [1, 1, -7, -7, 1], [0, 5, 5, -15, -15, 1], [-1, -1, 17, 17, -31, -31, 1], [0, -7, -7, 49, 49, -63, -63, 1], [1, 1, -31, -31, 129, 129, -127, -127, 1]] | +| RevAccTabl | [[1], [1, 0], [1, -1, -1], [1, -3, -3, 0], [1, -7, -7, 1, 1], [1, -15, -15, 5, 5, 0], [1, -31, -31, 17, 17, -1, -1], [1, -63, -63, 49, 49, -7, -7, 0], [1, -127, -127, 129, 129, -31, -31, 1, 1]] | +| AccRevTabl | [[1], [1, 1], [2, 2, 1], [4, 4, 1, 1], [8, 8, 0, 0, 1], [16, 16, -4, -4, 1, 1], [32, 32, -16, -16, 2, 2, 1], [64, 64, -48, -48, 8, 8, 1, 1], [128, 128, -128, -128, 32, 32, 0, 0, 1]] | ChebyshevT Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| EvenSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -| OddSum | [0, 1, 0, 1, 0, 1, 0, 1, 0] | -| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| AccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| AccRevSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| RevAccSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | -| DiagSum | [1, 0, 0, 0, 0, 0, 0, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| EvenSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | +| OddSum | [0, 1, 0, 1, 0, 1, 0, 1, 0] | +| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| AccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | +| AccRevSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | +| AntiDiagSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | ChebyshevT Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 8, 80, 288, 448, 1280] | -| RowGcd | [1, 1, 2, 1, 8, 1, 2, 1, 32] | -| RowMax | [1, 1, 2, 4, 8, 20, 48, 112, 256] | -| Middle | [1, 0, 0, -3, -8, 0, 0, 56, 160] | -| Central | [1, 0, -8, 0, 160] | -| LeftSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| RightSide | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| PosHalf | [1, 1, -2, -8, -8, 16, 64, 64, -128] | -| NegHalf | [1, 1, -2, -8, -8, 16, 64, 64, -128] | -| Bin | [1, 1, 1, -5, -39, -159, -419, -377, 3265] | -| InvBin | [1, 1, 1, -5, -39, -159, -419, -377, 3265] | -| TransSqrs | [0, 1, 8, 33, 96, 225, 456, 833, 1408] | -| TransNat0 | [0, 1, 4, 9, 16, 25, 36, 49, 64] | -| TransNat1 | [1, 2, 5, 10, 17, 26, 37, 50, 65] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 8, 80, 288, 448, 1280] | +| RowGcd | [1, 1, 2, 1, 8, 1, 2, 1, 32] | +| RowMax | [1, 1, 2, 4, 8, 20, 48, 112, 256] | +| ColMiddle | [1, 0, 0, -3, -8, 0, 0, 56, 160] | +| ColECenter | [1, 0, -8, 0, 160] | +| ColOCenter | [0, -3, 0, 56] | +| ColLeft | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| ColRight | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| TransSqrs | [0, 1, 8, 33, 96, 225, 456, 833, 1408] | +| TransNat0 | [0, 1, 4, 9, 16, 25, 36, 49, 64] | +| TransNat1 | [1, 2, 5, 10, 17, 26, 37, 50, 65] | ChebyshevT Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ ChebyshevT Polynomial values as columns | PolyCol7 | [1, 7, 97, 1351, 18817, 262087, 3650401, 50843527, 708158977] | | PolyCol8 | [1, 8, 127, 2024, 32257, 514088, 8193151, 130576328, 2081028097] | -# ChebyshevTRev -True +# ChebyshevT:Rev +[] -ChebyshevTRev Triangle view +ChebyshevT:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ ChebyshevTRev Triangle view | Row7 | [64, 0, -112, 0, 56, 0, -7, 0] | | Row8 | [128, 0, -256, 0, 160, 0, -32, 0, 1] | -ChebyshevTRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1] | -| TriRev | [1, 0, 1, -1, 0, 2, 0, -3, 0, 4, 1, 0, -8, 0, 8, 0, 5, 0, -20, 0, 16, -1, 0, 18, 0, -48, 0, 32, 0, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128] | -| TriAcc | [1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 8, 0, 0, 1, 16, 16, -4, -4, 1, 1, 32, 32, -16, -16, 2, 2, 1, 64, 64, -48, -48, 8, 8, 1, 1, 128, 128, -128, -128, 32, 32, 0, 0, 1] | -| TriRevAcc | [1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 1, 0, 0, 8, 8, 1, 1, -4, -4, 16, 16, 1, 2, 2, -16, -16, 32, 32, 1, 1, 8, 8, -48, -48, 64, 64, 1, 0, 0, 32, 32, -128, -128, 128, 128] | -| TriAccRev | [1, 0, 1, -1, -1, 1, 0, -3, -3, 1, 1, 1, -7, -7, 1, 0, 5, 5, -15, -15, 1, -1, -1, 17, 17, -31, -31, 1, 0, -7, -7, 49, 49, -63, -63, 1, 1, 1, -31, -31, 129, 129, -127, -127, 1] | -| TriDiag | [1, 1, 2, 0, 4, 0, 8, 0, -1, 16, 0, -3, 32, 0, -8, 0, 64, 0, -20, 0] | - -ChebyshevTRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| AccSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | -| AccRevSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | -| RevAccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| DiagSum | [1, 1, 2, 4, 7, 13, 24, 44] | - -ChebyshevTRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 8, 80, 288, 448, 1280] | -| RowGcd | [1, 1, 2, 1, 8, 1, 2, 1, 32] | -| RowMax | [1, 1, 2, 4, 8, 20, 48, 112, 256] | -| Middle | [1, 1, 0, 0, -8, -20, 0, 0, 160] | -| Central | [1, 0, -8, 0, 160] | -| LeftSide | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| RightSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| PosHalf | [1, 2, 7, 26, 97, 362, 1351, 5042, 18817] | -| NegHalf | [1, -2, 7, -26, 97, -362, 1351, -5042, 18817] | -| Bin | [1, 1, 1, -5, -39, -159, -419, -377, 3265] | -| InvBin | [1, -1, 1, 5, -39, 159, -419, 377, 3265] | -| TransSqrs | [0, 0, -4, -12, -16, 0, 60, 196, 448] | -| TransNat0 | [0, 0, -2, -6, -12, -20, -30, -42, -56] | -| TransNat1 | [1, 1, -1, -5, -11, -19, -29, -41, -55] | - -ChebyshevTRev Diagonals as rows +ChebyshevT:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 0, -1], [4, 0, -3, 0], [8, 0, -8, 0, 1], [16, 0, -20, 0, 5, 0], [32, 0, -48, 0, 18, 0, -1], [64, 0, -112, 0, 56, 0, -7, 0], [128, 0, -256, 0, 160, 0, -32, 0, 1]] | +| RevTabl | [[1], [0, 1], [-1, 0, 2], [0, -3, 0, 4], [1, 0, -8, 0, 8], [0, 5, 0, -20, 0, 16], [-1, 0, 18, 0, -48, 0, 32], [0, -7, 0, 56, 0, -112, 0, 64], [1, 0, -32, 0, 160, 0, -256, 0, 128]] | +| AntiDiag | [[1], [1], [2, 0], [4, 0], [8, 0, -1], [16, 0, -3], [32, 0, -8, 0], [64, 0, -20, 0], [128, 0, -48, 0, 1]] | +| AccTabl | [[1], [1, 1], [2, 2, 1], [4, 4, 1, 1], [8, 8, 0, 0, 1], [16, 16, -4, -4, 1, 1], [32, 32, -16, -16, 2, 2, 1], [64, 64, -48, -48, 8, 8, 1, 1], [128, 128, -128, -128, 32, 32, 0, 0, 1]] | +| RevAccTabl | [[1], [1, 1], [1, 2, 2], [1, 1, 4, 4], [1, 0, 0, 8, 8], [1, 1, -4, -4, 16, 16], [1, 2, 2, -16, -16, 32, 32], [1, 1, 8, 8, -48, -48, 64, 64], [1, 0, 0, 32, 32, -128, -128, 128, 128]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 1], [0, -3, -3, 1], [1, 1, -7, -7, 1], [0, 5, 5, -15, -15, 1], [-1, -1, 17, 17, -31, -31, 1], [0, -7, -7, 49, 49, -63, -63, 1], [1, 1, -31, -31, 129, 129, -127, -127, 1]] | + +ChebyshevT:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AccSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | +| AccRevSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | +| AntiDiagSum | [1, 1, 2, 4, 7, 13, 24, 44, 81] | + +ChebyshevT:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 8, 80, 288, 448, 1280] | +| RowGcd | [1, 1, 2, 1, 8, 1, 2, 1, 32] | +| RowMax | [1, 1, 2, 4, 8, 20, 48, 112, 256] | +| ColMiddle | [1, 1, 0, 0, -8, -20, 0, 0, 160] | +| ColECenter | [1, 0, -8, 0, 160] | +| ColOCenter | [1, 0, -20, 0] | +| ColLeft | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| ColRight | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| TransSqrs | [0, 0, -4, -12, -16, 0, 60, 196, 448] | +| TransNat0 | [0, 0, -2, -6, -12, -20, -30, -42, -56] | +| TransNat1 | [1, 1, -1, -5, -11, -19, -29, -41, -55] | + +ChebyshevT:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 1, 0, -1, 0, 1]| @@ -175,7 +167,7 @@ ChebyshevTRev Diagonals as rows | DiagRow7 | [64, 0, -576, 0, 2816, 0, -9984, 0, 28800]| | DiagRow8 | [128, 0, -1280, 0, 6912, 0, -26880, 0, 84480]| -ChebyshevTRev Diagonals as columns +ChebyshevT:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 4, 8, 16, 32, 64, 128] | @@ -188,7 +180,7 @@ ChebyshevTRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, 9, 50, 220, 840, 2912, 9408, 28800, 84480] | -ChebyshevTRev Polynomial values as rows +ChebyshevT:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ ChebyshevTRev Polynomial values as rows | PolyRow7 | [64, 1, 64, -1511, -16064, -77111, -257984, -694511, -1612736] | | PolyRow8 | [128, 1, -128, -5983, -28544, -15647, 384896, 2371777, 9027712] | -ChebyshevTRev Polynomial values as columns +ChebyshevT:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 4, 8, 16, 32, 64, 128] | diff --git a/data/md/ChebyshevU.md b/data/md/ChebyshevU.md index 0d0ae69..fb334a5 100644 --- a/data/md/ChebyshevU.md +++ b/data/md/ChebyshevU.md @@ -14,45 +14,41 @@ ChebyshevU Triangle view | Row7 | [0, -8, 0, 80, 0, -192, 0, 128] | | Row8 | [1, 0, -40, 0, 240, 0, -448, 0, 256] | -ChebyshevU Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256] | -| TriRev | [1, 2, 0, 4, 0, -1, 8, 0, -4, 0, 16, 0, -12, 0, 1, 32, 0, -32, 0, 6, 0, 64, 0, -80, 0, 24, 0, -1, 128, 0, -192, 0, 80, 0, -8, 0, 256, 0, -448, 0, 240, 0, -40, 0, 1] | -| TriAcc | [1, 0, 2, -1, -1, 3, 0, -4, -4, 4, 1, 1, -11, -11, 5, 0, 6, 6, -26, -26, 6, -1, -1, 23, 23, -57, -57, 7, 0, -8, -8, 72, 72, -120, -120, 8, 1, 1, -39, -39, 201, 201, -247, -247, 9] | -| TriRevAcc | [1, 2, 0, 3, -1, -1, 4, -4, -4, 0, 5, -11, -11, 1, 1, 6, -26, -26, 6, 6, 0, 7, -57, -57, 23, 23, -1, -1, 8, -120, -120, 72, 72, -8, -8, 0, 9, -247, -247, 201, 201, -39, -39, 1, 1] | -| TriAccRev | [1, 2, 2, 4, 4, 3, 8, 8, 4, 4, 16, 16, 4, 4, 5, 32, 32, 0, 0, 6, 6, 64, 64, -16, -16, 8, 8, 7, 128, 128, -64, -64, 16, 16, 8, 8, 256, 256, -192, -192, 48, 48, 8, 8, 9] | -| TriDiag | [1, 0, -1, 2, 0, 0, 1, -4, 4, 0, 0, 0, -1, 6, -12, 8, 0, 0, 0, 0] | +ChebyshevU Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 2], [-1, 0, 4], [0, -4, 0, 8], [1, 0, -12, 0, 16], [0, 6, 0, -32, 0, 32], [-1, 0, 24, 0, -80, 0, 64], [0, -8, 0, 80, 0, -192, 0, 128], [1, 0, -40, 0, 240, 0, -448, 0, 256]] | +| RevTabl | [[1], [2, 0], [4, 0, -1], [8, 0, -4, 0], [16, 0, -12, 0, 1], [32, 0, -32, 0, 6, 0], [64, 0, -80, 0, 24, 0, -1], [128, 0, -192, 0, 80, 0, -8, 0], [256, 0, -448, 0, 240, 0, -40, 0, 1]] | +| AntiDiag | [[1], [0], [-1, 2], [0, 0], [1, -4, 4], [0, 0, 0], [-1, 6, -12, 8], [0, 0, 0, 0], [1, -8, 24, -32, 16]] | +| AccTabl | [[1], [0, 2], [-1, -1, 3], [0, -4, -4, 4], [1, 1, -11, -11, 5], [0, 6, 6, -26, -26, 6], [-1, -1, 23, 23, -57, -57, 7], [0, -8, -8, 72, 72, -120, -120, 8], [1, 1, -39, -39, 201, 201, -247, -247, 9]] | +| RevAccTabl | [[1], [2, 0], [3, -1, -1], [4, -4, -4, 0], [5, -11, -11, 1, 1], [6, -26, -26, 6, 6, 0], [7, -57, -57, 23, 23, -1, -1], [8, -120, -120, 72, 72, -8, -8, 0], [9, -247, -247, 201, 201, -39, -39, 1, 1]] | +| AccRevTabl | [[1], [2, 2], [4, 4, 3], [8, 8, 4, 4], [16, 16, 4, 4, 5], [32, 32, 0, 0, 6, 6], [64, 64, -16, -16, 8, 8, 7], [128, 128, -64, -64, 16, 16, 8, 8], [256, 256, -192, -192, 48, 48, 8, 8, 9]] | ChebyshevU Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| EvenSum | [1, 0, 3, 0, 5, 0, 7, 0, 9] | -| OddSum | [0, 2, 0, 4, 0, 6, 0, 8, 0] | -| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| AccSum | [1, 2, 1, -4, -15, -34, -63, -104, -159] | -| AccRevSum | [1, 2, 1, -4, -15, -34, -63, -104, -159] | -| RevAccSum | [1, 4, 11, 24, 45, 76, 119, 176, 249] | -| DiagSum | [1, 0, 1, 0, 1, 0, 1, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| EvenSum | [1, 0, 3, 0, 5, 0, 7, 0, 9] | +| OddSum | [0, 2, 0, 4, 0, 6, 0, 8, 0] | +| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| AccSum | [1, 2, 1, -4, -15, -34, -63, -104, -159] | +| AccRevSum | [1, 4, 11, 24, 45, 76, 119, 176, 249] | +| AntiDiagSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | ChebyshevU Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 8, 48, 96, 960, 1920, 26880] | -| RowGcd | [1, 2, 4, 4, 4, 2, 8, 8, 8] | -| RowMax | [1, 2, 4, 8, 16, 32, 80, 192, 448] | -| Middle | [1, 0, 0, -4, -12, 0, 0, 80, 240] | -| Central | [1, 0, -12, 0, 240] | -| LeftSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| RightSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| PosHalf | [1, 2, 0, -8, -16, 0, 64, 128, 0] | -| NegHalf | [1, 2, 0, -8, -16, 0, 64, 128, 0] | -| Bin | [1, 2, 3, -4, -55, -258, -777, -1160, 3393] | -| InvBin | [1, 2, 3, -4, -55, -258, -777, -1160, 3393] | -| TransSqrs | [0, 2, 16, 68, 208, 518, 1120, 2184, 3936] | -| TransNat0 | [0, 2, 8, 20, 40, 70, 112, 168, 240] | -| TransNat1 | [1, 4, 11, 24, 45, 76, 119, 176, 249] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 8, 48, 96, 960, 1920, 26880] | +| RowGcd | [1, 2, 4, 4, 4, 2, 8, 8, 8] | +| RowMax | [1, 2, 4, 8, 16, 32, 80, 192, 448] | +| ColMiddle | [1, 0, 0, -4, -12, 0, 0, 80, 240] | +| ColECenter | [1, 0, -12, 0, 240] | +| ColOCenter | [0, -4, 0, 80] | +| ColLeft | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| ColRight | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| TransSqrs | [0, 2, 16, 68, 208, 518, 1120, 2184, 3936] | +| TransNat0 | [0, 2, 8, 20, 40, 70, 112, 168, 240] | +| TransNat1 | [1, 4, 11, 24, 45, 76, 119, 176, 249] | ChebyshevU Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ ChebyshevU Polynomial values as columns | PolyCol7 | [1, 14, 195, 2716, 37829, 526890, 7338631, 102213944, 1423656585] | | PolyCol8 | [1, 16, 255, 4064, 64769, 1032240, 16451071, 262184896, 4178507265] | -# ChebyshevURev -True +# ChebyshevU:Rev +[] -ChebyshevURev Triangle view +ChebyshevU:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ ChebyshevURev Triangle view | Row7 | [128, 0, -192, 0, 80, 0, -8, 0] | | Row8 | [256, 0, -448, 0, 240, 0, -40, 0, 1] | -ChebyshevURev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 0, 4, 0, -1, 8, 0, -4, 0, 16, 0, -12, 0, 1, 32, 0, -32, 0, 6, 0, 64, 0, -80, 0, 24, 0, -1, 128, 0, -192, 0, 80, 0, -8, 0, 256, 0, -448, 0, 240, 0, -40, 0, 1] | -| TriRev | [1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256] | -| TriAcc | [1, 2, 2, 4, 4, 3, 8, 8, 4, 4, 16, 16, 4, 4, 5, 32, 32, 0, 0, 6, 6, 64, 64, -16, -16, 8, 8, 7, 128, 128, -64, -64, 16, 16, 8, 8, 256, 256, -192, -192, 48, 48, 8, 8, 9] | -| TriRevAcc | [1, 2, 2, 3, 4, 4, 4, 4, 8, 8, 5, 4, 4, 16, 16, 6, 6, 0, 0, 32, 32, 7, 8, 8, -16, -16, 64, 64, 8, 8, 16, 16, -64, -64, 128, 128, 9, 8, 8, 48, 48, -192, -192, 256, 256] | -| TriAccRev | [1, 0, 2, -1, -1, 3, 0, -4, -4, 4, 1, 1, -11, -11, 5, 0, 6, 6, -26, -26, 6, -1, -1, 23, 23, -57, -57, 7, 0, -8, -8, 72, 72, -120, -120, 8, 1, 1, -39, -39, 201, 201, -247, -247, 9] | -| TriDiag | [1, 2, 4, 0, 8, 0, 16, 0, -1, 32, 0, -4, 64, 0, -12, 0, 128, 0, -32, 0] | - -ChebyshevURev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| EvenSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| AccSum | [1, 4, 11, 24, 45, 76, 119, 176, 249] | -| AccRevSum | [1, 4, 11, 24, 45, 76, 119, 176, 249] | -| RevAccSum | [1, 2, 1, -4, -15, -34, -63, -104, -159] | -| DiagSum | [1, 2, 4, 8, 15, 28, 52, 96] | - -ChebyshevURev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 8, 48, 96, 960, 1920, 26880] | -| RowGcd | [1, 2, 4, 4, 4, 2, 8, 8, 8] | -| RowMax | [1, 2, 4, 8, 16, 32, 80, 192, 448] | -| Middle | [1, 2, 0, 0, -12, -32, 0, 0, 240] | -| Central | [1, 0, -12, 0, 240] | -| LeftSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RightSide | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| PosHalf | [1, 4, 15, 56, 209, 780, 2911, 10864, 40545] | -| NegHalf | [1, -4, 15, -56, 209, -780, 2911, -10864, 40545] | -| Bin | [1, 2, 3, -4, -55, -258, -777, -1160, 3393] | -| InvBin | [1, -2, 3, 4, -55, 258, -777, 1160, 3393] | -| TransSqrs | [0, 0, -4, -16, -32, -32, 28, 224, 672] | -| TransNat0 | [0, 0, -2, -8, -20, -40, -70, -112, -168] | -| TransNat1 | [1, 2, 1, -4, -15, -34, -63, -104, -159] | - -ChebyshevURev Diagonals as rows +ChebyshevU:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 0], [4, 0, -1], [8, 0, -4, 0], [16, 0, -12, 0, 1], [32, 0, -32, 0, 6, 0], [64, 0, -80, 0, 24, 0, -1], [128, 0, -192, 0, 80, 0, -8, 0], [256, 0, -448, 0, 240, 0, -40, 0, 1]] | +| RevTabl | [[1], [0, 2], [-1, 0, 4], [0, -4, 0, 8], [1, 0, -12, 0, 16], [0, 6, 0, -32, 0, 32], [-1, 0, 24, 0, -80, 0, 64], [0, -8, 0, 80, 0, -192, 0, 128], [1, 0, -40, 0, 240, 0, -448, 0, 256]] | +| AntiDiag | [[1], [2], [4, 0], [8, 0], [16, 0, -1], [32, 0, -4], [64, 0, -12, 0], [128, 0, -32, 0], [256, 0, -80, 0, 1]] | +| AccTabl | [[1], [2, 2], [4, 4, 3], [8, 8, 4, 4], [16, 16, 4, 4, 5], [32, 32, 0, 0, 6, 6], [64, 64, -16, -16, 8, 8, 7], [128, 128, -64, -64, 16, 16, 8, 8], [256, 256, -192, -192, 48, 48, 8, 8, 9]] | +| RevAccTabl | [[1], [2, 2], [3, 4, 4], [4, 4, 8, 8], [5, 4, 4, 16, 16], [6, 6, 0, 0, 32, 32], [7, 8, 8, -16, -16, 64, 64], [8, 8, 16, 16, -64, -64, 128, 128], [9, 8, 8, 48, 48, -192, -192, 256, 256]] | +| AccRevTabl | [[1], [0, 2], [-1, -1, 3], [0, -4, -4, 4], [1, 1, -11, -11, 5], [0, 6, 6, -26, -26, 6], [-1, -1, 23, 23, -57, -57, 7], [0, -8, -8, 72, 72, -120, -120, 8], [1, 1, -39, -39, 201, 201, -247, -247, 9]] | + +ChebyshevU:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| EvenSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| AccSum | [1, 4, 11, 24, 45, 76, 119, 176, 249] | +| AccRevSum | [1, 2, 1, -4, -15, -34, -63, -104, -159] | +| AntiDiagSum | [1, 2, 4, 8, 15, 28, 52, 96, 177] | + +ChebyshevU:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 8, 48, 96, 960, 1920, 26880] | +| RowGcd | [1, 2, 4, 4, 4, 2, 8, 8, 8] | +| RowMax | [1, 2, 4, 8, 16, 32, 80, 192, 448] | +| ColMiddle | [1, 2, 0, 0, -12, -32, 0, 0, 240] | +| ColECenter | [1, 0, -12, 0, 240] | +| ColOCenter | [2, 0, -32, 0] | +| ColLeft | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| ColRight | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| TransSqrs | [0, 0, -4, -16, -32, -32, 28, 224, 672] | +| TransNat0 | [0, 0, -2, -8, -20, -40, -70, -112, -168] | +| TransNat1 | [1, 2, 1, -4, -15, -34, -63, -104, -159] | + +ChebyshevU:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 1, 0, -1, 0, 1]| @@ -175,7 +167,7 @@ ChebyshevURev Diagonals as rows | DiagRow7 | [128, 0, -1024, 0, 4608, 0, -15360, 0, 42240]| | DiagRow8 | [256, 0, -2304, 0, 11520, 0, -42240, 0, 126720]| -ChebyshevURev Diagonals as columns +ChebyshevU:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | @@ -188,7 +180,7 @@ ChebyshevURev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, 10, 60, 280, 1120, 4032, 13440, 42240, 126720] | -ChebyshevURev Polynomial values as rows +ChebyshevU:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ ChebyshevURev Polynomial values as rows | PolyRow7 | [128, 8, 128, -952, -15232, -79672, -276352, -758392, -1781632] | | PolyRow8 | [256, 9, 0, -6935, -43776, -95319, 108544, 1613385, 7246080] | -ChebyshevURev Polynomial values as columns +ChebyshevU:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | diff --git a/data/md/ChristTree.md b/data/md/ChristTree.md index 13082db..8f85fa1 100644 --- a/data/md/ChristTree.md +++ b/data/md/ChristTree.md @@ -14,45 +14,41 @@ ChristTree Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -ChristTree Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] | -| TriRev | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] | -| TriAcc | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 2, 2, 3, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 2, 3, 3, 4, 4, 5] | -| TriRevAcc | [1, 2, 1, 2, 1, 1, 4, 3, 2, 1, 3, 2, 2, 1, 1, 6, 5, 4, 3, 2, 1, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 5, 4, 3, 2, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1] | -| TriAccRev | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 2, 2, 3, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 2, 3, 3, 4, 4, 5] | -| TriDiag | [1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0] | +ChristTree Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0, 1]] | +| RevTabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 0], [1, 1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0], [1, 1, 1, 1, 1]] | +| AccTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 2, 2, 3], [1, 2, 3, 4, 5, 6], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 2, 2, 3, 3, 4, 4, 5]] | +| RevAccTabl | [[1], [2, 1], [2, 1, 1], [4, 3, 2, 1], [3, 2, 2, 1, 1], [6, 5, 4, 3, 2, 1], [4, 3, 3, 2, 2, 1, 1], [8, 7, 6, 5, 4, 3, 2, 1], [5, 4, 4, 3, 3, 2, 2, 1, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 2, 2, 3], [1, 2, 3, 4, 5, 6], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 2, 2, 3, 3, 4, 4, 5]] | ChristTree Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 2, 4, 3, 6, 4, 8, 5] | -| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | -| OddSum | [0, 1, 0, 2, 0, 3, 0, 4, 0] | -| AltSum | [1, 0, 2, 0, 3, 0, 4, 0, 5] | -| AccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| AccRevSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| RevAccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| DiagSum | [1, 1, 2, 1, 3, 2, 4, 2] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 2, 4, 3, 6, 4, 8, 5] | +| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | +| OddSum | [0, 1, 0, 2, 0, 3, 0, 4, 0] | +| AltSum | [1, 0, 2, 0, 3, 0, 4, 0, 5] | +| AccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | +| AccRevSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | +| AntiDiagSum | [1, 1, 2, 1, 3, 2, 4, 2, 5] | ChristTree Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, 1, 1, 1, 0, 1, 1] | -| Central | [1, 0, 1, 0, 1] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 5, 15, 21, 63, 85, 255, 341] | -| NegHalf | [1, -1, 5, -5, 21, -21, 85, -85, 341] | -| Bin | [1, 2, 2, 8, 8, 32, 32, 128, 128] | -| InvBin | [1, 0, 2, 0, 8, 0, 32, 0, 128] | -| TransSqrs | [0, 1, 4, 14, 20, 55, 56, 140, 120] | -| TransNat0 | [0, 1, 2, 6, 6, 15, 12, 28, 20] | -| TransNat1 | [1, 3, 4, 10, 9, 21, 16, 36, 25] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, 1, 1, 1, 0, 1, 1] | +| ColECenter | [1, 0, 1, 0, 1] | +| ColOCenter | [1, 1, 1, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 14, 20, 55, 56, 140, 120] | +| TransNat0 | [0, 1, 2, 6, 6, 15, 12, 28, 20] | +| TransNat1 | [1, 3, 4, 10, 9, 21, 16, 36, 25] | ChristTree Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ ChristTree Polynomial values as columns | PolyCol7 | [1, 8, 50, 400, 2451, 19608, 120100, 960800, 5884901] | | PolyCol8 | [1, 9, 65, 585, 4161, 37449, 266305, 2396745, 17043521] | -# ChristTreeInv -True +# ChristTree:Inv +[] -ChristTreeInv Triangle view +ChristTree:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ ChristTreeInv Triangle view | Row7 | [0, 0, 0, 0, 1, -1, -1, 1] | | Row8 | [0, 0, 0, 0, 0, 0, -1, 0, 1] | -ChristTreeInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, -1, 0, 1, 1, -1, -1, 1, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1] | -| TriRev | [1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 0, -1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0] | -| TriAcc | [1, -1, 0, -1, -1, 0, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, -1, 0, -1, 0, 1, 0, -1, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0] | -| TriAccRev | [1, 1, 0, 1, 1, 0, 1, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0] | -| TriDiag | [1, -1, -1, 1, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, -1, 1, 0, 0, 1, 0] | - -ChristTreeInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | -| AccRevSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | -| RevAccSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | -| DiagSum | [1, -1, 0, 1, 0, -1, 0, 1] | - -ChristTreeInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, -1, 0, -1, -1, 1, 0, 0, 0] | -| Central | [1, 0, -1, 0, 0] | -| LeftSide | [1, -1, -1, 1, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, 3, -3, 3, -3, 3, -3] | -| NegHalf | [1, 3, -3, -9, -3, -9, -3, -9, -3] | -| Bin | [1, 0, 0, -4, -5, -4, -14, 8, -27] | -| InvBin | [1, 2, 0, 0, -5, -14, -14, -48, -27] | -| TransSqrs | [0, 1, 4, 4, 12, 4, 20, 4, 28] | -| TransNat0 | [0, 1, 2, 0, 2, 0, 2, 0, 2] | -| TransNat1 | [1, 1, 2, 0, 2, 0, 2, 0, 2] | - -ChristTreeInv Diagonals as rows +ChristTree:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [0, 0, -1, 0, 1], [0, 0, 1, -1, -1, 1], [0, 0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, -1, 0, 1]] | +| RevTabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, -1, 0, 0], [1, -1, -1, 1, 0, 0], [1, 0, -1, 0, 0, 0, 0], [1, -1, -1, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [-1, 1], [1, 0], [0, -1, 1], [0, 0, -1], [0, 0, -1, 1], [0, 0, 1, 0], [0, 0, 0, -1, 1]] | +| AccTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [0, 0, -1, -1, 0], [0, 0, 1, 0, -1, 0], [0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, -1], [0, -1, 0, 1], [0, -1, -1, 0, 0], [0, -1, 0, 1, 0, 0], [0, -1, -1, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 0, 0, 0], [1, 0, -1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0]] | + +ChristTree:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | +| AccRevSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | +| AntiDiagSum | [1, -1, 0, 1, 0, -1, 0, 1, 0] | + +ChristTree:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, 0, -1, -1, 1, 0, 0, 0] | +| ColECenter | [1, 0, -1, 0, 0] | +| ColOCenter | [-1, -1, 1, 0] | +| ColLeft | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 4, 12, 4, 20, 4, 28] | +| TransNat0 | [0, 1, 2, 0, 2, 0, 2, 0, 2] | +| TransNat1 | [1, 1, 2, 0, 2, 0, 2, 0, 2] | + +ChristTree:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ ChristTreeInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| -ChristTreeInv Diagonals as columns +ChristTree:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ ChristTreeInv Diagonals as columns | DiagCol7 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | -ChristTreeInv Polynomial values as rows +ChristTree:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ ChristTreeInv Polynomial values as rows | PolyRow7 | [0, 0, 48, 1296, 11520, 60000, 226800, 691488, 1806336] | | PolyRow8 | [0, 0, 192, 5832, 61440, 375000, 1632960, 5647152, 16515072] | -ChristTreeInv Polynomial values as columns +ChristTree:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ ChristTreeInv Polynomial values as columns | PolyCol7 | [1, 6, 48, 288, 2352, 14112, 115248, 691488, 5647152] | | PolyCol8 | [1, 7, 63, 441, 4032, 28224, 258048, 1806336, 16515072] | -# ChristTreeRev -True +# ChristTree:Rev +[] -ChristTreeRev Triangle view +ChristTree:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ ChristTreeRev Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -ChristTreeRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] | -| TriRev | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] | -| TriAcc | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 2, 2, 3, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 2, 3, 3, 4, 4, 5] | -| TriRevAcc | [1, 2, 1, 2, 1, 1, 4, 3, 2, 1, 3, 2, 2, 1, 1, 6, 5, 4, 3, 2, 1, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 5, 4, 3, 2, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1] | -| TriAccRev | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 2, 2, 3, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 2, 3, 3, 4, 4, 5] | -| TriDiag | [1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0] | - -ChristTreeRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 2, 4, 3, 6, 4, 8, 5] | -| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | -| OddSum | [0, 1, 0, 2, 0, 3, 0, 4, 0] | -| AltSum | [1, 0, 2, 0, 3, 0, 4, 0, 5] | -| AccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| AccRevSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| RevAccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | -| DiagSum | [1, 1, 2, 1, 3, 2, 4, 2] | - -ChristTreeRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, 1, 1, 1, 0, 1, 1] | -| Central | [1, 0, 1, 0, 1] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 5, 15, 21, 63, 85, 255, 341] | -| NegHalf | [1, -1, 5, -5, 21, -21, 85, -85, 341] | -| Bin | [1, 2, 2, 8, 8, 32, 32, 128, 128] | -| InvBin | [1, 0, 2, 0, 8, 0, 32, 0, 128] | -| TransSqrs | [0, 1, 4, 14, 20, 55, 56, 140, 120] | -| TransNat0 | [0, 1, 2, 6, 6, 15, 12, 28, 20] | -| TransNat1 | [1, 3, 4, 10, 9, 21, 16, 36, 25] | - -ChristTreeRev Diagonals as rows +ChristTree:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0, 1]] | +| RevTabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 0], [1, 1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 1, 0], [1, 1, 1, 1, 1]] | +| AccTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 2, 2, 3], [1, 2, 3, 4, 5, 6], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 2, 2, 3, 3, 4, 4, 5]] | +| RevAccTabl | [[1], [2, 1], [2, 1, 1], [4, 3, 2, 1], [3, 2, 2, 1, 1], [6, 5, 4, 3, 2, 1], [4, 3, 3, 2, 2, 1, 1], [8, 7, 6, 5, 4, 3, 2, 1], [5, 4, 4, 3, 3, 2, 2, 1, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 2, 2, 3], [1, 2, 3, 4, 5, 6], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 2, 2, 3, 3, 4, 4, 5]] | + +ChristTree:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 2, 4, 3, 6, 4, 8, 5] | +| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | +| OddSum | [0, 1, 0, 2, 0, 3, 0, 4, 0] | +| AltSum | [1, 0, 2, 0, 3, 0, 4, 0, 5] | +| AccSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | +| AccRevSum | [1, 3, 4, 10, 9, 21, 16, 36, 25] | +| AntiDiagSum | [1, 1, 2, 1, 3, 2, 4, 2, 5] | + +ChristTree:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, 1, 1, 1, 0, 1, 1] | +| ColECenter | [1, 0, 1, 0, 1] | +| ColOCenter | [1, 1, 1, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 14, 20, 55, 56, 140, 120] | +| TransNat0 | [0, 1, 2, 6, 6, 15, 12, 28, 20] | +| TransNat1 | [1, 3, 4, 10, 9, 21, 16, 36, 25] | + +ChristTree:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ ChristTreeRev Diagonals as rows | DiagRow7 | [1, 0, 1, 0, 1, 0, 1, 0, 1]| | DiagRow8 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| -ChristTreeRev Diagonals as columns +ChristTree:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ ChristTreeRev Diagonals as columns | DiagCol7 | [1, 0, 1, 0, 1, 0, 1, 0, 1] | | DiagCol8 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -ChristTreeRev Polynomial values as rows +ChristTree:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ ChristTreeRev Polynomial values as rows | PolyRow7 | [1, 8, 255, 3280, 21845, 97656, 335923, 960800, 2396745] | | PolyRow8 | [1, 5, 341, 7381, 69905, 406901, 1727605, 5884901, 17043521] | -ChristTreeRev Polynomial values as columns +ChristTree:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ ChristTreeRev Polynomial values as columns | PolyCol7 | [1, 8, 50, 400, 2451, 19608, 120100, 960800, 5884901] | | PolyCol8 | [1, 9, 65, 585, 4161, 37449, 266305, 2396745, 17043521] | -# ChristTreeInvRev -True +# ChristTree:Inv:Rev +[] -ChristTreeInvRev Triangle view +ChristTree:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ ChristTreeInvRev Triangle view | Row7 | [1, -1, -1, 1, 0, 0, 0, 0] | | Row8 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | -ChristTreeInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 0, -1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0] | -| TriRev | [1, -1, 1, -1, 0, 1, 1, -1, -1, 1, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1] | -| TriAcc | [1, 1, 0, 1, 1, 0, 1, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0] | -| TriRevAcc | [1, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1] | -| TriAccRev | [1, -1, 0, -1, -1, 0, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 0, -1, 0] | - -ChristTreeInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, -1, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 2, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | -| AccRevSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | -| RevAccSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | -| DiagSum | [1, 1, 0, 1, -1, 0, 0, 0] | - -ChristTreeInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, -1, -1, -1, 0, 1, 0] | -| Central | [1, 0, -1, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, -1, 1, 0, 0, 0, 0, 0] | -| PosHalf | [1, 1, 3, 3, 12, 12, 48, 48, 192] | -| NegHalf | [1, -3, 3, -9, 12, -36, 48, -144, 192] | -| Bin | [1, 0, 0, -4, -5, -4, -14, 8, -27] | -| InvBin | [1, -2, 0, 0, -5, 14, -14, 48, -27] | -| TransSqrs | [0, -1, -4, 4, -4, 4, -4, 4, -4] | -| TransNat0 | [0, -1, -2, 0, -2, 0, -2, 0, -2] | -| TransNat1 | [1, -1, -2, 0, -2, 0, -2, 0, -2] | - -ChristTreeInvRev Diagonals as rows +ChristTree:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, -1, 0, 0], [1, -1, -1, 1, 0, 0], [1, 0, -1, 0, 0, 0, 0], [1, -1, -1, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0]] | +| RevTabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [0, 0, -1, 0, 1], [0, 0, 1, -1, -1, 1], [0, 0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, -1, 0, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, 0], [1, -1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, -1, 0], [1, -1, -1, 1, 0]] | +| AccTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 0, 0, 0], [1, 0, -1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0]] | +| RevAccTabl | [[1], [0, 1], [0, 1, 1], [0, -1, 0, 1], [0, 0, 0, 1, 1], [0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 1]] | +| AccRevTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [0, 0, -1, -1, 0], [0, 0, 1, 0, -1, 0], [0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0]] | + +ChristTree:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, -1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | +| AccRevSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | +| AntiDiagSum | [1, 1, 0, 1, -1, 0, 0, 0, 0] | + +ChristTree:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, -1, -1, -1, 0, 1, 0] | +| ColECenter | [1, 0, -1, 0, 0] | +| ColOCenter | [1, -1, -1, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| TransSqrs | [0, -1, -4, 4, -4, 4, -4, 4, -4] | +| TransNat0 | [0, -1, -2, 0, -2, 0, -2, 0, -2] | +| TransNat1 | [1, -1, -2, 0, -2, 0, -2, 0, -2] | + +ChristTree:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, -1, 1, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ ChristTreeInvRev Diagonals as rows | DiagRow7 | [1, 0, -1, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, -1, -1, 1, 0, 0, 0, 0, 0]| -ChristTreeInvRev Diagonals as columns +ChristTree:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ ChristTreeInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -ChristTreeInvRev Polynomial values as rows +ChristTree:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ ChristTreeInvRev Polynomial values as rows | PolyRow7 | [1, 0, 3, 16, 45, 96, 175, 288, 441] | | PolyRow8 | [1, 0, -3, -8, -15, -24, -35, -48, -63] | -ChristTreeInvRev Polynomial values as columns +ChristTree:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ ChristTreeInvRev Polynomial values as columns | PolyCol7 | [1, -6, -48, 288, -48, 288, -48, 288, -48] | | PolyCol8 | [1, -7, -63, 441, -63, 441, -63, 441, -63] | +# ChristTree:Rev:Inv +[] + +ChristTree:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [-1, 0, 1] | +| Row3 | [1, -1, -1, 1] | +| Row4 | [0, 0, -1, 0, 1] | +| Row5 | [0, 0, 1, -1, -1, 1] | +| Row6 | [0, 0, 0, 0, -1, 0, 1] | +| Row7 | [0, 0, 0, 0, 1, -1, -1, 1] | +| Row8 | [0, 0, 0, 0, 0, 0, -1, 0, 1] | + +ChristTree:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [0, 0, -1, 0, 1], [0, 0, 1, -1, -1, 1], [0, 0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, -1, 0, 1]] | +| RevTabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, -1, 0, 0], [1, -1, -1, 1, 0, 0], [1, 0, -1, 0, 0, 0, 0], [1, -1, -1, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [-1, 1], [1, 0], [0, -1, 1], [0, 0, -1], [0, 0, -1, 1], [0, 0, 1, 0], [0, 0, 0, -1, 1]] | +| AccTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [0, 0, -1, -1, 0], [0, 0, 1, 0, -1, 0], [0, 0, 0, 0, -1, -1, 0], [0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, -1], [0, -1, 0, 1], [0, -1, -1, 0, 0], [0, -1, 0, 1, 0, 0], [0, -1, -1, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 0, 0, 0], [1, 0, -1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0]] | + +ChristTree:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, -1, -2, 0, -2, 0, -2, 0, -2] | +| AccRevSum | [1, 1, 2, 0, 2, 0, 2, 0, 2] | +| AntiDiagSum | [1, -1, 0, 1, 0, -1, 0, 1, 0] | + +ChristTree:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, 0, -1, -1, 1, 0, 0, 0] | +| ColECenter | [1, 0, -1, 0, 0] | +| ColOCenter | [-1, -1, 1, 0] | +| ColLeft | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 4, 12, 4, 20, 4, 28] | +| TransNat0 | [0, 1, 2, 0, 2, 0, 2, 0, 2] | +| TransNat1 | [1, 1, 2, 0, 2, 0, 2, 0, 2] | + +ChristTree:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, 0, -1, 0, -1, 0, -1, 0, -1]| +| DiagRow2 | [-1, -1, -1, -1, -1, -1, -1, -1, -1]| +| DiagRow3 | [1, 0, 1, 0, 1, 0, 1, 0, 1]| +| DiagRow4 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow5 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow6 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| + +ChristTree:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | +| DiagCol2 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| DiagCol3 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | +| DiagCol4 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| DiagCol5 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | +| DiagCol6 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| DiagCol7 | [1, 0, -1, 0, 0, 0, 0, 0, 0] | +| DiagCol8 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | + +ChristTree:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [-1, 0, 3, 8, 15, 24, 35, 48, 63] | +| PolyRow3 | [1, 0, 3, 16, 45, 96, 175, 288, 441] | +| PolyRow4 | [0, 0, 12, 72, 240, 600, 1260, 2352, 4032] | +| PolyRow5 | [0, 0, 12, 144, 720, 2400, 6300, 14112, 28224] | +| PolyRow6 | [0, 0, 48, 648, 3840, 15000, 45360, 115248, 258048] | +| PolyRow7 | [0, 0, 48, 1296, 11520, 60000, 226800, 691488, 1806336] | +| PolyRow8 | [0, 0, 192, 5832, 61440, 375000, 1632960, 5647152, 16515072] | + +ChristTree:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, -1, 1, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 3, 3, 12, 12, 48, 48, 192] | +| PolyCol3 | [1, 2, 8, 16, 72, 144, 648, 1296, 5832] | +| PolyCol4 | [1, 3, 15, 45, 240, 720, 3840, 11520, 61440] | +| PolyCol5 | [1, 4, 24, 96, 600, 2400, 15000, 60000, 375000] | +| PolyCol6 | [1, 5, 35, 175, 1260, 6300, 45360, 226800, 1632960] | +| PolyCol7 | [1, 6, 48, 288, 2352, 14112, 115248, 691488, 5647152] | +| PolyCol8 | [1, 7, 63, 441, 4032, 28224, 258048, 1806336, 16515072] | + diff --git a/data/md/Composition.md b/data/md/Composition.md index 9332b0f..1551356 100644 --- a/data/md/Composition.md +++ b/data/md/Composition.md @@ -14,45 +14,41 @@ Composition Triangle view | Row7 | [0, 1, 20, 23, 12, 5, 2, 1] | | Row8 | [0, 1, 33, 47, 27, 12, 5, 2, 1] | -Composition Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 2, 1, 0, 1, 7, 5, 2, 1, 0, 1, 12, 11, 5, 2, 1, 0, 1, 20, 23, 12, 5, 2, 1, 0, 1, 33, 47, 27, 12, 5, 2, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 5, 7, 1, 0, 1, 2, 5, 11, 12, 1, 0, 1, 2, 5, 12, 23, 20, 1, 0, 1, 2, 5, 12, 27, 47, 33, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 7, 8, 0, 1, 8, 13, 15, 16, 0, 1, 13, 24, 29, 31, 32, 0, 1, 21, 44, 56, 61, 63, 64, 0, 1, 34, 81, 108, 120, 125, 127, 128] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 7, 5, 1, 0, 16, 15, 13, 8, 1, 0, 32, 31, 29, 24, 13, 1, 0, 64, 63, 61, 56, 44, 21, 1, 0, 128, 127, 125, 120, 108, 81, 34, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 3, 7, 8, 8, 1, 3, 8, 15, 16, 16, 1, 3, 8, 19, 31, 32, 32, 1, 3, 8, 20, 43, 63, 64, 64, 1, 3, 8, 20, 47, 94, 127, 128, 128] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 4, 1, 0, 1, 7, 2] | +Composition Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 2, 1], [0, 1, 4, 2, 1], [0, 1, 7, 5, 2, 1], [0, 1, 12, 11, 5, 2, 1], [0, 1, 20, 23, 12, 5, 2, 1], [0, 1, 33, 47, 27, 12, 5, 2, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 2, 1, 0], [1, 2, 4, 1, 0], [1, 2, 5, 7, 1, 0], [1, 2, 5, 11, 12, 1, 0], [1, 2, 5, 12, 23, 20, 1, 0], [1, 2, 5, 12, 27, 47, 33, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 2], [0, 1, 4, 1], [0, 1, 7, 2], [0, 1, 12, 5, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 4], [0, 1, 5, 7, 8], [0, 1, 8, 13, 15, 16], [0, 1, 13, 24, 29, 31, 32], [0, 1, 21, 44, 56, 61, 63, 64], [0, 1, 34, 81, 108, 120, 125, 127, 128]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [4, 3, 1, 0], [8, 7, 5, 1, 0], [16, 15, 13, 8, 1, 0], [32, 31, 29, 24, 13, 1, 0], [64, 63, 61, 56, 44, 21, 1, 0], [128, 127, 125, 120, 108, 81, 34, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 3, 4, 4], [1, 3, 7, 8, 8], [1, 3, 8, 15, 16, 16], [1, 3, 8, 19, 31, 32, 32], [1, 3, 8, 20, 43, 63, 64, 64], [1, 3, 8, 20, 47, 94, 127, 128, 128]] | Composition Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| EvenSum | [1, 0, 1, 2, 5, 9, 18, 34, 66] | -| OddSum | [0, 1, 1, 2, 3, 7, 14, 30, 62] | -| AltSum | [1, -1, 0, 0, 2, 2, 4, 4, 4] | -| AccSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | -| AccRevSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | -| RevAccSum | [1, 2, 5, 12, 27, 59, 126, 266, 556] | -| DiagSum | [1, 0, 1, 1, 2, 3, 6, 10] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| EvenSum | [1, 0, 1, 2, 5, 9, 18, 34, 66] | +| OddSum | [0, 1, 1, 2, 3, 7, 14, 30, 62] | +| AltSum | [1, -1, 0, 0, 2, 2, 4, 4, 4] | +| AccSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | +| AccRevSum | [1, 2, 5, 12, 27, 59, 126, 266, 556] | +| AntiDiagSum | [1, 0, 1, 1, 2, 3, 6, 10, 19] | Composition Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 4, 70, 660, 1380, 279180] | -| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 4, 7, 12, 23, 47] | -| Middle | [1, 0, 1, 1, 4, 7, 11, 23, 27] | -| Central | [1, 1, 4, 11, 27] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 9, 29, 97, 337, 1193, 4297] | -| NegHalf | [1, 1, -1, 1, 5, -23, 89, -287, 833] | -| Bin | [1, 1, 3, 10, 37, 136, 494, 1772, 6283] | -| InvBin | [1, 1, -1, -2, 13, -24, 18, 64, -373] | -| TransSqrs | [0, 1, 5, 18, 51, 131, 314, 726, 1630] | -| TransNat0 | [0, 1, 3, 8, 19, 43, 94, 202, 428] | -| TransNat1 | [1, 2, 5, 12, 27, 59, 126, 266, 556] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 4, 70, 660, 1380, 279180] | +| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 4, 7, 12, 23, 47] | +| ColMiddle | [1, 0, 1, 1, 4, 7, 11, 23, 27] | +| ColECenter | [1, 1, 4, 11, 27] | +| ColOCenter | [0, 1, 7, 23] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 18, 51, 131, 314, 726, 1630] | +| TransNat0 | [0, 1, 3, 8, 19, 43, 94, 202, 428] | +| TransNat1 | [1, 2, 5, 12, 27, 59, 126, 266, 556] | Composition Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Composition Polynomial values as columns | PolyCol7 | [1, 7, 56, 448, 3290, 23674, 167636, 1180564, 8284388] | | PolyCol8 | [1, 8, 72, 648, 5384, 43976, 354568, 2847496, 22812232] | -# CompositionInv -True +# Composition:Inv +[] -CompositionInv Triangle view +Composition:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ CompositionInv Triangle view | Row7 | [0, -5, 3, 4, 0, -1, -2, 1] | | Row8 | [0, -5, -1, 6, 2, 0, -1, -2, 1] | -CompositionInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, 1, 0, -2, 1, 0, -1, 3, -1, -2, 1, 0, -3, 4, 1, -1, -2, 1, 0, -5, 3, 4, 0, -1, -2, 1, 0, -5, -1, 6, 2, 0, -1, -2, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -2, 0, 1, 0, 1, -2, -1, 3, -1, 0, 1, -2, -1, 1, 4, -3, 0, 1, -2, -1, 0, 4, 3, -5, 0, 1, -2, -1, 0, 2, 6, -1, -5, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 1, -1, 0, 0, -1, 2, 1, -1, 0, 0, -3, 1, 2, 1, -1, 0, 0, -5, -2, 2, 2, 1, -1, 0, 0, -5, -6, 0, 2, 2, 1, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, -1, 1, 2, -1, 0, 0, -1, 1, 2, 1, -3, 0, 0, -1, 1, 2, 2, -2, -5, 0, 0, -1, 1, 2, 2, 0, -6, -5, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -1, 0, 0, 1, -1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 1, -1, -2, -1, 3, 0, 0, 1, -1, -2, -2, 2, 5, 0, 0, 1, -1, -2, -2, 0, 6, 5, 0, 0] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 1, 1, 0, 1, -2, 0, -1, 0, 1, 0, -3, 3, -2] | - -CompositionInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 0, 1, -2, 1, 1, 4, 1, 1] | -| OddSum | [0, 1, -1, 2, -1, -1, -4, -1, -1] | -| AltSum | [1, -1, 2, -4, 2, 2, 8, 2, 2] | -| AccSum | [1, 1, -1, 0, 1, 1, 0, -3, -7] | -| AccRevSum | [1, 1, -1, 0, 1, 1, 0, -3, -7] | -| RevAccSum | [1, 2, 1, 0, -1, -1, 0, 3, 7] | -| DiagSum | [1, 0, 1, -1, 2, -1, 0, -2] | - -CompositionInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -2, 6, 12, 60, 30] | -| RowGcd | [1, 1, 1, -2, -2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 2, 3, 4, 5, 6] | -| Middle | [1, 0, -1, 1, 0, 3, 1, 4, 2] | -| Central | [1, -1, 0, 1, 2] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, 1, 5, 1, -31, -167, -487] | -| NegHalf | [1, 1, 3, 9, -3, -39, 153, -351, 417] | -| Bin | [1, 1, -1, -2, -3, 6, 36, 134, 365] | -| InvBin | [1, 1, 3, 10, 5, -34, 56, 36, -195] | -| TransSqrs | [0, 1, 3, 2, -1, -5, -8, -5, 7] | -| TransNat0 | [0, 1, 1, 0, -1, -1, 0, 3, 7] | -| TransNat1 | [1, 2, 1, 0, -1, -1, 0, 3, 7] | - -CompositionInv Diagonals as rows +Composition:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 1, -2, 1], [0, 1, 0, -2, 1], [0, -1, 3, -1, -2, 1], [0, -3, 4, 1, -1, -2, 1], [0, -5, 3, 4, 0, -1, -2, 1], [0, -5, -1, 6, 2, 0, -1, -2, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, 0], [1, -2, 0, 1, 0], [1, -2, -1, 3, -1, 0], [1, -2, -1, 1, 4, -3, 0], [1, -2, -1, 0, 4, 3, -5, 0], [1, -2, -1, 0, 2, 6, -1, -5, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 1, 1], [0, 1, -2], [0, -1, 0, 1], [0, -3, 3, -2], [0, -5, 4, -1, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 1, -1, 0], [0, 1, 1, -1, 0], [0, -1, 2, 1, -1, 0], [0, -3, 1, 2, 1, -1, 0], [0, -5, -2, 2, 2, 1, -1, 0], [0, -5, -6, 0, 2, 2, 1, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 1, 0], [0, -1, 1, 1, 0], [0, -1, 1, 2, -1, 0], [0, -1, 1, 2, 1, -3, 0], [0, -1, 1, 2, 2, -2, -5, 0], [0, -1, 1, 2, 2, 0, -6, -5, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -1, 0, 0], [1, -1, -1, 0, 0], [1, -1, -2, 1, 0, 0], [1, -1, -2, -1, 3, 0, 0], [1, -1, -2, -2, 2, 5, 0, 0], [1, -1, -2, -2, 0, 6, 5, 0, 0]] | + +Composition:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -2, 1, 1, 4, 1, 1] | +| OddSum | [0, 1, -1, 2, -1, -1, -4, -1, -1] | +| AltSum | [1, -1, 2, -4, 2, 2, 8, 2, 2] | +| AccSum | [1, 1, -1, 0, 1, 1, 0, -3, -7] | +| AccRevSum | [1, 2, 1, 0, -1, -1, 0, 3, 7] | +| AntiDiagSum | [1, 0, 1, -1, 2, -1, 0, -2, -1] | + +Composition:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 2, 6, 12, 60, 30] | +| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 2, 3, 4, 5, 6] | +| ColMiddle | [1, 0, -1, 1, 0, 3, 1, 4, 2] | +| ColECenter | [1, -1, 0, 1, 2] | +| ColOCenter | [0, 1, 3, 4] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 2, -1, -5, -8, -5, 7] | +| TransNat0 | [0, 1, 1, 0, -1, -1, 0, 3, 7] | +| TransNat1 | [1, 2, 1, 0, -1, -1, 0, 3, 7] | + +Composition:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ CompositionInv Diagonals as rows | DiagRow7 | [0, -5, -11, 10, 10, 8, 7, 5, 4]| | DiagRow8 | [0, -1, -28, 9, 15, 11, 9, 8, 6]| -CompositionInv Diagonals as columns +Composition:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ CompositionInv Diagonals as columns | DiagCol7 | [1, -2, -1, 0, 1, 2, 3, 5, 8] | | DiagCol8 | [1, -2, -1, 0, 1, 2, 3, 4, 6] | -CompositionInv Polynomial values as rows +Composition:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ CompositionInv Polynomial values as rows | PolyRow7 | [0, 0, 2, 606, 7452, 44300, 179790, 572922, 1542296] | | PolyRow8 | [0, 0, 2, 1758, 29532, 220700, 1076910, 4006842, 12331928] | -CompositionInv Polynomial values as columns +Composition:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ CompositionInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 252, 1722, 11802, 82152, 572922, 4006842] | | PolyCol8 | [1, 8, 56, 392, 3080, 24248, 193256, 1542296, 12331928] | -# CompositionRev -True +# Composition:Rev +[] -CompositionRev Triangle view +Composition:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ CompositionRev Triangle view | Row7 | [1, 2, 5, 12, 23, 20, 1, 0] | | Row8 | [1, 2, 5, 12, 27, 47, 33, 1, 0] | -CompositionRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 5, 7, 1, 0, 1, 2, 5, 11, 12, 1, 0, 1, 2, 5, 12, 23, 20, 1, 0, 1, 2, 5, 12, 27, 47, 33, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 2, 1, 0, 1, 7, 5, 2, 1, 0, 1, 12, 11, 5, 2, 1, 0, 1, 20, 23, 12, 5, 2, 1, 0, 1, 33, 47, 27, 12, 5, 2, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 3, 7, 8, 8, 1, 3, 8, 15, 16, 16, 1, 3, 8, 19, 31, 32, 32, 1, 3, 8, 20, 43, 63, 64, 64, 1, 3, 8, 20, 47, 94, 127, 128, 128] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 7, 3, 1, 16, 16, 15, 8, 3, 1, 32, 32, 31, 19, 8, 3, 1, 64, 64, 63, 43, 20, 8, 3, 1, 128, 128, 127, 94, 47, 20, 8, 3, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 7, 8, 0, 1, 8, 13, 15, 16, 0, 1, 13, 24, 29, 31, 32, 0, 1, 21, 44, 56, 61, 63, 64, 0, 1, 34, 81, 108, 120, 125, 127, 128] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 1, 2, 4, 0, 1, 2, 5, 1] | - -CompositionRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| EvenSum | [1, 1, 1, 2, 5, 7, 18, 30, 66] | -| OddSum | [0, 0, 1, 2, 3, 9, 14, 34, 62] | -| AltSum | [1, 1, 0, 0, 2, -2, 4, -4, 4] | -| AccSum | [1, 2, 5, 12, 27, 59, 126, 266, 556] | -| AccRevSum | [1, 2, 5, 12, 27, 59, 126, 266, 556] | -| RevAccSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | -| DiagSum | [1, 1, 1, 2, 3, 4, 7, 9] | - -CompositionRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 4, 70, 660, 1380, 279180] | -| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 4, 7, 12, 23, 47] | -| Middle | [1, 1, 1, 2, 4, 5, 11, 12, 27] | -| Central | [1, 1, 4, 11, 27] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 18, 50, 134, 346, 874, 2158] | -| NegHalf | [1, -2, 2, -2, 14, -14, 38, -74, 122] | -| Bin | [1, 1, 3, 10, 37, 136, 494, 1772, 6283] | -| InvBin | [1, -1, -1, 2, 13, 24, 18, -64, -373] | -| TransSqrs | [0, 0, 1, 6, 27, 101, 338, 1034, 2974] | -| TransNat0 | [0, 0, 1, 4, 13, 37, 98, 246, 596] | -| TransNat1 | [1, 1, 3, 8, 21, 53, 130, 310, 724] | - -CompositionRev Diagonals as rows +Composition:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 2, 1, 0], [1, 2, 4, 1, 0], [1, 2, 5, 7, 1, 0], [1, 2, 5, 11, 12, 1, 0], [1, 2, 5, 12, 23, 20, 1, 0], [1, 2, 5, 12, 27, 47, 33, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 2, 1], [0, 1, 4, 2, 1], [0, 1, 7, 5, 2, 1], [0, 1, 12, 11, 5, 2, 1], [0, 1, 20, 23, 12, 5, 2, 1], [0, 1, 33, 47, 27, 12, 5, 2, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 2, 0], [1, 2, 1], [1, 2, 4, 0], [1, 2, 5, 1], [1, 2, 5, 7, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 3, 4, 4], [1, 3, 7, 8, 8], [1, 3, 8, 15, 16, 16], [1, 3, 8, 19, 31, 32, 32], [1, 3, 8, 20, 43, 63, 64, 64], [1, 3, 8, 20, 47, 94, 127, 128, 128]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [4, 4, 3, 1], [8, 8, 7, 3, 1], [16, 16, 15, 8, 3, 1], [32, 32, 31, 19, 8, 3, 1], [64, 64, 63, 43, 20, 8, 3, 1], [128, 128, 127, 94, 47, 20, 8, 3, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 4], [0, 1, 5, 7, 8], [0, 1, 8, 13, 15, 16], [0, 1, 13, 24, 29, 31, 32], [0, 1, 21, 44, 56, 61, 63, 64], [0, 1, 34, 81, 108, 120, 125, 127, 128]] | + +Composition:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| EvenSum | [1, 1, 1, 2, 5, 7, 18, 30, 66] | +| OddSum | [0, 0, 1, 2, 3, 9, 14, 34, 62] | +| AltSum | [1, 1, 0, 0, 2, -2, 4, -4, 4] | +| AccSum | [1, 2, 5, 12, 27, 59, 126, 266, 556] | +| AccRevSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | +| AntiDiagSum | [1, 1, 1, 2, 3, 4, 7, 9, 15] | + +Composition:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 4, 70, 660, 1380, 279180] | +| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 4, 7, 12, 23, 47] | +| ColMiddle | [1, 1, 1, 2, 4, 5, 11, 12, 27] | +| ColECenter | [1, 1, 4, 11, 27] | +| ColOCenter | [1, 2, 5, 12] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 6, 27, 101, 338, 1034, 2974] | +| TransNat0 | [0, 0, 1, 4, 13, 37, 98, 246, 596] | +| TransNat1 | [1, 1, 3, 8, 21, 53, 130, 310, 724] | + +Composition:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ CompositionRev Diagonals as rows | DiagRow7 | [1, 2, 5, 12, 28, 64, 144, 319, 699]| | DiagRow8 | [1, 2, 5, 12, 28, 64, 144, 320, 703]| -CompositionRev Diagonals as columns +Composition:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ CompositionRev Diagonals as columns | DiagCol7 | [0, 1, 54, 185, 269, 303, 315, 319, 320] | | DiagCol8 | [0, 1, 88, 360, 563, 653, 687, 699, 703] | -CompositionRev Polynomial values as rows +Composition:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ CompositionRev Polynomial values as rows | PolyRow7 | [1, 64, 1193, 7828, 31321, 94136, 234769, 513388, 1018193] | | PolyRow8 | [1, 128, 4297, 40228, 207449, 759136, 2222833, 5565092, 12405073] | -CompositionRev Polynomial values as columns +Composition:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ CompositionRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 64, 554, 5062, 49652, 513388, 5565092] | | PolyCol8 | [1, 1, 9, 81, 785, 8017, 87889, 1018193, 12405073] | -# CompositionInvRev -True +# Composition:Inv:Rev +[] -CompositionInvRev Triangle view +Composition:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ CompositionInvRev Triangle view | Row7 | [1, -2, -1, 0, 4, 3, -5, 0] | | Row8 | [1, -2, -1, 0, 2, 6, -1, -5, 0] | -CompositionInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -2, 0, 1, 0, 1, -2, -1, 3, -1, 0, 1, -2, -1, 1, 4, -3, 0, 1, -2, -1, 0, 4, 3, -5, 0, 1, -2, -1, 0, 2, 6, -1, -5, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, 1, 0, -2, 1, 0, -1, 3, -1, -2, 1, 0, -3, 4, 1, -1, -2, 1, 0, -5, 3, 4, 0, -1, -2, 1, 0, -5, -1, 6, 2, 0, -1, -2, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -1, 0, 0, 1, -1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 1, -1, -2, -1, 3, 0, 0, 1, -1, -2, -2, 2, 5, 0, 0, 1, -1, -2, -2, 0, 6, 5, 0, 0] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, -1, 1, 0, 0, 1, -2, -1, 1, 0, 0, 3, -1, -2, -1, 1, 0, 0, 5, 2, -2, -2, -1, 1, 0, 0, 5, 6, 0, -2, -2, -1, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 1, -1, 0, 0, -1, 2, 1, -1, 0, 0, -3, 1, 2, 1, -1, 0, 0, -5, -2, 2, 2, 1, -1, 0, 0, -5, -6, 0, 2, 2, 1, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -2, 0, 1, -2, 1, 1, -2, 0, 0, 1, -2, -1, 1] | - -CompositionInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 2, 1, -1, 4, -1, 1] | -| OddSum | [0, 0, -1, -2, -1, 1, -4, 1, -1] | -| AltSum | [1, 1, 2, 4, 2, -2, 8, -2, 2] | -| AccSum | [1, 2, 1, 0, -1, -1, 0, 3, 7] | -| AccRevSum | [1, 2, 1, 0, -1, -1, 0, 3, 7] | -| RevAccSum | [1, 1, -1, 0, 1, 1, 0, -3, -7] | -| DiagSum | [1, 1, 1, 0, -1, 0, -1, -1] | - -CompositionInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, -2, 6, 12, 60, 30] | -| RowGcd | [1, 1, 1, -2, -2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 2, 3, 4, 5, 6] | -| Middle | [1, 1, -1, -2, 0, -1, 1, 0, 2] | -| Central | [1, -1, 0, 1, 2] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, 2, 2, 2, 2, 2, 2] | -| NegHalf | [1, -2, 6, -18, 30, -42, 126, -234, 438] | -| Bin | [1, 1, -1, -2, -3, 6, 36, 134, 365] | -| InvBin | [1, -1, 3, -10, 5, 34, 56, -36, -195] | -| TransSqrs | [0, 0, -1, 2, 7, 5, -8, -47, -105] | -| TransNat0 | [0, 0, -1, 0, 1, 1, 0, -3, -7] | -| TransNat1 | [1, 1, -1, 0, 1, 1, 0, -3, -7] | - -CompositionInvRev Diagonals as rows +Composition:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, 0], [1, -2, 0, 1, 0], [1, -2, -1, 3, -1, 0], [1, -2, -1, 1, 4, -3, 0], [1, -2, -1, 0, 4, 3, -5, 0], [1, -2, -1, 0, 2, 6, -1, -5, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 1, -2, 1], [0, 1, 0, -2, 1], [0, -1, 3, -1, -2, 1], [0, -3, 4, 1, -1, -2, 1], [0, -5, 3, 4, 0, -1, -2, 1], [0, -5, -1, 6, 2, 0, -1, -2, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -2, 0], [1, -2, 1], [1, -2, 0, 0], [1, -2, -1, 1], [1, -2, -1, 3, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -1, 0, 0], [1, -1, -1, 0, 0], [1, -1, -2, 1, 0, 0], [1, -1, -2, -1, 3, 0, 0], [1, -1, -2, -2, 2, 5, 0, 0], [1, -1, -2, -2, 0, 6, 5, 0, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [0, 0, -1, 1], [0, 0, -1, -1, 1], [0, 0, 1, -2, -1, 1], [0, 0, 3, -1, -2, -1, 1], [0, 0, 5, 2, -2, -2, -1, 1], [0, 0, 5, 6, 0, -2, -2, -1, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 1, -1, 0], [0, 1, 1, -1, 0], [0, -1, 2, 1, -1, 0], [0, -3, 1, 2, 1, -1, 0], [0, -5, -2, 2, 2, 1, -1, 0], [0, -5, -6, 0, 2, 2, 1, -1, 0]] | + +Composition:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 2, 1, -1, 4, -1, 1] | +| OddSum | [0, 0, -1, -2, -1, 1, -4, 1, -1] | +| AltSum | [1, 1, 2, 4, 2, -2, 8, -2, 2] | +| AccSum | [1, 2, 1, 0, -1, -1, 0, 3, 7] | +| AccRevSum | [1, 1, -1, 0, 1, 1, 0, -3, -7] | +| AntiDiagSum | [1, 1, 1, 0, -1, 0, -1, -1, 1] | + +Composition:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 2, 6, 12, 60, 30] | +| RowGcd | [1, 1, 1, 2, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 2, 3, 4, 5, 6] | +| ColMiddle | [1, 1, -1, -2, 0, -1, 1, 0, 2] | +| ColECenter | [1, -1, 0, 1, 2] | +| ColOCenter | [1, -2, -1, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, 2, 7, 5, -8, -47, -105] | +| TransNat0 | [0, 0, -1, 0, 1, 1, 0, -3, -7] | +| TransNat1 | [1, 1, -1, 0, 1, 1, 0, -3, -7] | + +Composition:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ CompositionInvRev Diagonals as rows | DiagRow7 | [1, -2, -1, 0, 1, 2, 3, 5, 8]| | DiagRow8 | [1, -2, -1, 0, 1, 2, 3, 4, 6]| -CompositionInvRev Diagonals as columns +Composition:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ CompositionInvRev Diagonals as columns | DiagCol7 | [0, -5, -11, 10, 10, 8, 7, 5, 4] | | DiagCol8 | [0, -1, -28, 9, 15, 11, 9, 8, 6] | -CompositionInvRev Polynomial values as rows +Composition:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ CompositionInvRev Polynomial values as rows | PolyRow7 | [1, 0, -167, -2606, -16407, -66284, -204815, -528282, -1196111] | | PolyRow8 | [1, 0, -487, -10058, -79383, -386284, -1397135, -4129782, -10543183] | -CompositionInvRev Polynomial values as columns +Composition:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/CompositionMax.md b/data/md/CompositionMax.md index 40b6d2e..97ab355 100644 --- a/data/md/CompositionMax.md +++ b/data/md/CompositionMax.md @@ -14,45 +14,41 @@ CompositionMax Triangle view | Row7 | [0, 1, 21, 44, 56, 61, 63, 64] | | Row8 | [0, 1, 34, 81, 108, 120, 125, 127, 128] | -CompositionMax Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 7, 8, 0, 1, 8, 13, 15, 16, 0, 1, 13, 24, 29, 31, 32, 0, 1, 21, 44, 56, 61, 63, 64, 0, 1, 34, 81, 108, 120, 125, 127, 128] | -| TriRev | [1, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 7, 5, 1, 0, 16, 15, 13, 8, 1, 0, 32, 31, 29, 24, 13, 1, 0, 64, 63, 61, 56, 44, 21, 1, 0, 128, 127, 125, 120, 108, 81, 34, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 3, 0, 1, 4, 8, 0, 1, 6, 13, 21, 0, 1, 9, 22, 37, 53, 0, 1, 14, 38, 67, 98, 130, 0, 1, 22, 66, 122, 183, 246, 310, 0, 1, 35, 116, 224, 344, 469, 596, 724] | -| TriRevAcc | [1, 1, 0, 3, 1, 0, 8, 4, 1, 0, 21, 13, 6, 1, 0, 53, 37, 22, 9, 1, 0, 130, 98, 67, 38, 14, 1, 0, 310, 246, 183, 122, 66, 22, 1, 0, 724, 596, 469, 344, 224, 116, 35, 1, 0] | -| TriAccRev | [1, 1, 1, 2, 3, 3, 4, 7, 8, 8, 8, 15, 20, 21, 21, 16, 31, 44, 52, 53, 53, 32, 63, 92, 116, 129, 130, 130, 64, 127, 188, 244, 288, 309, 310, 310, 128, 255, 380, 500, 608, 689, 723, 724, 724] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 5, 4, 0, 1, 8, 7] | +CompositionMax Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 4], [0, 1, 5, 7, 8], [0, 1, 8, 13, 15, 16], [0, 1, 13, 24, 29, 31, 32], [0, 1, 21, 44, 56, 61, 63, 64], [0, 1, 34, 81, 108, 120, 125, 127, 128]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [4, 3, 1, 0], [8, 7, 5, 1, 0], [16, 15, 13, 8, 1, 0], [32, 31, 29, 24, 13, 1, 0], [64, 63, 61, 56, 44, 21, 1, 0], [128, 127, 125, 120, 108, 81, 34, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 3], [0, 1, 5, 4], [0, 1, 8, 7], [0, 1, 13, 13, 8]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 4, 8], [0, 1, 6, 13, 21], [0, 1, 9, 22, 37, 53], [0, 1, 14, 38, 67, 98, 130], [0, 1, 22, 66, 122, 183, 246, 310], [0, 1, 35, 116, 224, 344, 469, 596, 724]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [8, 4, 1, 0], [21, 13, 6, 1, 0], [53, 37, 22, 9, 1, 0], [130, 98, 67, 38, 14, 1, 0], [310, 246, 183, 122, 66, 22, 1, 0], [724, 596, 469, 344, 224, 116, 35, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [4, 7, 8, 8], [8, 15, 20, 21, 21], [16, 31, 44, 52, 53, 53], [32, 63, 92, 116, 129, 130, 130], [64, 127, 188, 244, 288, 309, 310, 310], [128, 255, 380, 500, 608, 689, 723, 724, 724]] | CompositionMax Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | -| EvenSum | [1, 0, 2, 3, 13, 23, 74, 140, 395] | -| OddSum | [0, 1, 1, 5, 8, 30, 56, 170, 329] | -| AltSum | [1, -1, 1, -2, 5, -7, 18, -30, 66] | -| AccSum | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | -| AccRevSum | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | -| RevAccSum | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | -| DiagSum | [1, 0, 1, 1, 3, 4, 10, 16] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | +| EvenSum | [1, 0, 2, 3, 13, 23, 74, 140, 395] | +| OddSum | [0, 1, 1, 5, 8, 30, 56, 170, 329] | +| AltSum | [1, -1, 1, -2, 5, -7, 18, -30, 66] | +| AccSum | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | +| AccRevSum | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | +| AntiDiagSum | [1, 0, 1, 1, 3, 4, 10, 16, 35] | CompositionMax Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 280, 3120, 1121952, 2705472, 2798064000] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| Middle | [1, 0, 1, 1, 5, 8, 24, 44, 108] | -| Central | [1, 1, 5, 24, 108] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| PosHalf | [1, 1, 4, 14, 50, 178, 642, 2322, 8466] | -| NegHalf | [1, 1, 0, 2, 6, -10, 70, -170, 598] | -| Bin | [1, 1, 4, 16, 70, 306, 1334, 5734, 24420] | -| InvBin | [1, 1, 0, -2, 6, -4, -10, 50, -140] | -| TransSqrs | [0, 1, 9, 49, 212, 790, 2660, 8306, 24509] | -| TransNat0 | [0, 1, 5, 19, 64, 196, 562, 1530, 4007] | -| TransNat1 | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 280, 3120, 1121952, 2705472, 2798064000] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| ColMiddle | [1, 0, 1, 1, 5, 8, 24, 44, 108] | +| ColECenter | [1, 1, 5, 24, 108] | +| ColOCenter | [0, 1, 8, 44] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| TransSqrs | [0, 1, 9, 49, 212, 790, 2660, 8306, 24509] | +| TransNat0 | [0, 1, 5, 19, 64, 196, 562, 1530, 4007] | +| TransNat1 | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | CompositionMax Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ CompositionMax Polynomial values as columns | PolyCol7 | [1, 7, 105, 1526, 21861, 309785, 4364290, 61294450, 859496218] | | PolyCol8 | [1, 8, 136, 2248, 36680, 592904, 9536328, 152984904, 2451008136] | -# CompositionMaxRev -True +# CompositionMax:Rev +[] -CompositionMaxRev Triangle view +CompositionMax:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ CompositionMaxRev Triangle view | Row7 | [64, 63, 61, 56, 44, 21, 1, 0] | | Row8 | [128, 127, 125, 120, 108, 81, 34, 1, 0] | -CompositionMaxRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 7, 5, 1, 0, 16, 15, 13, 8, 1, 0, 32, 31, 29, 24, 13, 1, 0, 64, 63, 61, 56, 44, 21, 1, 0, 128, 127, 125, 120, 108, 81, 34, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 7, 8, 0, 1, 8, 13, 15, 16, 0, 1, 13, 24, 29, 31, 32, 0, 1, 21, 44, 56, 61, 63, 64, 0, 1, 34, 81, 108, 120, 125, 127, 128] | -| TriAcc | [1, 1, 1, 2, 3, 3, 4, 7, 8, 8, 8, 15, 20, 21, 21, 16, 31, 44, 52, 53, 53, 32, 63, 92, 116, 129, 130, 130, 64, 127, 188, 244, 288, 309, 310, 310, 128, 255, 380, 500, 608, 689, 723, 724, 724] | -| TriRevAcc | [1, 1, 1, 3, 3, 2, 8, 8, 7, 4, 21, 21, 20, 15, 8, 53, 53, 52, 44, 31, 16, 130, 130, 129, 116, 92, 63, 32, 310, 310, 309, 288, 244, 188, 127, 64, 724, 724, 723, 689, 608, 500, 380, 255, 128] | -| TriAccRev | [1, 0, 1, 0, 1, 3, 0, 1, 4, 8, 0, 1, 6, 13, 21, 0, 1, 9, 22, 37, 53, 0, 1, 14, 38, 67, 98, 130, 0, 1, 22, 66, 122, 183, 246, 310, 0, 1, 35, 116, 224, 344, 469, 596, 724] | -| TriDiag | [1, 1, 2, 0, 4, 1, 8, 3, 0, 16, 7, 1, 32, 15, 5, 0, 64, 31, 13, 1] | - -CompositionMaxRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | -| EvenSum | [1, 1, 2, 5, 13, 30, 74, 170, 395] | -| OddSum | [0, 0, 1, 3, 8, 23, 56, 140, 329] | -| AltSum | [1, 1, 1, 2, 5, 7, 18, 30, 66] | -| AccSum | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | -| AccRevSum | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | -| RevAccSum | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | -| DiagSum | [1, 1, 2, 5, 11, 24, 52, 109] | - -CompositionMaxRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 280, 3120, 1121952, 2705472, 2798064000] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| Middle | [1, 1, 1, 3, 5, 13, 24, 56, 108] | -| Central | [1, 1, 5, 24, 108] | -| LeftSide | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 10, 46, 206, 890, 3750, 15510, 63378] | -| NegHalf | [1, -2, 6, -22, 90, -346, 1378, -5486, 21886] | -| Bin | [1, 1, 4, 16, 70, 306, 1334, 5734, 24420] | -| InvBin | [1, -1, 0, 2, 6, 4, -10, -50, -140] | -| TransSqrs | [0, 0, 1, 7, 36, 155, 596, 2076, 6733] | -| TransNat0 | [0, 0, 1, 5, 20, 69, 218, 640, 1785] | -| TransNat1 | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | - -CompositionMaxRev Diagonals as rows +CompositionMax:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [4, 3, 1, 0], [8, 7, 5, 1, 0], [16, 15, 13, 8, 1, 0], [32, 31, 29, 24, 13, 1, 0], [64, 63, 61, 56, 44, 21, 1, 0], [128, 127, 125, 120, 108, 81, 34, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 4], [0, 1, 5, 7, 8], [0, 1, 8, 13, 15, 16], [0, 1, 13, 24, 29, 31, 32], [0, 1, 21, 44, 56, 61, 63, 64], [0, 1, 34, 81, 108, 120, 125, 127, 128]] | +| AntiDiag | [[1], [1], [2, 0], [4, 1], [8, 3, 0], [16, 7, 1], [32, 15, 5, 0], [64, 31, 13, 1], [128, 63, 29, 8, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [4, 7, 8, 8], [8, 15, 20, 21, 21], [16, 31, 44, 52, 53, 53], [32, 63, 92, 116, 129, 130, 130], [64, 127, 188, 244, 288, 309, 310, 310], [128, 255, 380, 500, 608, 689, 723, 724, 724]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [8, 8, 7, 4], [21, 21, 20, 15, 8], [53, 53, 52, 44, 31, 16], [130, 130, 129, 116, 92, 63, 32], [310, 310, 309, 288, 244, 188, 127, 64], [724, 724, 723, 689, 608, 500, 380, 255, 128]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 4, 8], [0, 1, 6, 13, 21], [0, 1, 9, 22, 37, 53], [0, 1, 14, 38, 67, 98, 130], [0, 1, 22, 66, 122, 183, 246, 310], [0, 1, 35, 116, 224, 344, 469, 596, 724]] | + +CompositionMax:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 8, 21, 53, 130, 310, 724] | +| EvenSum | [1, 1, 2, 5, 13, 30, 74, 170, 395] | +| OddSum | [0, 0, 1, 3, 8, 23, 56, 140, 329] | +| AltSum | [1, 1, 1, 2, 5, 7, 18, 30, 66] | +| AccSum | [1, 2, 8, 27, 85, 249, 692, 1840, 4731] | +| AccRevSum | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | +| AntiDiagSum | [1, 1, 2, 5, 11, 24, 52, 109, 228] | + +CompositionMax:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 280, 3120, 1121952, 2705472, 2798064000] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| ColMiddle | [1, 1, 1, 3, 5, 13, 24, 56, 108] | +| ColECenter | [1, 1, 5, 24, 108] | +| ColOCenter | [1, 3, 13, 56] | +| ColLeft | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 7, 36, 155, 596, 2076, 6733] | +| TransNat0 | [0, 0, 1, 5, 20, 69, 218, 640, 1785] | +| TransNat1 | [1, 1, 4, 13, 41, 122, 348, 950, 2509] | + +CompositionMax:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ CompositionMaxRev Diagonals as rows | DiagRow7 | [64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808]| | DiagRow8 | [128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192]| -CompositionMaxRev Diagonals as columns +CompositionMax:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 4, 8, 16, 32, 64, 128] | @@ -188,7 +180,7 @@ CompositionMaxRev Diagonals as columns | DiagCol7 | [0, 1, 55, 274, 773, 1793, 3840, 7936, 16128] | | DiagCol8 | [0, 1, 89, 504, 1490, 3525, 7617, 15808, 32192] | -CompositionMaxRev Polynomial values as rows +CompositionMax:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ CompositionMaxRev Polynomial values as rows | PolyRow7 | [64, 310, 2322, 11710, 41740, 117654, 281710, 598942, 1163640] | | PolyRow8 | [128, 724, 8466, 60278, 276556, 948888, 2667374, 6492586, 14177208] | -CompositionMaxRev Polynomial values as columns +CompositionMax:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 4, 8, 16, 32, 64, 128] | diff --git a/data/md/Delannoy.md b/data/md/Delannoy.md index bd8c1c8..f0ca71e 100644 --- a/data/md/Delannoy.md +++ b/data/md/Delannoy.md @@ -14,45 +14,41 @@ Delannoy Triangle view | Row7 | [1, 13, 61, 129, 129, 61, 13, 1] | | Row8 | [1, 15, 85, 231, 321, 231, 85, 15, 1] | -Delannoy Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1] | -| TriRev | [1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1] | -| TriAcc | [1, 1, 2, 1, 4, 5, 1, 6, 11, 12, 1, 8, 21, 28, 29, 1, 10, 35, 60, 69, 70, 1, 12, 53, 116, 157, 168, 169, 1, 14, 75, 204, 333, 394, 407, 408, 1, 16, 101, 332, 653, 884, 969, 984, 985] | -| TriRevAcc | [1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1] | -| TriAccRev | [1, 1, 2, 1, 4, 5, 1, 6, 11, 12, 1, 8, 21, 28, 29, 1, 10, 35, 60, 69, 70, 1, 12, 53, 116, 157, 168, 169, 1, 14, 75, 204, 333, 394, 407, 408, 1, 16, 101, 332, 653, 884, 969, 984, 985] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 7, 5, 1, 9, 13, 1, 1, 11, 25, 7] | +Delannoy Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 7, 13, 7, 1], [1, 9, 25, 25, 9, 1], [1, 11, 41, 63, 41, 11, 1], [1, 13, 61, 129, 129, 61, 13, 1], [1, 15, 85, 231, 321, 231, 85, 15, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 7, 13, 7, 1], [1, 9, 25, 25, 9, 1], [1, 11, 41, 63, 41, 11, 1], [1, 13, 61, 129, 129, 61, 13, 1], [1, 15, 85, 231, 321, 231, 85, 15, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 5, 1], [1, 7, 5], [1, 9, 13, 1], [1, 11, 25, 7], [1, 13, 41, 25, 1]] | +| AccTabl | [[1], [1, 2], [1, 4, 5], [1, 6, 11, 12], [1, 8, 21, 28, 29], [1, 10, 35, 60, 69, 70], [1, 12, 53, 116, 157, 168, 169], [1, 14, 75, 204, 333, 394, 407, 408], [1, 16, 101, 332, 653, 884, 969, 984, 985]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 1], [12, 11, 6, 1], [29, 28, 21, 8, 1], [70, 69, 60, 35, 10, 1], [169, 168, 157, 116, 53, 12, 1], [408, 407, 394, 333, 204, 75, 14, 1], [985, 984, 969, 884, 653, 332, 101, 16, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 5], [1, 6, 11, 12], [1, 8, 21, 28, 29], [1, 10, 35, 60, 69, 70], [1, 12, 53, 116, 157, 168, 169], [1, 14, 75, 204, 333, 394, 407, 408], [1, 16, 101, 332, 653, 884, 969, 984, 985]] | Delannoy Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 12, 29, 70, 169, 408, 985] | -| EvenSum | [1, 1, 2, 6, 15, 35, 84, 204, 493] | -| OddSum | [0, 1, 3, 6, 14, 35, 85, 204, 492] | -| AltSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| AccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| AccRevSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| RevAccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| DiagSum | [1, 1, 2, 4, 7, 13, 24, 44] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 12, 29, 70, 169, 408, 985] | +| EvenSum | [1, 1, 2, 6, 15, 35, 84, 204, 493] | +| OddSum | [0, 1, 3, 6, 14, 35, 85, 204, 492] | +| AltSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| AccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | +| AccRevSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | +| AntiDiagSum | [1, 1, 2, 4, 7, 13, 24, 44, 81] | Delannoy Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 5, 91, 225, 28413, 102297, 2100945] | -| RowGcd | [1, 1, 3, 5, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 5, 13, 25, 63, 129, 321] | -| Middle | [1, 1, 3, 5, 13, 25, 63, 129, 321] | -| Central | [1, 3, 13, 63, 321] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 11, 39, 139, 495, 1763, 6279, 22363] | -| NegHalf | [1, -1, -1, 3, -1, -5, 7, 3, -17] | -| Bin | [1, 2, 8, 32, 136, 592, 2624, 11776, 53344] | -| InvBin | [1, 0, -4, 0, 24, 0, -160, 0, 1120] | -| TransSqrs | [0, 1, 7, 34, 138, 503, 1709, 5524, 17204] | -| TransNat0 | [0, 1, 5, 18, 58, 175, 507, 1428, 3940] | -| TransNat1 | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 5, 91, 225, 28413, 102297, 2100945] | +| RowGcd | [1, 1, 3, 5, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 5, 13, 25, 63, 129, 321] | +| ColMiddle | [1, 1, 3, 5, 13, 25, 63, 129, 321] | +| ColECenter | [1, 3, 13, 63, 321] | +| ColOCenter | [1, 5, 25, 129] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 34, 138, 503, 1709, 5524, 17204] | +| TransNat0 | [0, 1, 5, 18, 58, 175, 507, 1428, 3940] | +| TransNat1 | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | Delannoy Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,114 @@ Delannoy Polynomial values as columns | PolyCol7 | [1, 8, 71, 624, 5489, 48280, 424663, 3735264, 32854753] | | PolyCol8 | [1, 9, 89, 873, 8569, 84105, 825497, 8102313, 79524793] | -# DelannoyRev -True +# Delannoy:Inv +[] -DelannoyRev Triangle view +Delannoy:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [2, -3, 1] | +| Row3 | [-6, 10, -5, 1] | +| Row4 | [22, -38, 22, -7, 1] | +| Row5 | [-90, 158, -98, 38, -9, 1] | +| Row6 | [394, -698, 450, -194, 58, -11, 1] | +| Row7 | [-1806, 3218, -2126, 978, -334, 82, -13, 1] | +| Row8 | [8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1] | + +Delannoy:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [2, -3, 1], [-6, 10, -5, 1], [22, -38, 22, -7, 1], [-90, 158, -98, 38, -9, 1], [394, -698, 450, -194, 58, -11, 1], [-1806, 3218, -2126, 978, -334, 82, -13, 1], [8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 2], [1, -5, 10, -6], [1, -7, 22, -38, 22], [1, -9, 38, -98, 158, -90], [1, -11, 58, -194, 450, -698, 394], [1, -13, 82, -334, 978, -2126, 3218, -1806], [1, -15, 110, -526, 1838, -4942, 10286, -15310, 8558]] | +| AntiDiag | [[1], [-1], [2, 1], [-6, -3], [22, 10, 1], [-90, -38, -5], [394, 158, 22, 1], [-1806, -698, -98, -7], [8558, 3218, 450, 38, 1]] | +| AccTabl | [[1], [-1, 0], [2, -1, 0], [-6, 4, -1, 0], [22, -16, 6, -1, 0], [-90, 68, -30, 8, -1, 0], [394, -304, 146, -48, 10, -1, 0], [-1806, 1412, -714, 264, -70, 12, -1, 0], [8558, -6752, 3534, -1408, 430, -96, 14, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 2], [0, -1, 4, -6], [0, -1, 6, -16, 22], [0, -1, 8, -30, 68, -90], [0, -1, 10, -48, 146, -304, 394], [0, -1, 12, -70, 264, -714, 1412, -1806], [0, -1, 14, -96, 430, -1408, 3534, -6752, 8558]] | +| AccRevTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 6, 0], [1, -6, 16, -22, 0], [1, -8, 30, -68, 90, 0], [1, -10, 48, -146, 304, -394, 0], [1, -12, 70, -264, 714, -1412, 1806, 0], [1, -14, 96, -430, 1408, -3534, 6752, -8558, 0]] | + +Delannoy:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 3, -11, 45, -197, 903, -4279, 20793] | +| OddSum | [0, 1, -3, 11, -45, 197, -903, 4279, -20793] | +| AltSum | [1, -2, 6, -22, 90, -394, 1806, -8558, 41586] | +| AccSum | [1, -1, 1, -3, 11, -45, 197, -903, 4279] | +| AccRevSum | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | +| AntiDiagSum | [1, -1, 3, -9, 33, -133, 575, -2609, 12265] | + +Delannoy:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 30, 2926, 6619410, 957340835550, 44816541527197986, 603669538270861563270] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 10, 38, 158, 698, 3218, 15310] | +| ColMiddle | [1, -1, -3, 10, 22, -98, -194, 978, 1838] | +| ColECenter | [1, -3, 22, -194, 1838] | +| ColOCenter | [-1, 10, -98, 978] | +| ColLeft | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -1, 3, -11, 45, -197, 903] | +| TransNat0 | [0, 1, -1, 3, -11, 45, -197, 903, -4279] | +| TransNat1 | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | + +Delannoy:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -3, -5, -7, -9, -11, -13, -15, -17]| +| DiagRow2 | [2, 10, 22, 38, 58, 82, 110, 142, 178]| +| DiagRow3 | [-6, -38, -98, -194, -334, -526, -778, -1098, -1494]| +| DiagRow4 | [22, 158, 450, 978, 1838, 3142, 5018, 7610, 11078]| +| DiagRow5 | [-90, -698, -2126, -4942, -9922, -18082, -30710, -49398, -76074]| +| DiagRow6 | [394, 3218, 10286, 25150, 53154, 101946, 182054, 307526, 496634]| +| DiagRow7 | [-1806, -15310, -50746, -129050, -284150, -568150, -1057730, -1861890, -3131166]| +| DiagRow8 | [8558, 74614, 254410, 667610, 1519910, 3145790, 6065410, 11058466, 19258078]| + +Delannoy:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| DiagCol1 | [1, -3, 10, -38, 158, -698, 3218, -15310, 74614] | +| DiagCol2 | [1, -5, 22, -98, 450, -2126, 10286, -50746, 254410] | +| DiagCol3 | [1, -7, 38, -194, 978, -4942, 25150, -129050, 667610] | +| DiagCol4 | [1, -9, 58, -334, 1838, -9922, 53154, -284150, 1519910] | +| DiagCol5 | [1, -11, 82, -526, 3142, -18082, 101946, -568150, 3145790] | +| DiagCol6 | [1, -13, 110, -778, 5018, -30710, 182054, -1057730, 6065410] | +| DiagCol7 | [1, -15, 142, -1098, 7610, -49398, 307526, -1861890, 11058466] | +| DiagCol8 | [1, -17, 178, -1494, 11078, -76074, 496634, -3131166, 19258078] | + +Delannoy:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [2, 0, 0, 2, 6, 12, 20, 30, 42] | +| PolyRow3 | [-6, 0, 2, 6, 18, 44, 90, 162, 266] | +| PolyRow4 | [22, 0, -6, -2, 30, 132, 370, 834, 1638] | +| PolyRow5 | [-90, 0, 26, 42, 126, 500, 1650, 4446, 10262] | +| PolyRow6 | [394, 0, -114, -134, 66, 1404, 6790, 23046, 63546] | +| PolyRow7 | [-1806, 0, 526, 702, 1242, 5884, 30390, 122346, 396802] | +| PolyRow8 | [8558, 0, -2502, -3226, -2154, 13908, 124130, 635898, 2462334] | + +Delannoy:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 0, 2, -6, 26, -114, 526, -2502] | +| PolyCol3 | [1, 2, 2, 6, -2, 42, -134, 702, -3226] | +| PolyCol4 | [1, 3, 6, 18, 30, 126, 66, 1242, -2154] | +| PolyCol5 | [1, 4, 12, 44, 132, 500, 1404, 5884, 13908] | +| PolyCol6 | [1, 5, 20, 90, 370, 1650, 6790, 30390, 124130] | +| PolyCol7 | [1, 6, 30, 162, 834, 4446, 23046, 122346, 635898] | +| PolyCol8 | [1, 7, 42, 266, 1638, 10262, 63546, 396802, 2462334] | + +# Delannoy:Rev +[] + +Delannoy:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +222,43 @@ DelannoyRev Triangle view | Row7 | [1, 13, 61, 129, 129, 61, 13, 1] | | Row8 | [1, 15, 85, 231, 321, 231, 85, 15, 1] | -DelannoyRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1] | -| TriRev | [1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1] | -| TriAcc | [1, 1, 2, 1, 4, 5, 1, 6, 11, 12, 1, 8, 21, 28, 29, 1, 10, 35, 60, 69, 70, 1, 12, 53, 116, 157, 168, 169, 1, 14, 75, 204, 333, 394, 407, 408, 1, 16, 101, 332, 653, 884, 969, 984, 985] | -| TriRevAcc | [1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1] | -| TriAccRev | [1, 1, 2, 1, 4, 5, 1, 6, 11, 12, 1, 8, 21, 28, 29, 1, 10, 35, 60, 69, 70, 1, 12, 53, 116, 157, 168, 169, 1, 14, 75, 204, 333, 394, 407, 408, 1, 16, 101, 332, 653, 884, 969, 984, 985] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 7, 5, 1, 9, 13, 1, 1, 11, 25, 7] | - -DelannoyRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 12, 29, 70, 169, 408, 985] | -| EvenSum | [1, 1, 2, 6, 15, 35, 84, 204, 493] | -| OddSum | [0, 1, 3, 6, 14, 35, 85, 204, 492] | -| AltSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| AccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| AccRevSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| RevAccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | -| DiagSum | [1, 1, 2, 4, 7, 13, 24, 44] | - -DelannoyRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 5, 91, 225, 28413, 102297, 2100945] | -| RowGcd | [1, 1, 3, 5, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 5, 13, 25, 63, 129, 321] | -| Middle | [1, 1, 3, 5, 13, 25, 63, 129, 321] | -| Central | [1, 3, 13, 63, 321] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 11, 39, 139, 495, 1763, 6279, 22363] | -| NegHalf | [1, -1, -1, 3, -1, -5, 7, 3, -17] | -| Bin | [1, 2, 8, 32, 136, 592, 2624, 11776, 53344] | -| InvBin | [1, 0, -4, 0, 24, 0, -160, 0, 1120] | -| TransSqrs | [0, 1, 7, 34, 138, 503, 1709, 5524, 17204] | -| TransNat0 | [0, 1, 5, 18, 58, 175, 507, 1428, 3940] | -| TransNat1 | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | - -DelannoyRev Diagonals as rows +Delannoy:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 7, 13, 7, 1], [1, 9, 25, 25, 9, 1], [1, 11, 41, 63, 41, 11, 1], [1, 13, 61, 129, 129, 61, 13, 1], [1, 15, 85, 231, 321, 231, 85, 15, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 1], [1, 5, 5, 1], [1, 7, 13, 7, 1], [1, 9, 25, 25, 9, 1], [1, 11, 41, 63, 41, 11, 1], [1, 13, 61, 129, 129, 61, 13, 1], [1, 15, 85, 231, 321, 231, 85, 15, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 5, 1], [1, 7, 5], [1, 9, 13, 1], [1, 11, 25, 7], [1, 13, 41, 25, 1]] | +| AccTabl | [[1], [1, 2], [1, 4, 5], [1, 6, 11, 12], [1, 8, 21, 28, 29], [1, 10, 35, 60, 69, 70], [1, 12, 53, 116, 157, 168, 169], [1, 14, 75, 204, 333, 394, 407, 408], [1, 16, 101, 332, 653, 884, 969, 984, 985]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 1], [12, 11, 6, 1], [29, 28, 21, 8, 1], [70, 69, 60, 35, 10, 1], [169, 168, 157, 116, 53, 12, 1], [408, 407, 394, 333, 204, 75, 14, 1], [985, 984, 969, 884, 653, 332, 101, 16, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 5], [1, 6, 11, 12], [1, 8, 21, 28, 29], [1, 10, 35, 60, 69, 70], [1, 12, 53, 116, 157, 168, 169], [1, 14, 75, 204, 333, 394, 407, 408], [1, 16, 101, 332, 653, 884, 969, 984, 985]] | + +Delannoy:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 12, 29, 70, 169, 408, 985] | +| EvenSum | [1, 1, 2, 6, 15, 35, 84, 204, 493] | +| OddSum | [0, 1, 3, 6, 14, 35, 85, 204, 492] | +| AltSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| AccSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | +| AccRevSum | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | +| AntiDiagSum | [1, 1, 2, 4, 7, 13, 24, 44, 81] | + +Delannoy:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 5, 91, 225, 28413, 102297, 2100945] | +| RowGcd | [1, 1, 3, 5, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 5, 13, 25, 63, 129, 321] | +| ColMiddle | [1, 1, 3, 5, 13, 25, 63, 129, 321] | +| ColECenter | [1, 3, 13, 63, 321] | +| ColOCenter | [1, 5, 25, 129] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 34, 138, 503, 1709, 5524, 17204] | +| TransNat0 | [0, 1, 5, 18, 58, 175, 507, 1428, 3940] | +| TransNat1 | [1, 3, 10, 30, 87, 245, 676, 1836, 4925] | + +Delannoy:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +271,7 @@ DelannoyRev Diagonals as rows | DiagRow7 | [1, 15, 113, 575, 2241, 7183, 19825, 48639, 108545]| | DiagRow8 | [1, 17, 145, 833, 3649, 13073, 40081, 108545, 265729]| -DelannoyRev Diagonals as columns +Delannoy:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -188,7 +284,7 @@ DelannoyRev Diagonals as columns | DiagCol7 | [1, 15, 113, 575, 2241, 7183, 19825, 48639, 108545] | | DiagCol8 | [1, 17, 145, 833, 3649, 13073, 40081, 108545, 265729] | -DelannoyRev Polynomial values as rows +Delannoy:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +297,7 @@ DelannoyRev Polynomial values as rows | PolyRow7 | [1, 408, 6279, 41008, 174405, 570216, 1558123, 3735264, 8102313] | | PolyRow8 | [1, 985, 22363, 190513, 994381, 3844201, 12109735, 32854753, 79524793] | -DelannoyRev Polynomial values as columns +Delannoy:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -214,3 +310,211 @@ DelannoyRev Polynomial values as columns | PolyCol7 | [1, 8, 71, 624, 5489, 48280, 424663, 3735264, 32854753] | | PolyCol8 | [1, 9, 89, 873, 8569, 84105, 825497, 8102313, 79524793] | +# Delannoy:Inv:Rev +[] + +Delannoy:Inv:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, -1] | +| Row2 | [1, -3, 2] | +| Row3 | [1, -5, 10, -6] | +| Row4 | [1, -7, 22, -38, 22] | +| Row5 | [1, -9, 38, -98, 158, -90] | +| Row6 | [1, -11, 58, -194, 450, -698, 394] | +| Row7 | [1, -13, 82, -334, 978, -2126, 3218, -1806] | +| Row8 | [1, -15, 110, -526, 1838, -4942, 10286, -15310, 8558] | + +Delannoy:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -3, 2], [1, -5, 10, -6], [1, -7, 22, -38, 22], [1, -9, 38, -98, 158, -90], [1, -11, 58, -194, 450, -698, 394], [1, -13, 82, -334, 978, -2126, 3218, -1806], [1, -15, 110, -526, 1838, -4942, 10286, -15310, 8558]] | +| RevTabl | [[1], [-1, 1], [2, -3, 1], [-6, 10, -5, 1], [22, -38, 22, -7, 1], [-90, 158, -98, 38, -9, 1], [394, -698, 450, -194, 58, -11, 1], [-1806, 3218, -2126, 978, -334, 82, -13, 1], [8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -3], [1, -5, 2], [1, -7, 10], [1, -9, 22, -6], [1, -11, 38, -38], [1, -13, 58, -98, 22]] | +| AccTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 6, 0], [1, -6, 16, -22, 0], [1, -8, 30, -68, 90, 0], [1, -10, 48, -146, 304, -394, 0], [1, -12, 70, -264, 714, -1412, 1806, 0], [1, -14, 96, -430, 1408, -3534, 6752, -8558, 0]] | +| RevAccTabl | [[1], [0, 1], [0, -2, 1], [0, 6, -4, 1], [0, -22, 16, -6, 1], [0, 90, -68, 30, -8, 1], [0, -394, 304, -146, 48, -10, 1], [0, 1806, -1412, 714, -264, 70, -12, 1], [0, -8558, 6752, -3534, 1408, -430, 96, -14, 1]] | +| AccRevTabl | [[1], [-1, 0], [2, -1, 0], [-6, 4, -1, 0], [22, -16, 6, -1, 0], [-90, 68, -30, 8, -1, 0], [394, -304, 146, -48, 10, -1, 0], [-1806, 1412, -714, 264, -70, 12, -1, 0], [8558, -6752, 3534, -1408, 430, -96, 14, -1, 0]] | + +Delannoy:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| OddSum | [0, -1, -3, -11, -45, -197, -903, -4279, -20793] | +| AltSum | [1, 2, 6, 22, 90, 394, 1806, 8558, 41586] | +| AccSum | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | +| AccRevSum | [1, -1, 1, -3, 11, -45, 197, -903, 4279] | +| AntiDiagSum | [1, 1, 0, -2, -2, 4, 8, -10, -30] | + +Delannoy:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 30, 2926, 6619410, 957340835550, 44816541527197986, 603669538270861563270] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 10, 38, 158, 698, 3218, 15310] | +| ColMiddle | [1, 1, -3, -5, 22, 38, -194, -334, 1838] | +| ColECenter | [1, -3, 22, -194, 1838] | +| ColOCenter | [1, -5, 38, -334] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| TransSqrs | [0, -1, 5, -19, 91, -461, 2409, -12839, 69367] | +| TransNat0 | [0, -1, 1, -3, 11, -45, 197, -903, 4279] | +| TransNat1 | [1, -1, 1, -3, 11, -45, 197, -903, 4279] | + +Delannoy:Inv:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, -1, 2, -6, 22, -90, 394, -1806, 8558]| +| DiagRow1 | [1, -3, 10, -38, 158, -698, 3218, -15310, 74614]| +| DiagRow2 | [1, -5, 22, -98, 450, -2126, 10286, -50746, 254410]| +| DiagRow3 | [1, -7, 38, -194, 978, -4942, 25150, -129050, 667610]| +| DiagRow4 | [1, -9, 58, -334, 1838, -9922, 53154, -284150, 1519910]| +| DiagRow5 | [1, -11, 82, -526, 3142, -18082, 101946, -568150, 3145790]| +| DiagRow6 | [1, -13, 110, -778, 5018, -30710, 182054, -1057730, 6065410]| +| DiagRow7 | [1, -15, 142, -1098, 7610, -49398, 307526, -1861890, 11058466]| +| DiagRow8 | [1, -17, 178, -1494, 11078, -76074, 496634, -3131166, 19258078]| + +Delannoy:Inv:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [-1, -3, -5, -7, -9, -11, -13, -15, -17] | +| DiagCol2 | [2, 10, 22, 38, 58, 82, 110, 142, 178] | +| DiagCol3 | [-6, -38, -98, -194, -334, -526, -778, -1098, -1494] | +| DiagCol4 | [22, 158, 450, 978, 1838, 3142, 5018, 7610, 11078] | +| DiagCol5 | [-90, -698, -2126, -4942, -9922, -18082, -30710, -49398, -76074] | +| DiagCol6 | [394, 3218, 10286, 25150, 53154, 101946, 182054, 307526, 496634] | +| DiagCol7 | [-1806, -15310, -50746, -129050, -284150, -568150, -1057730, -1861890, -3131166] | +| DiagCol8 | [8558, 74614, 254410, 667610, 1519910, 3145790, 6065410, 11058466, 19258078] | + +Delannoy:Inv:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| PolyRow2 | [1, 0, 3, 10, 21, 36, 55, 78, 105] | +| PolyRow3 | [1, 0, -17, -86, -243, -524, -965, -1602, -2471] | +| PolyRow4 | [1, 0, 123, 934, 3525, 9516, 21055, 40818, 72009] | +| PolyRow5 | [1, 0, -1001, -11402, -57411, -193844, -514925, -1165086, -2349767] | +| PolyRow6 | [1, 0, 8739, 149314, 1002741, 4233396, 13498135, 35639094, 82160169] | +| PolyRow7 | [1, 0, -79969, -2049518, -18355347, -96884764, -370758965, -1142218314, -3009701863] | +| PolyRow8 | [1, 0, 756939, 29099278, 347527461, 2293235676, 10532080495, 37858038954, 114013757193] | + +Delannoy:Inv:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, -1, 3, -17, 123, -1001, 8739, -79969, 756939] | +| PolyCol3 | [1, -2, 10, -86, 934, -11402, 149314, -2049518, 29099278] | +| PolyCol4 | [1, -3, 21, -243, 3525, -57411, 1002741, -18355347, 347527461] | +| PolyCol5 | [1, -4, 36, -524, 9516, -193844, 4233396, -96884764, 2293235676] | +| PolyCol6 | [1, -5, 55, -965, 21055, -514925, 13498135, -370758965, 10532080495] | +| PolyCol7 | [1, -6, 78, -1602, 40818, -1165086, 35639094, -1142218314, 37858038954] | +| PolyCol8 | [1, -7, 105, -2471, 72009, -2349767, 82160169, -3009701863, 114013757193] | + +# Delannoy:Rev:Inv +[] + +Delannoy:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [2, -3, 1] | +| Row3 | [-6, 10, -5, 1] | +| Row4 | [22, -38, 22, -7, 1] | +| Row5 | [-90, 158, -98, 38, -9, 1] | +| Row6 | [394, -698, 450, -194, 58, -11, 1] | +| Row7 | [-1806, 3218, -2126, 978, -334, 82, -13, 1] | +| Row8 | [8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1] | + +Delannoy:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [2, -3, 1], [-6, 10, -5, 1], [22, -38, 22, -7, 1], [-90, 158, -98, 38, -9, 1], [394, -698, 450, -194, 58, -11, 1], [-1806, 3218, -2126, 978, -334, 82, -13, 1], [8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 2], [1, -5, 10, -6], [1, -7, 22, -38, 22], [1, -9, 38, -98, 158, -90], [1, -11, 58, -194, 450, -698, 394], [1, -13, 82, -334, 978, -2126, 3218, -1806], [1, -15, 110, -526, 1838, -4942, 10286, -15310, 8558]] | +| AntiDiag | [[1], [-1], [2, 1], [-6, -3], [22, 10, 1], [-90, -38, -5], [394, 158, 22, 1], [-1806, -698, -98, -7], [8558, 3218, 450, 38, 1]] | +| AccTabl | [[1], [-1, 0], [2, -1, 0], [-6, 4, -1, 0], [22, -16, 6, -1, 0], [-90, 68, -30, 8, -1, 0], [394, -304, 146, -48, 10, -1, 0], [-1806, 1412, -714, 264, -70, 12, -1, 0], [8558, -6752, 3534, -1408, 430, -96, 14, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 2], [0, -1, 4, -6], [0, -1, 6, -16, 22], [0, -1, 8, -30, 68, -90], [0, -1, 10, -48, 146, -304, 394], [0, -1, 12, -70, 264, -714, 1412, -1806], [0, -1, 14, -96, 430, -1408, 3534, -6752, 8558]] | +| AccRevTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 6, 0], [1, -6, 16, -22, 0], [1, -8, 30, -68, 90, 0], [1, -10, 48, -146, 304, -394, 0], [1, -12, 70, -264, 714, -1412, 1806, 0], [1, -14, 96, -430, 1408, -3534, 6752, -8558, 0]] | + +Delannoy:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 3, -11, 45, -197, 903, -4279, 20793] | +| OddSum | [0, 1, -3, 11, -45, 197, -903, 4279, -20793] | +| AltSum | [1, -2, 6, -22, 90, -394, 1806, -8558, 41586] | +| AccSum | [1, -1, 1, -3, 11, -45, 197, -903, 4279] | +| AccRevSum | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | +| AntiDiagSum | [1, -1, 3, -9, 33, -133, 575, -2609, 12265] | + +Delannoy:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 30, 2926, 6619410, 957340835550, 44816541527197986, 603669538270861563270] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 10, 38, 158, 698, 3218, 15310] | +| ColMiddle | [1, -1, -3, 10, 22, -98, -194, 978, 1838] | +| ColECenter | [1, -3, 22, -194, 1838] | +| ColOCenter | [-1, 10, -98, 978] | +| ColLeft | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -1, 3, -11, 45, -197, 903] | +| TransNat0 | [0, 1, -1, 3, -11, 45, -197, 903, -4279] | +| TransNat1 | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | + +Delannoy:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -3, -5, -7, -9, -11, -13, -15, -17]| +| DiagRow2 | [2, 10, 22, 38, 58, 82, 110, 142, 178]| +| DiagRow3 | [-6, -38, -98, -194, -334, -526, -778, -1098, -1494]| +| DiagRow4 | [22, 158, 450, 978, 1838, 3142, 5018, 7610, 11078]| +| DiagRow5 | [-90, -698, -2126, -4942, -9922, -18082, -30710, -49398, -76074]| +| DiagRow6 | [394, 3218, 10286, 25150, 53154, 101946, 182054, 307526, 496634]| +| DiagRow7 | [-1806, -15310, -50746, -129050, -284150, -568150, -1057730, -1861890, -3131166]| +| DiagRow8 | [8558, 74614, 254410, 667610, 1519910, 3145790, 6065410, 11058466, 19258078]| + +Delannoy:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| DiagCol1 | [1, -3, 10, -38, 158, -698, 3218, -15310, 74614] | +| DiagCol2 | [1, -5, 22, -98, 450, -2126, 10286, -50746, 254410] | +| DiagCol3 | [1, -7, 38, -194, 978, -4942, 25150, -129050, 667610] | +| DiagCol4 | [1, -9, 58, -334, 1838, -9922, 53154, -284150, 1519910] | +| DiagCol5 | [1, -11, 82, -526, 3142, -18082, 101946, -568150, 3145790] | +| DiagCol6 | [1, -13, 110, -778, 5018, -30710, 182054, -1057730, 6065410] | +| DiagCol7 | [1, -15, 142, -1098, 7610, -49398, 307526, -1861890, 11058466] | +| DiagCol8 | [1, -17, 178, -1494, 11078, -76074, 496634, -3131166, 19258078] | + +Delannoy:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [2, 0, 0, 2, 6, 12, 20, 30, 42] | +| PolyRow3 | [-6, 0, 2, 6, 18, 44, 90, 162, 266] | +| PolyRow4 | [22, 0, -6, -2, 30, 132, 370, 834, 1638] | +| PolyRow5 | [-90, 0, 26, 42, 126, 500, 1650, 4446, 10262] | +| PolyRow6 | [394, 0, -114, -134, 66, 1404, 6790, 23046, 63546] | +| PolyRow7 | [-1806, 0, 526, 702, 1242, 5884, 30390, 122346, 396802] | +| PolyRow8 | [8558, 0, -2502, -3226, -2154, 13908, 124130, 635898, 2462334] | + +Delannoy:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 2, -6, 22, -90, 394, -1806, 8558] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 0, 2, -6, 26, -114, 526, -2502] | +| PolyCol3 | [1, 2, 2, 6, -2, 42, -134, 702, -3226] | +| PolyCol4 | [1, 3, 6, 18, 30, 126, 66, 1242, -2154] | +| PolyCol5 | [1, 4, 12, 44, 132, 500, 1404, 5884, 13908] | +| PolyCol6 | [1, 5, 20, 90, 370, 1650, 6790, 30390, 124130] | +| PolyCol7 | [1, 6, 30, 162, 834, 4446, 23046, 122346, 635898] | +| PolyCol8 | [1, 7, 42, 266, 1638, 10262, 63546, 396802, 2462334] | + diff --git a/data/md/Euler.md b/data/md/Euler.md index 7ea8e9f..036b5e6 100644 --- a/data/md/Euler.md +++ b/data/md/Euler.md @@ -1,5 +1,5 @@ # Euler -['A109449', 'A247453'] +['A247453', 'A109449'] Euler Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Euler Triangle view | Row7 | [-272, 427, -336, 175, -70, 21, -7, 1] | | Row8 | [1385, -2176, 1708, -896, 350, -112, 28, -8, 1] | -Euler Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 5, -8, 6, -4, 1, -16, 25, -20, 10, -5, 1, 61, -96, 75, -40, 15, -6, 1, -272, 427, -336, 175, -70, 21, -7, 1, 1385, -2176, 1708, -896, 350, -112, 28, -8, 1] | -| TriRev | [1, 1, -1, 1, -2, 1, 1, -3, 3, -2, 1, -4, 6, -8, 5, 1, -5, 10, -20, 25, -16, 1, -6, 15, -40, 75, -96, 61, 1, -7, 21, -70, 175, -336, 427, -272, 1, -8, 28, -112, 350, -896, 1708, -2176, 1385] | -| TriAcc | [1, -1, 0, 1, -1, 0, -2, 1, -2, -1, 5, -3, 3, -1, 0, -16, 9, -11, -1, -6, -5, 61, -35, 40, 0, 15, 9, 10, -272, 155, -181, -6, -76, -55, -62, -61, 1385, -791, 917, 21, 371, 259, 287, 279, 280] | -| TriRevAcc | [1, 0, -1, 0, -1, 1, -1, -2, 1, -2, 0, -1, 3, -3, 5, -5, -6, -1, -11, 9, -16, 10, 9, 15, 0, 40, -35, 61, -61, -62, -55, -76, -6, -181, 155, -272, 280, 279, 287, 259, 371, 21, 917, -791, 1385] | -| TriAccRev | [1, 1, 0, 1, -1, 0, 1, -2, 1, -1, 1, -3, 3, -5, 0, 1, -4, 6, -14, 11, -5, 1, -5, 10, -30, 45, -51, 10, 1, -6, 15, -55, 120, -216, 211, -61, 1, -7, 21, -91, 259, -637, 1071, -1105, 280] | -| TriDiag | [1, -1, 1, 1, -2, -2, 5, 3, 1, -16, -8, -3, 61, 25, 6, 1, -272, -96, -20, -4] | +Euler Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -2, 1], [-2, 3, -3, 1], [5, -8, 6, -4, 1], [-16, 25, -20, 10, -5, 1], [61, -96, 75, -40, 15, -6, 1], [-272, 427, -336, 175, -70, 21, -7, 1], [1385, -2176, 1708, -896, 350, -112, 28, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 1], [1, -3, 3, -2], [1, -4, 6, -8, 5], [1, -5, 10, -20, 25, -16], [1, -6, 15, -40, 75, -96, 61], [1, -7, 21, -70, 175, -336, 427, -272], [1, -8, 28, -112, 350, -896, 1708, -2176, 1385]] | +| AntiDiag | [[1], [-1], [1, 1], [-2, -2], [5, 3, 1], [-16, -8, -3], [61, 25, 6, 1], [-272, -96, -20, -4], [1385, 427, 75, 10, 1]] | +| AccTabl | [[1], [-1, 0], [1, -1, 0], [-2, 1, -2, -1], [5, -3, 3, -1, 0], [-16, 9, -11, -1, -6, -5], [61, -35, 40, 0, 15, 9, 10], [-272, 155, -181, -6, -76, -55, -62, -61], [1385, -791, 917, 21, 371, 259, 287, 279, 280]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 1], [-1, -2, 1, -2], [0, -1, 3, -3, 5], [-5, -6, -1, -11, 9, -16], [10, 9, 15, 0, 40, -35, 61], [-61, -62, -55, -76, -6, -181, 155, -272], [280, 279, 287, 259, 371, 21, 917, -791, 1385]] | +| AccRevTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, -1], [1, -3, 3, -5, 0], [1, -4, 6, -14, 11, -5], [1, -5, 10, -30, 45, -51, 10], [1, -6, 15, -55, 120, -216, 211, -61], [1, -7, 21, -91, 259, -637, 1071, -1105, 280]] | Euler Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | -| EvenSum | [1, -1, 2, -5, 12, -41, 152, -685, 3472] | -| OddSum | [0, 1, -2, 4, -12, 36, -142, 624, -3192] | -| AltSum | [1, -2, 4, -9, 24, -77, 294, -1309, 6664] | -| AccSum | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | -| AccRevSum | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | -| RevAccSum | [1, 1, 0, -1, -4, -5, -20, 9, -208] | -| DiagSum | [1, -1, 2, -4, 9, -27, 93, -392] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | +| EvenSum | [1, -1, 2, -5, 12, -41, 152, -685, 3472] | +| OddSum | [0, 1, -2, 4, -12, 36, -142, 624, -3192] | +| AltSum | [1, -2, 4, -9, 24, -77, 294, -1309, 6664] | +| AccSum | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | +| AccRevSum | [1, 1, 0, -1, -4, -5, -20, 9, -208] | +| AntiDiagSum | [1, -1, 2, -4, 9, -27, 93, -392, 1898] | Euler Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 120, 400, 146400, 8710800, 6434377600] | -| RowGcd | [1, 1, -2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | -| Middle | [1, -1, -2, 3, 6, -20, -40, 175, 350] | -| Central | [1, -2, 6, -40, 350] | -| LeftSide | [1, -1, 1, -2, 5, -16, 61, -272, 1385] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, 1, -9, 33, -241, 1761, -15929, 161473] | -| NegHalf | [1, 3, 9, 35, 177, 1123, 8569, 76355, 777697] | -| Bin | [1, 0, -2, -1, -6, -15, 0, -271, 574] | -| InvBin | [1, 2, 6, 21, 90, 467, 2824, 19383, 148414] | -| TransSqrs | [0, 1, 2, 0, -4, -20, -30, -140, 72] | -| TransNat0 | [0, 1, 0, 0, -4, 0, -30, 70, -488] | -| TransNat1 | [1, 1, 0, -1, -4, -5, -20, 9, -208] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | +| ColMiddle | [1, -1, -2, 3, 6, -20, -40, 175, 350] | +| ColECenter | [1, -2, 6, -40, 350] | +| ColOCenter | [-1, 3, -20, 175] | +| ColLeft | [1, -1, 1, -2, 5, -16, 61, -272, 1385] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, 0, -4, -20, -30, -140, 72] | +| TransNat0 | [0, 1, 0, 0, -4, 0, -30, 70, -488] | +| TransNat1 | [1, 1, 0, -1, -4, -5, -20, 9, -208] | Euler Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Euler Polynomial values as columns | PolyCol7 | [1, 6, 36, 215, 1272, 7411, 42166, 231155, 1191112] | | PolyCol8 | [1, 7, 49, 342, 2373, 16312, 110589, 734792, 4738153] | -# EulerInv -True +# Euler:Inv +[] -EulerInv Triangle view +Euler:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ EulerInv Triangle view | Row7 | [272, 427, 336, 175, 70, 21, 7, 1] | | Row8 | [1385, 2176, 1708, 896, 350, 112, 28, 8, 1] | -EulerInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 8, 5, 1, 5, 10, 20, 25, 16, 1, 6, 15, 40, 75, 96, 61, 1, 7, 21, 70, 175, 336, 427, 272, 1, 8, 28, 112, 350, 896, 1708, 2176, 1385] | -| TriAcc | [1, 1, 2, 1, 3, 4, 2, 5, 8, 9, 5, 13, 19, 23, 24, 16, 41, 61, 71, 76, 77, 61, 157, 232, 272, 287, 293, 294, 272, 699, 1035, 1210, 1280, 1301, 1308, 1309, 1385, 3561, 5269, 6165, 6515, 6627, 6655, 6663, 6664] | -| TriRevAcc | [1, 2, 1, 4, 3, 1, 9, 8, 5, 2, 24, 23, 19, 13, 5, 77, 76, 71, 61, 41, 16, 294, 293, 287, 272, 232, 157, 61, 1309, 1308, 1301, 1280, 1210, 1035, 699, 272, 6664, 6663, 6655, 6627, 6515, 6165, 5269, 3561, 1385] | -| TriAccRev | [1, 1, 2, 1, 3, 4, 1, 4, 7, 9, 1, 5, 11, 19, 24, 1, 6, 16, 36, 61, 77, 1, 7, 22, 62, 137, 233, 294, 1, 8, 29, 99, 274, 610, 1037, 1309, 1, 9, 37, 149, 499, 1395, 3103, 5279, 6664] | -| TriDiag | [1, 1, 1, 1, 2, 2, 5, 3, 1, 16, 8, 3, 61, 25, 6, 1, 272, 96, 20, 4] | - -EulerInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | -| EvenSum | [1, 1, 2, 5, 12, 41, 152, 685, 3472] | -| OddSum | [0, 1, 2, 4, 12, 36, 142, 624, 3192] | -| AltSum | [1, 0, 0, 1, 0, 5, 10, 61, 280] | -| AccSum | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | -| AccRevSum | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | -| RevAccSum | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | -| DiagSum | [1, 1, 2, 4, 9, 27, 93, 392] | - -EulerInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | -| Middle | [1, 1, 2, 3, 6, 20, 40, 175, 350] | -| Central | [1, 2, 6, 40, 350] | -| LeftSide | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 9, 35, 177, 1123, 8569, 76355, 777697] | -| NegHalf | [1, -1, 1, -9, 33, -241, 1761, -15929, 161473] | -| Bin | [1, 2, 6, 21, 90, 467, 2824, 19383, 148414] | -| InvBin | [1, 0, -2, -1, -6, -15, 0, -271, 574] | -| TransSqrs | [0, 1, 6, 24, 84, 300, 1182, 5292, 26936] | -| TransNat0 | [0, 1, 4, 12, 36, 120, 462, 2058, 10472] | -| TransNat1 | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | - -EulerInv Diagonals as rows +Euler:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 1], [2, 3, 3, 1], [5, 8, 6, 4, 1], [16, 25, 20, 10, 5, 1], [61, 96, 75, 40, 15, 6, 1], [272, 427, 336, 175, 70, 21, 7, 1], [1385, 2176, 1708, 896, 350, 112, 28, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 2], [1, 4, 6, 8, 5], [1, 5, 10, 20, 25, 16], [1, 6, 15, 40, 75, 96, 61], [1, 7, 21, 70, 175, 336, 427, 272], [1, 8, 28, 112, 350, 896, 1708, 2176, 1385]] | +| AntiDiag | [[1], [1], [1, 1], [2, 2], [5, 3, 1], [16, 8, 3], [61, 25, 6, 1], [272, 96, 20, 4], [1385, 427, 75, 10, 1]] | +| AccTabl | [[1], [1, 2], [1, 3, 4], [2, 5, 8, 9], [5, 13, 19, 23, 24], [16, 41, 61, 71, 76, 77], [61, 157, 232, 272, 287, 293, 294], [272, 699, 1035, 1210, 1280, 1301, 1308, 1309], [1385, 3561, 5269, 6165, 6515, 6627, 6655, 6663, 6664]] | +| RevAccTabl | [[1], [2, 1], [4, 3, 1], [9, 8, 5, 2], [24, 23, 19, 13, 5], [77, 76, 71, 61, 41, 16], [294, 293, 287, 272, 232, 157, 61], [1309, 1308, 1301, 1280, 1210, 1035, 699, 272], [6664, 6663, 6655, 6627, 6515, 6165, 5269, 3561, 1385]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 9], [1, 5, 11, 19, 24], [1, 6, 16, 36, 61, 77], [1, 7, 22, 62, 137, 233, 294], [1, 8, 29, 99, 274, 610, 1037, 1309], [1, 9, 37, 149, 499, 1395, 3103, 5279, 6664]] | + +Euler:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | +| EvenSum | [1, 1, 2, 5, 12, 41, 152, 685, 3472] | +| OddSum | [0, 1, 2, 4, 12, 36, 142, 624, 3192] | +| AltSum | [1, 0, 0, 1, 0, 5, 10, 61, 280] | +| AccSum | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | +| AccRevSum | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | +| AntiDiagSum | [1, 1, 2, 4, 9, 27, 93, 392, 1898] | + +Euler:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | +| ColMiddle | [1, 1, 2, 3, 6, 20, 40, 175, 350] | +| ColECenter | [1, 2, 6, 40, 350] | +| ColOCenter | [1, 3, 20, 175] | +| ColLeft | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 24, 84, 300, 1182, 5292, 26936] | +| TransNat0 | [0, 1, 4, 12, 36, 120, 462, 2058, 10472] | +| TransNat1 | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | + +Euler:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ EulerInv Diagonals as rows | DiagRow7 | [272, 2176, 9792, 32640, 89760, 215424, 466752, 933504, 1750320]| | DiagRow8 | [1385, 12465, 62325, 228525, 685575, 1782495, 4159155, 8912475, 17824950]| -EulerInv Diagonals as columns +Euler:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | @@ -188,7 +180,7 @@ EulerInv Diagonals as columns | DiagCol7 | [1, 8, 36, 240, 1650, 12672, 104676, 933504, 8912475] | | DiagCol8 | [1, 9, 45, 330, 2475, 20592, 183183, 1750320, 17824950] | -EulerInv Polynomial values as rows +Euler:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ EulerInv Polynomial values as rows | PolyRow7 | [272, 1309, 6238, 27365, 103036, 329557, 913274, 2247853, 5021800] | | PolyRow8 | [1385, 6664, 31993, 147512, 610345, 2188840, 6819449, 18777688, 46584937] | -EulerInv Polynomial values as columns +Euler:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | @@ -214,10 +206,10 @@ EulerInv Polynomial values as columns | PolyCol7 | [1, 8, 64, 513, 4128, 33413, 272634, 2247853, 18777688] | | PolyCol8 | [1, 9, 81, 730, 6597, 59864, 546301, 5021800, 46584937] | -# EulerRev -True +# Euler:Rev +[] -EulerRev Triangle view +Euler:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ EulerRev Triangle view | Row7 | [1, -7, 21, -70, 175, -336, 427, -272] | | Row8 | [1, -8, 28, -112, 350, -896, 1708, -2176, 1385] | -EulerRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -2, 1, 1, -3, 3, -2, 1, -4, 6, -8, 5, 1, -5, 10, -20, 25, -16, 1, -6, 15, -40, 75, -96, 61, 1, -7, 21, -70, 175, -336, 427, -272, 1, -8, 28, -112, 350, -896, 1708, -2176, 1385] | -| TriRev | [1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 5, -8, 6, -4, 1, -16, 25, -20, 10, -5, 1, 61, -96, 75, -40, 15, -6, 1, -272, 427, -336, 175, -70, 21, -7, 1, 1385, -2176, 1708, -896, 350, -112, 28, -8, 1] | -| TriAcc | [1, 1, 0, 1, -1, 0, 1, -2, 1, -1, 1, -3, 3, -5, 0, 1, -4, 6, -14, 11, -5, 1, -5, 10, -30, 45, -51, 10, 1, -6, 15, -55, 120, -216, 211, -61, 1, -7, 21, -91, 259, -637, 1071, -1105, 280] | -| TriRevAcc | [1, 0, 1, 0, -1, 1, -1, 1, -2, 1, 0, -5, 3, -3, 1, -5, 11, -14, 6, -4, 1, 10, -51, 45, -30, 10, -5, 1, -61, 211, -216, 120, -55, 15, -6, 1, 280, -1105, 1071, -637, 259, -91, 21, -7, 1] | -| TriAccRev | [1, -1, 0, 1, -1, 0, -2, 1, -2, -1, 5, -3, 3, -1, 0, -16, 9, -11, -1, -6, -5, 61, -35, 40, 0, 15, 9, 10, -272, 155, -181, -6, -76, -55, -62, -61, 1385, -791, 917, 21, 371, 259, 287, 279, 280] | -| TriDiag | [1, 1, 1, -1, 1, -2, 1, -3, 1, 1, -4, 3, 1, -5, 6, -2, 1, -6, 10, -8] | - -EulerRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | -| EvenSum | [1, 1, 2, 4, 12, 36, 152, 624, 3472] | -| OddSum | [0, -1, -2, -5, -12, -41, -142, -685, -3192] | -| AltSum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | -| AccSum | [1, 1, 0, -1, -4, -5, -20, 9, -208] | -| AccRevSum | [1, 1, 0, -1, -4, -5, -20, 9, -208] | -| RevAccSum | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | -| DiagSum | [1, 1, 0, -1, -1, 0, 0, -3] | - -EulerRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 120, 400, 146400, 8710800, 6434377600] | -| RowGcd | [1, 1, -2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | -| Middle | [1, 1, -2, -3, 6, 10, -40, -70, 350] | -| Central | [1, -2, 6, -40, 350] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 1, -2, 5, -16, 61, -272, 1385] | -| PosHalf | [1, 1, 1, 0, -3, -14, -39, -130, -263] | -| NegHalf | [1, -3, 9, -28, 93, -338, 1369, -6238, 31993] | -| Bin | [1, 0, -2, -1, -6, -15, 0, -271, 574] | -| InvBin | [1, -2, 6, -21, 90, -467, 2824, -19383, 148414] | -| TransSqrs | [0, -1, 2, -9, 28, -145, 690, -4109, 25800] | -| TransNat0 | [0, -1, 0, -3, 4, -25, 90, -497, 2728] | -| TransNat1 | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | - -EulerRev Diagonals as rows +Euler:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -2, 1], [1, -3, 3, -2], [1, -4, 6, -8, 5], [1, -5, 10, -20, 25, -16], [1, -6, 15, -40, 75, -96, 61], [1, -7, 21, -70, 175, -336, 427, -272], [1, -8, 28, -112, 350, -896, 1708, -2176, 1385]] | +| RevTabl | [[1], [-1, 1], [1, -2, 1], [-2, 3, -3, 1], [5, -8, 6, -4, 1], [-16, 25, -20, 10, -5, 1], [61, -96, 75, -40, 15, -6, 1], [-272, 427, -336, 175, -70, 21, -7, 1], [1385, -2176, 1708, -896, 350, -112, 28, -8, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -2], [1, -3, 1], [1, -4, 3], [1, -5, 6, -2], [1, -6, 10, -8], [1, -7, 15, -20, 5]] | +| AccTabl | [[1], [1, 0], [1, -1, 0], [1, -2, 1, -1], [1, -3, 3, -5, 0], [1, -4, 6, -14, 11, -5], [1, -5, 10, -30, 45, -51, 10], [1, -6, 15, -55, 120, -216, 211, -61], [1, -7, 21, -91, 259, -637, 1071, -1105, 280]] | +| RevAccTabl | [[1], [0, 1], [0, -1, 1], [-1, 1, -2, 1], [0, -5, 3, -3, 1], [-5, 11, -14, 6, -4, 1], [10, -51, 45, -30, 10, -5, 1], [-61, 211, -216, 120, -55, 15, -6, 1], [280, -1105, 1071, -637, 259, -91, 21, -7, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -1, 0], [-2, 1, -2, -1], [5, -3, 3, -1, 0], [-16, 9, -11, -1, -6, -5], [61, -35, 40, 0, 15, 9, 10], [-272, 155, -181, -6, -76, -55, -62, -61], [1385, -791, 917, 21, 371, 259, 287, 279, 280]] | + +Euler:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | +| EvenSum | [1, 1, 2, 4, 12, 36, 152, 624, 3472] | +| OddSum | [0, -1, -2, -5, -12, -41, -142, -685, -3192] | +| AltSum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | +| AccSum | [1, 1, 0, -1, -4, -5, -20, 9, -208] | +| AccRevSum | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | +| AntiDiagSum | [1, 1, 0, -1, -1, 0, 0, -3, -6] | + +Euler:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | +| ColMiddle | [1, 1, -2, -3, 6, 10, -40, -70, 350] | +| ColECenter | [1, -2, 6, -40, 350] | +| ColOCenter | [1, -3, 10, -70] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -2, 5, -16, 61, -272, 1385] | +| TransSqrs | [0, -1, 2, -9, 28, -145, 690, -4109, 25800] | +| TransNat0 | [0, -1, 0, -3, 4, -25, 90, -497, 2728] | +| TransNat1 | [1, -1, 0, -4, 4, -30, 100, -558, 3008] | + +Euler:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 1, -2, 5, -16, 61, -272, 1385]| @@ -283,7 +271,7 @@ EulerRev Diagonals as rows | DiagRow7 | [1, -8, 36, -240, 1650, -12672, 104676, -933504, 8912475]| | DiagRow8 | [1, -9, 45, -330, 2475, -20592, 183183, -1750320, 17824950]| -EulerRev Diagonals as columns +Euler:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ EulerRev Diagonals as columns | DiagCol7 | [-272, -2176, -9792, -32640, -89760, -215424, -466752, -933504, -1750320] | | DiagCol8 | [1385, 12465, 62325, 228525, 685575, 1782495, 4159155, 8912475, 17824950] | -EulerRev Polynomial values as rows +Euler:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ EulerRev Polynomial values as rows | PolyRow7 | [1, -61, -15929, -352775, -3010891, -15527009, -58620821, -179017579, -468817655] | | PolyRow8 | [1, 280, 161473, 5381032, 61277089, 395108536, 1790278945, 6378908488, 19092801217] | -EulerRev Polynomial values as columns +Euler:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ EulerRev Polynomial values as columns | PolyCol7 | [1, -6, 36, -559, 9528, -215291, 5730166, -179017579, 6378908488] | | PolyCol8 | [1, -7, 49, -855, 16737, -431527, 13132689, -468817655, 19092801217] | -# EulerInvRev -True +# Euler:Inv:Rev +[] -EulerInvRev Triangle view +Euler:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ EulerInvRev Triangle view | Row7 | [1, 7, 21, 70, 175, 336, 427, 272] | | Row8 | [1, 8, 28, 112, 350, 896, 1708, 2176, 1385] | -EulerInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 8, 5, 1, 5, 10, 20, 25, 16, 1, 6, 15, 40, 75, 96, 61, 1, 7, 21, 70, 175, 336, 427, 272, 1, 8, 28, 112, 350, 896, 1708, 2176, 1385] | -| TriRev | [1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 4, 1, 4, 7, 9, 1, 5, 11, 19, 24, 1, 6, 16, 36, 61, 77, 1, 7, 22, 62, 137, 233, 294, 1, 8, 29, 99, 274, 610, 1037, 1309, 1, 9, 37, 149, 499, 1395, 3103, 5279, 6664] | -| TriRevAcc | [1, 2, 1, 4, 3, 1, 9, 7, 4, 1, 24, 19, 11, 5, 1, 77, 61, 36, 16, 6, 1, 294, 233, 137, 62, 22, 7, 1, 1309, 1037, 610, 274, 99, 29, 8, 1, 6664, 5279, 3103, 1395, 499, 149, 37, 9, 1] | -| TriAccRev | [1, 1, 2, 1, 3, 4, 2, 5, 8, 9, 5, 13, 19, 23, 24, 16, 41, 61, 71, 76, 77, 61, 157, 232, 272, 287, 293, 294, 272, 699, 1035, 1210, 1280, 1301, 1308, 1309, 1385, 3561, 5269, 6165, 6515, 6627, 6655, 6663, 6664] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 3, 1, 5, 6, 2, 1, 6, 10, 8] | - -EulerInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | -| EvenSum | [1, 1, 2, 4, 12, 36, 152, 624, 3472] | -| OddSum | [0, 1, 2, 5, 12, 41, 142, 685, 3192] | -| AltSum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | -| AccSum | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | -| AccRevSum | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | -| RevAccSum | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | -| DiagSum | [1, 1, 2, 3, 5, 8, 14, 25] | - -EulerInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | -| Middle | [1, 1, 2, 3, 6, 10, 40, 70, 350] | -| Central | [1, 2, 6, 40, 350] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| PosHalf | [1, 3, 9, 28, 93, 338, 1369, 6238, 31993] | -| NegHalf | [1, -1, 1, 0, -3, 14, -39, 130, -263] | -| Bin | [1, 2, 6, 21, 90, 467, 2824, 19383, 148414] | -| InvBin | [1, 0, -2, 1, -6, 15, 0, 271, 574] | -| TransSqrs | [0, 1, 6, 33, 180, 1025, 6222, 40621, 285880] | -| TransNat0 | [0, 1, 4, 15, 60, 265, 1302, 7105, 42840] | -| TransNat1 | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | - -EulerInvRev Diagonals as rows +Euler:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 1], [1, 3, 3, 2], [1, 4, 6, 8, 5], [1, 5, 10, 20, 25, 16], [1, 6, 15, 40, 75, 96, 61], [1, 7, 21, 70, 175, 336, 427, 272], [1, 8, 28, 112, 350, 896, 1708, 2176, 1385]] | +| RevTabl | [[1], [1, 1], [1, 2, 1], [2, 3, 3, 1], [5, 8, 6, 4, 1], [16, 25, 20, 10, 5, 1], [61, 96, 75, 40, 15, 6, 1], [272, 427, 336, 175, 70, 21, 7, 1], [1385, 2176, 1708, 896, 350, 112, 28, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 1], [1, 4, 3], [1, 5, 6, 2], [1, 6, 10, 8], [1, 7, 15, 20, 5]] | +| AccTabl | [[1], [1, 2], [1, 3, 4], [1, 4, 7, 9], [1, 5, 11, 19, 24], [1, 6, 16, 36, 61, 77], [1, 7, 22, 62, 137, 233, 294], [1, 8, 29, 99, 274, 610, 1037, 1309], [1, 9, 37, 149, 499, 1395, 3103, 5279, 6664]] | +| RevAccTabl | [[1], [2, 1], [4, 3, 1], [9, 7, 4, 1], [24, 19, 11, 5, 1], [77, 61, 36, 16, 6, 1], [294, 233, 137, 62, 22, 7, 1], [1309, 1037, 610, 274, 99, 29, 8, 1], [6664, 5279, 3103, 1395, 499, 149, 37, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 4], [2, 5, 8, 9], [5, 13, 19, 23, 24], [16, 41, 61, 71, 76, 77], [61, 157, 232, 272, 287, 293, 294], [272, 699, 1035, 1210, 1280, 1301, 1308, 1309], [1385, 3561, 5269, 6165, 6515, 6627, 6655, 6663, 6664]] | + +Euler:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 4, 9, 24, 77, 294, 1309, 6664] | +| EvenSum | [1, 1, 2, 4, 12, 36, 152, 624, 3472] | +| OddSum | [0, 1, 2, 5, 12, 41, 142, 685, 3192] | +| AltSum | [1, 0, 0, -1, 0, -5, 10, -61, 280] | +| AccSum | [1, 3, 8, 21, 60, 197, 756, 3367, 17136] | +| AccRevSum | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | +| AntiDiagSum | [1, 1, 2, 3, 5, 8, 14, 25, 48] | + +Euler:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 120, 400, 146400, 8710800, 6434377600] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 8, 25, 96, 427, 2176] | +| ColMiddle | [1, 1, 2, 3, 6, 10, 40, 70, 350] | +| ColECenter | [1, 2, 6, 40, 350] | +| ColOCenter | [1, 3, 10, 70] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| TransSqrs | [0, 1, 6, 33, 180, 1025, 6222, 40621, 285880] | +| TransNat0 | [0, 1, 4, 15, 60, 265, 1302, 7105, 42840] | +| TransNat1 | [1, 3, 8, 24, 84, 342, 1596, 8414, 49504] | + +Euler:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 2, 5, 16, 61, 272, 1385]| @@ -391,7 +375,7 @@ EulerInvRev Diagonals as rows | DiagRow7 | [1, 8, 36, 240, 1650, 12672, 104676, 933504, 8912475]| | DiagRow8 | [1, 9, 45, 330, 2475, 20592, 183183, 1750320, 17824950]| -EulerInvRev Diagonals as columns +Euler:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ EulerInvRev Diagonals as columns | DiagCol7 | [272, 2176, 9792, 32640, 89760, 215424, 466752, 933504, 1750320] | | DiagCol8 | [1385, 12465, 62325, 228525, 685575, 1782495, 4159155, 8912475, 17824950] | -EulerInvRev Polynomial values as rows +Euler:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ EulerInvRev Polynomial values as rows | PolyRow7 | [1, 1309, 76355, 1004071, 6599149, 29090561, 98920159, 280332235, 694124921] | | PolyRow8 | [1, 6664, 777697, 15340408, 134429665, 740736616, 3022543489, 9993162712, 28278441793] | -EulerInvRev Polynomial values as columns +Euler:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/EulerSec.md b/data/md/EulerSec.md index 9ebe153..817b1df 100644 --- a/data/md/EulerSec.md +++ b/data/md/EulerSec.md @@ -14,45 +14,41 @@ EulerSec Triangle view | Row7 | [0, -427, 0, 175, 0, -21, 0, 1] | | Row8 | [1385, 0, -1708, 0, 350, 0, -28, 0, 1] | -EulerSec Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 5, 1, 0, -10, 0, 25, 0, 1, 0, -15, 0, 75, 0, -61, 1, 0, -21, 0, 175, 0, -427, 0, 1, 0, -28, 0, 350, 0, -1708, 0, 1385] | -| TriAcc | [1, 0, 1, -1, -1, 0, 0, -3, -3, -2, 5, 5, -1, -1, 0, 0, 25, 25, 15, 15, 16, -61, -61, 14, 14, -1, -1, 0, 0, -427, -427, -252, -252, -273, -273, -272, 1385, 1385, -323, -323, 27, 27, -1, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, -1, -2, -3, -3, 0, 0, -1, -1, 5, 5, 16, 15, 15, 25, 25, 0, 0, -1, -1, 14, 14, -61, -61, -272, -273, -273, -252, -252, -427, -427, 0, 0, -1, -1, 27, 27, -323, -323, 1385, 1385] | -| TriAccRev | [1, 1, 1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -5, -5, 0, 1, 1, -9, -9, 16, 16, 1, 1, -14, -14, 61, 61, 0, 1, 1, -20, -20, 155, 155, -272, -272, 1, 1, -27, -27, 323, 323, -1385, -1385, 0] | -| TriDiag | [1, 0, -1, 1, 0, 0, 5, -3, 1, 0, 0, 0, -61, 25, -6, 1, 0, 0, 0, 0] | +EulerSec Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [5, 0, -6, 0, 1], [0, 25, 0, -10, 0, 1], [-61, 0, 75, 0, -15, 0, 1], [0, -427, 0, 175, 0, -21, 0, 1], [1385, 0, -1708, 0, 350, 0, -28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 5], [1, 0, -10, 0, 25, 0], [1, 0, -15, 0, 75, 0, -61], [1, 0, -21, 0, 175, 0, -427, 0], [1, 0, -28, 0, 350, 0, -1708, 0, 1385]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [5, -3, 1], [0, 0, 0], [-61, 25, -6, 1], [0, 0, 0, 0], [1385, -427, 75, -10, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [5, 5, -1, -1, 0], [0, 25, 25, 15, 15, 16], [-61, -61, 14, 14, -1, -1, 0], [0, -427, -427, -252, -252, -273, -273, -272], [1385, 1385, -323, -323, 27, 27, -1, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-2, -3, -3, 0], [0, -1, -1, 5, 5], [16, 15, 15, 25, 25, 0], [0, -1, -1, 14, 14, -61, -61], [-272, -273, -273, -252, -252, -427, -427, 0], [0, -1, -1, 27, 27, -323, -323, 1385, 1385]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, 0], [1, 1, -9, -9, 16, 16], [1, 1, -14, -14, 61, 61, 0], [1, 1, -20, -20, 155, 155, -272, -272], [1, 1, -27, -27, 323, 323, -1385, -1385, 0]] | EulerSec Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| EvenSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, 1, 0, -2, 0, 16, 0, -272, 0] | -| AltSum | [1, -1, 0, 2, 0, -16, 0, 272, 0] | -| AccSum | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | -| AccRevSum | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | -| RevAccSum | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | -| DiagSum | [1, 0, 0, 0, 3, 0, -41, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | +| EvenSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 0, -2, 0, 16, 0, -272, 0] | +| AltSum | [1, -1, 0, 2, 0, -16, 0, 272, 0] | +| AccSum | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | +| AccRevSum | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | +| AntiDiagSum | [1, 0, 0, 0, 3, 0, -41, 0, 1024] | EulerSec Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 30, 50, 4575, 32025, 11827900] | -| RowGcd | [1, 1, 1, -3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 6, 25, 75, 427, 1708] | -| Middle | [1, 0, 0, -3, -6, 0, 0, 175, 350] | -| Central | [1, 0, -6, 0, 350] | -| LeftSide | [1, 0, -1, 0, 5, 0, -61, 0, 1385] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, -11, 57, 361, -2763, -24611, 250737] | -| NegHalf | [1, 1, -3, -11, 57, 361, -2763, -24611, 250737] | -| Bin | [1, 1, 0, -8, -30, 26, 840, 2696, -22722] | -| InvBin | [1, 1, 0, -8, -30, 26, 840, 2696, -22722] | -| TransSqrs | [0, 1, 4, 6, -8, -40, 96, 672, -2176] | -| TransNat0 | [0, 1, 2, 0, -8, 0, 96, 0, -2176] | -| TransNat1 | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 30, 50, 4575, 32025, 11827900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 6, 25, 75, 427, 1708] | +| ColMiddle | [1, 0, 0, -3, -6, 0, 0, 175, 350] | +| ColECenter | [1, 0, -6, 0, 350] | +| ColOCenter | [0, -3, 0, 175] | +| ColLeft | [1, 0, -1, 0, 5, 0, -61, 0, 1385] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 6, -8, -40, 96, 672, -2176] | +| TransNat0 | [0, 1, 2, 0, -8, 0, 96, 0, -2176] | +| TransNat1 | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | EulerSec Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ EulerSec Polynomial values as columns | PolyCol7 | [1, 7, 48, 322, 2112, 13552, 85248, 527632, 3228672] | | PolyCol8 | [1, 8, 63, 488, 3717, 27848, 205443, 1495208, 10762857] | -# EulerSecInv -True +# EulerSec:Inv +[] -EulerSecInv Triangle view +EulerSec:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ EulerSecInv Triangle view | Row7 | [0, 7, 0, 35, 0, 21, 0, 1] | | Row8 | [1, 0, 28, 0, 70, 0, 28, 0, 1] | -EulerSecInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 1, 1, 0, 10, 0, 5, 0, 1, 0, 15, 0, 15, 0, 1, 1, 0, 21, 0, 35, 0, 7, 0, 1, 0, 28, 0, 70, 0, 28, 0, 1] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 1, 1, 7, 7, 8, 0, 5, 5, 15, 15, 16, 1, 1, 16, 16, 31, 31, 32, 0, 7, 7, 42, 42, 63, 63, 64, 1, 1, 29, 29, 99, 99, 127, 127, 128] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 4, 3, 3, 0, 8, 7, 7, 1, 1, 16, 15, 15, 5, 5, 0, 32, 31, 31, 16, 16, 1, 1, 64, 63, 63, 42, 42, 7, 7, 0, 128, 127, 127, 99, 99, 29, 29, 1, 1] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 8, 1, 1, 11, 11, 16, 16, 1, 1, 16, 16, 31, 31, 32, 1, 1, 22, 22, 57, 57, 64, 64, 1, 1, 29, 29, 99, 99, 127, 127, 128] | -| TriDiag | [1, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 0] | - -EulerSecInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| EvenSum | [1, 0, 2, 0, 8, 0, 32, 0, 128] | -| OddSum | [0, 1, 0, 4, 0, 16, 0, 64, 0] | -| AltSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | -| AccSum | [1, 1, 4, 10, 24, 56, 128, 288, 640] | -| AccRevSum | [1, 1, 4, 10, 24, 56, 128, 288, 640] | -| RevAccSum | [1, 2, 4, 10, 24, 56, 128, 288, 640] | -| DiagSum | [1, 0, 2, 0, 5, 0, 13, 0] | - -EulerSecInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 10, 15, 105, 140] | -| RowGcd | [1, 1, 1, 3, 6, 5, 15, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 10, 15, 35, 70] | -| Middle | [1, 0, 0, 3, 6, 0, 0, 35, 70] | -| Central | [1, 0, 6, 0, 70] | -| LeftSide | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 13, 41, 121, 365, 1093, 3281] | -| NegHalf | [1, 1, 5, 13, 41, 121, 365, 1093, 3281] | -| Bin | [1, 1, 2, 10, 38, 126, 452, 1716, 6470] | -| InvBin | [1, 1, 2, 10, 38, 126, 452, 1716, 6470] | -| TransSqrs | [0, 1, 4, 12, 40, 120, 336, 896, 2304] | -| TransNat0 | [0, 1, 2, 6, 16, 40, 96, 224, 512] | -| TransNat1 | [1, 2, 4, 10, 24, 56, 128, 288, 640] | - -EulerSecInv Diagonals as rows +EulerSec:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [1, 0, 6, 0, 1], [0, 5, 0, 10, 0, 1], [1, 0, 15, 0, 15, 0, 1], [0, 7, 0, 35, 0, 21, 0, 1], [1, 0, 28, 0, 70, 0, 28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 1], [1, 0, 10, 0, 5, 0], [1, 0, 15, 0, 15, 0, 1], [1, 0, 21, 0, 35, 0, 7, 0], [1, 0, 28, 0, 70, 0, 28, 0, 1]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [1, 3, 1], [0, 0, 0], [1, 5, 6, 1], [0, 0, 0, 0], [1, 7, 15, 10, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [1, 1, 7, 7, 8], [0, 5, 5, 15, 15, 16], [1, 1, 16, 16, 31, 31, 32], [0, 7, 7, 42, 42, 63, 63, 64], [1, 1, 29, 29, 99, 99, 127, 127, 128]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [4, 3, 3, 0], [8, 7, 7, 1, 1], [16, 15, 15, 5, 5, 0], [32, 31, 31, 16, 16, 1, 1], [64, 63, 63, 42, 42, 7, 7, 0], [128, 127, 127, 99, 99, 29, 29, 1, 1]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 8], [1, 1, 11, 11, 16, 16], [1, 1, 16, 16, 31, 31, 32], [1, 1, 22, 22, 57, 57, 64, 64], [1, 1, 29, 29, 99, 99, 127, 127, 128]] | + +EulerSec:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| EvenSum | [1, 0, 2, 0, 8, 0, 32, 0, 128] | +| OddSum | [0, 1, 0, 4, 0, 16, 0, 64, 0] | +| AltSum | [1, -1, 2, -4, 8, -16, 32, -64, 128] | +| AccSum | [1, 1, 4, 10, 24, 56, 128, 288, 640] | +| AccRevSum | [1, 2, 4, 10, 24, 56, 128, 288, 640] | +| AntiDiagSum | [1, 0, 2, 0, 5, 0, 13, 0, 34] | + +EulerSec:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 10, 15, 105, 140] | +| RowGcd | [1, 1, 1, 3, 6, 5, 15, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 10, 15, 35, 70] | +| ColMiddle | [1, 0, 0, 3, 6, 0, 0, 35, 70] | +| ColECenter | [1, 0, 6, 0, 70] | +| ColOCenter | [0, 3, 0, 35] | +| ColLeft | [1, 0, 1, 0, 1, 0, 1, 0, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 12, 40, 120, 336, 896, 2304] | +| TransNat0 | [0, 1, 2, 6, 16, 40, 96, 224, 512] | +| TransNat1 | [1, 2, 4, 10, 24, 56, 128, 288, 640] | + +EulerSec:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ EulerSecInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870]| -EulerSecInv Diagonals as columns +EulerSec:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 1, 0, 1, 0, 1, 0, 1] | @@ -188,7 +180,7 @@ EulerSecInv Diagonals as columns | DiagCol7 | [1, 0, 36, 0, 330, 0, 1716, 0, 6435] | | DiagCol8 | [1, 0, 45, 0, 495, 0, 3003, 0, 12870] | -EulerSecInv Polynomial values as rows +EulerSec:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ EulerSecInv Polynomial values as rows | PolyRow7 | [0, 64, 1094, 8256, 40156, 148160, 450834, 1188544, 2803256] | | PolyRow8 | [1, 128, 3281, 32896, 198593, 872576, 3077713, 9228416, 24405761] | -EulerSecInv Polynomial values as columns +EulerSec:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 1, 0, 1, 0, 1, 0, 1] | @@ -214,10 +206,10 @@ EulerSecInv Polynomial values as columns | PolyCol7 | [1, 7, 50, 364, 2696, 20272, 154400, 1188544, 9228416] | | PolyCol8 | [1, 8, 65, 536, 4481, 37928, 324545, 2803256, 24405761] | -# EulerSecRev -True +# EulerSec:Rev +[] -EulerSecRev Triangle view +EulerSec:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ EulerSecRev Triangle view | Row7 | [1, 0, -21, 0, 175, 0, -427, 0] | | Row8 | [1, 0, -28, 0, 350, 0, -1708, 0, 1385] | -EulerSecRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 5, 1, 0, -10, 0, 25, 0, 1, 0, -15, 0, 75, 0, -61, 1, 0, -21, 0, 175, 0, -427, 0, 1, 0, -28, 0, 350, 0, -1708, 0, 1385] | -| TriRev | [1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -5, -5, 0, 1, 1, -9, -9, 16, 16, 1, 1, -14, -14, 61, 61, 0, 1, 1, -20, -20, 155, 155, -272, -272, 1, 1, -27, -27, 323, 323, -1385, -1385, 0] | -| TriRevAcc | [1, 1, 1, 0, 1, 1, -2, -2, 1, 1, 0, -5, -5, 1, 1, 16, 16, -9, -9, 1, 1, 0, 61, 61, -14, -14, 1, 1, -272, -272, 155, 155, -20, -20, 1, 1, 0, -1385, -1385, 323, 323, -27, -27, 1, 1] | -| TriAccRev | [1, 0, 1, -1, -1, 0, 0, -3, -3, -2, 5, 5, -1, -1, 0, 0, 25, 25, 15, 15, 16, -61, -61, 14, 14, -1, -1, 0, 0, -427, -427, -252, -252, -273, -273, -272, 1385, 1385, -323, -323, 27, 27, -1, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -3, 1, 0, -6, 0, 1, 0, -10, 0] | - -EulerSecRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| EvenSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| AccSum | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | -| AccRevSum | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | -| RevAccSum | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | -| DiagSum | [1, 1, 1, 1, 0, -2, -5, -9] | - -EulerSecRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 30, 50, 4575, 32025, 11827900] | -| RowGcd | [1, 1, 1, -3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 6, 25, 75, 427, 1708] | -| Middle | [1, 1, 0, 0, -6, -10, 0, 0, 350] | -| Central | [1, 0, -6, 0, 350] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, -1, 0, 5, 0, -61, 0, 1385] | -| PosHalf | [1, 2, 3, 2, -3, 2, 63, 2, -1383] | -| NegHalf | [1, -2, 3, -2, -3, -2, 63, -2, -1383] | -| Bin | [1, 1, 0, -8, -30, 26, 840, 2696, -22722] | -| InvBin | [1, -1, 0, 8, -30, -26, 840, -2696, -22722] | -| TransSqrs | [0, 0, -4, -12, 56, 360, -1056, -12656, 32640] | -| TransNat0 | [0, 0, -2, -6, 8, 80, -96, -1904, 2176] | -| TransNat1 | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | - -EulerSecRev Diagonals as rows +EulerSec:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 5], [1, 0, -10, 0, 25, 0], [1, 0, -15, 0, 75, 0, -61], [1, 0, -21, 0, 175, 0, -427, 0], [1, 0, -28, 0, 350, 0, -1708, 0, 1385]] | +| RevTabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [5, 0, -6, 0, 1], [0, 25, 0, -10, 0, 1], [-61, 0, 75, 0, -15, 0, 1], [0, -427, 0, 175, 0, -21, 0, 1], [1385, 0, -1708, 0, 350, 0, -28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, -1], [1, 0, -3], [1, 0, -6, 0], [1, 0, -10, 0], [1, 0, -15, 0, 5]] | +| AccTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, 0], [1, 1, -9, -9, 16, 16], [1, 1, -14, -14, 61, 61, 0], [1, 1, -20, -20, 155, 155, -272, -272], [1, 1, -27, -27, 323, 323, -1385, -1385, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 1, 1], [-2, -2, 1, 1], [0, -5, -5, 1, 1], [16, 16, -9, -9, 1, 1], [0, 61, 61, -14, -14, 1, 1], [-272, -272, 155, 155, -20, -20, 1, 1], [0, -1385, -1385, 323, 323, -27, -27, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [5, 5, -1, -1, 0], [0, 25, 25, 15, 15, 16], [-61, -61, 14, 14, -1, -1, 0], [0, -427, -427, -252, -252, -273, -273, -272], [1385, 1385, -323, -323, 27, 27, -1, -1, 0]] | + +EulerSec:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | +| EvenSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | +| AccSum | [1, 2, 2, -2, -8, 16, 96, -272, -2176] | +| AccRevSum | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | +| AntiDiagSum | [1, 1, 1, 1, 0, -2, -5, -9, -9] | + +EulerSec:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 30, 50, 4575, 32025, 11827900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 6, 25, 75, 427, 1708] | +| ColMiddle | [1, 1, 0, 0, -6, -10, 0, 0, 350] | +| ColECenter | [1, 0, -6, 0, 350] | +| ColOCenter | [1, 0, -10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, -1, 0, 5, 0, -61, 0, 1385] | +| TransSqrs | [0, 0, -4, -12, 56, 360, -1056, -12656, 32640] | +| TransNat0 | [0, 0, -2, -6, 8, 80, -96, -1904, 2176] | +| TransNat1 | [1, 1, -2, -8, 8, 96, -96, -2176, 2176] | + +EulerSec:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 5, 0, -61, 0, 1385]| @@ -283,7 +271,7 @@ EulerSecRev Diagonals as rows | DiagRow7 | [1, 0, -36, 0, 1650, 0, -104676, 0, 8912475]| | DiagRow8 | [1, 0, -45, 0, 2475, 0, -183183, 0, 17824950]| -EulerSecRev Diagonals as columns +EulerSec:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ EulerSecRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1385, 12465, 62325, 228525, 685575, 1782495, 4159155, 8912475, 17824950] | -EulerSecRev Polynomial values as rows +EulerSec:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ EulerSecRev Polynomial values as rows | PolyRow7 | [1, -272, -24611, -297296, -1704527, -6563024, -19696067, -49816976, -111220031] | | PolyRow8 | [1, 0, 250737, 7869952, 83860545, 514546176, 2247032305, 7784143872, 22790134017] | -EulerSecRev Polynomial values as columns +EulerSec:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ EulerSecRev Polynomial values as columns | PolyCol7 | [1, 1, -48, -146, 11712, 59536, -6997248, -49816976, 7784143872] | | PolyCol8 | [1, 1, -63, -191, 20097, 101761, -15684543, -111220031, 22790134017] | -# EulerSecInvRev -True +# EulerSec:Inv:Rev +[] -EulerSecInvRev Triangle view +EulerSec:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ EulerSecInvRev Triangle view | Row7 | [1, 0, 21, 0, 35, 0, 7, 0] | | Row8 | [1, 0, 28, 0, 70, 0, 28, 0, 1] | -EulerSecInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 1, 1, 0, 10, 0, 5, 0, 1, 0, 15, 0, 15, 0, 1, 1, 0, 21, 0, 35, 0, 7, 0, 1, 0, 28, 0, 70, 0, 28, 0, 1] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 8, 1, 1, 11, 11, 16, 16, 1, 1, 16, 16, 31, 31, 32, 1, 1, 22, 22, 57, 57, 64, 64, 1, 1, 29, 29, 99, 99, 127, 127, 128] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 7, 7, 1, 1, 16, 16, 11, 11, 1, 1, 32, 31, 31, 16, 16, 1, 1, 64, 64, 57, 57, 22, 22, 1, 1, 128, 127, 127, 99, 99, 29, 29, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 1, 1, 7, 7, 8, 0, 5, 5, 15, 15, 16, 1, 1, 16, 16, 31, 31, 32, 0, 7, 7, 42, 42, 63, 63, 64, 1, 1, 29, 29, 99, 99, 127, 127, 128] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 6, 0, 1, 0, 10, 0] | - -EulerSecInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| AccSum | [1, 2, 4, 10, 24, 56, 128, 288, 640] | -| AccRevSum | [1, 2, 4, 10, 24, 56, 128, 288, 640] | -| RevAccSum | [1, 1, 4, 10, 24, 56, 128, 288, 640] | -| DiagSum | [1, 1, 1, 1, 2, 4, 7, 11] | - -EulerSecInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 10, 15, 105, 140] | -| RowGcd | [1, 1, 1, 3, 6, 5, 15, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 10, 15, 35, 70] | -| Middle | [1, 1, 0, 0, 6, 10, 0, 0, 70] | -| Central | [1, 0, 6, 0, 70] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -| PosHalf | [1, 2, 5, 14, 41, 122, 365, 1094, 3281] | -| NegHalf | [1, -2, 5, -14, 41, -122, 365, -1094, 3281] | -| Bin | [1, 1, 2, 10, 38, 126, 452, 1716, 6470] | -| InvBin | [1, -1, 2, -10, 38, -126, 452, -1716, 6470] | -| TransSqrs | [0, 0, 4, 12, 40, 120, 336, 896, 2304] | -| TransNat0 | [0, 0, 2, 6, 16, 40, 96, 224, 512] | -| TransNat1 | [1, 1, 4, 10, 24, 56, 128, 288, 640] | - -EulerSecInvRev Diagonals as rows +EulerSec:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 1], [1, 0, 10, 0, 5, 0], [1, 0, 15, 0, 15, 0, 1], [1, 0, 21, 0, 35, 0, 7, 0], [1, 0, 28, 0, 70, 0, 28, 0, 1]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [1, 0, 6, 0, 1], [0, 5, 0, 10, 0, 1], [1, 0, 15, 0, 15, 0, 1], [0, 7, 0, 35, 0, 21, 0, 1], [1, 0, 28, 0, 70, 0, 28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 3], [1, 0, 6, 0], [1, 0, 10, 0], [1, 0, 15, 0, 1]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 8], [1, 1, 11, 11, 16, 16], [1, 1, 16, 16, 31, 31, 32], [1, 1, 22, 22, 57, 57, 64, 64], [1, 1, 29, 29, 99, 99, 127, 127, 128]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [4, 4, 1, 1], [8, 7, 7, 1, 1], [16, 16, 11, 11, 1, 1], [32, 31, 31, 16, 16, 1, 1], [64, 64, 57, 57, 22, 22, 1, 1], [128, 127, 127, 99, 99, 29, 29, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [1, 1, 7, 7, 8], [0, 5, 5, 15, 15, 16], [1, 1, 16, 16, 31, 31, 32], [0, 7, 7, 42, 42, 63, 63, 64], [1, 1, 29, 29, 99, 99, 127, 127, 128]] | + +EulerSec:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| EvenSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| AccSum | [1, 2, 4, 10, 24, 56, 128, 288, 640] | +| AccRevSum | [1, 1, 4, 10, 24, 56, 128, 288, 640] | +| AntiDiagSum | [1, 1, 1, 1, 2, 4, 7, 11, 17] | + +EulerSec:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 10, 15, 105, 140] | +| RowGcd | [1, 1, 1, 3, 6, 5, 15, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 10, 15, 35, 70] | +| ColMiddle | [1, 1, 0, 0, 6, 10, 0, 0, 70] | +| ColECenter | [1, 0, 6, 0, 70] | +| ColOCenter | [1, 0, 10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 1, 0, 1, 0, 1] | +| TransSqrs | [0, 0, 4, 12, 40, 120, 336, 896, 2304] | +| TransNat0 | [0, 0, 2, 6, 16, 40, 96, 224, 512] | +| TransNat1 | [1, 1, 4, 10, 24, 56, 128, 288, 640] | + +EulerSec:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 0, 1, 0, 1, 0, 1]| @@ -391,7 +375,7 @@ EulerSecInvRev Diagonals as rows | DiagRow7 | [1, 0, 36, 0, 330, 0, 1716, 0, 6435]| | DiagRow8 | [1, 0, 45, 0, 495, 0, 3003, 0, 12870]| -EulerSecInvRev Diagonals as columns +EulerSec:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ EulerSecInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, 9, 45, 165, 495, 1287, 3003, 6435, 12870] | -EulerSecInvRev Polynomial values as rows +EulerSec:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ EulerSecInvRev Polynomial values as rows | PolyRow7 | [1, 64, 1093, 8128, 37969, 131776, 372709, 908608, 1979713] | | PolyRow8 | [1, 128, 3281, 32896, 198593, 872576, 3077713, 9228416, 24405761] | -EulerSecInvRev Polynomial values as columns +EulerSec:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/EulerTan.md b/data/md/EulerTan.md index 465d5b5..79b1d68 100644 --- a/data/md/EulerTan.md +++ b/data/md/EulerTan.md @@ -14,45 +14,41 @@ EulerTan Triangle view | Row7 | [-272, 0, 336, 0, -70, 0, 7, 0] | | Row8 | [0, -2176, 0, 896, 0, -112, 0, 8, 0] | -EulerTan Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0] | -| TriRev | [0, 0, 1, 0, 2, 0, 0, 3, 0, -2, 0, 4, 0, -8, 0, 0, 5, 0, -20, 0, 16, 0, 6, 0, -40, 0, 96, 0, 0, 7, 0, -70, 0, 336, 0, -272, 0, 8, 0, -112, 0, 896, 0, -2176, 0] | -| TriAcc | [0, 1, 1, 0, 2, 2, -2, -2, 1, 1, 0, -8, -8, -4, -4, 16, 16, -4, -4, 1, 1, 0, 96, 96, 56, 56, 62, 62, -272, -272, 64, 64, -6, -6, 1, 1, 0, -2176, -2176, -1280, -1280, -1392, -1392, -1384, -1384] | -| TriRevAcc | [0, 1, 1, 2, 2, 0, 1, 1, -2, -2, -4, -4, -8, -8, 0, 1, 1, -4, -4, 16, 16, 62, 62, 56, 56, 96, 96, 0, 1, 1, -6, -6, 64, 64, -272, -272, -1384, -1384, -1392, -1392, -1280, -1280, -2176, -2176, 0] | -| TriAccRev | [0, 0, 1, 0, 2, 2, 0, 3, 3, 1, 0, 4, 4, -4, -4, 0, 5, 5, -15, -15, 1, 0, 6, 6, -34, -34, 62, 62, 0, 7, 7, -63, -63, 273, 273, 1, 0, 8, 8, -104, -104, 792, 792, -1384, -1384] | -| TriDiag | [0, 1, 0, 0, -2, 2, 0, 0, 0, 16, -8, 3, 0, 0, 0, 0, -272, 96, -20, 4] | +EulerTan Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [1, 0], [0, 2, 0], [-2, 0, 3, 0], [0, -8, 0, 4, 0], [16, 0, -20, 0, 5, 0], [0, 96, 0, -40, 0, 6, 0], [-272, 0, 336, 0, -70, 0, 7, 0], [0, -2176, 0, 896, 0, -112, 0, 8, 0]] | +| RevTabl | [[0], [0, 1], [0, 2, 0], [0, 3, 0, -2], [0, 4, 0, -8, 0], [0, 5, 0, -20, 0, 16], [0, 6, 0, -40, 0, 96, 0], [0, 7, 0, -70, 0, 336, 0, -272], [0, 8, 0, -112, 0, 896, 0, -2176, 0]] | +| AntiDiag | [[0], [1], [0, 0], [-2, 2], [0, 0, 0], [16, -8, 3], [0, 0, 0, 0], [-272, 96, -20, 4], [0, 0, 0, 0, 0]] | +| AccTabl | [[0], [1, 1], [0, 2, 2], [-2, -2, 1, 1], [0, -8, -8, -4, -4], [16, 16, -4, -4, 1, 1], [0, 96, 96, 56, 56, 62, 62], [-272, -272, 64, 64, -6, -6, 1, 1], [0, -2176, -2176, -1280, -1280, -1392, -1392, -1384, -1384]] | +| RevAccTabl | [[0], [1, 1], [2, 2, 0], [1, 1, -2, -2], [-4, -4, -8, -8, 0], [1, 1, -4, -4, 16, 16], [62, 62, 56, 56, 96, 96, 0], [1, 1, -6, -6, 64, 64, -272, -272], [-1384, -1384, -1392, -1392, -1280, -1280, -2176, -2176, 0]] | +| AccRevTabl | [[0], [0, 1], [0, 2, 2], [0, 3, 3, 1], [0, 4, 4, -4, -4], [0, 5, 5, -15, -15, 1], [0, 6, 6, -34, -34, 62, 62], [0, 7, 7, -63, -63, 273, 273, 1], [0, 8, 8, -104, -104, 792, 792, -1384, -1384]] | EulerTan Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | -| EvenSum | [0, 1, 0, 1, 0, 1, 0, 1, 0] | -| OddSum | [0, 0, 2, 0, -4, 0, 62, 0, -1384] | -| AltSum | [0, 1, -2, 1, 4, 1, -62, 1, 1384] | -| AccSum | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | -| AccRevSum | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | -| RevAccSum | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | -| DiagSum | [0, 1, 0, 0, 0, 11, 0, -192] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | +| EvenSum | [0, 1, 0, 1, 0, 1, 0, 1, 0] | +| OddSum | [0, 0, 2, 0, -4, 0, 62, 0, -1384] | +| AltSum | [0, 1, -2, 1, 4, 1, -62, 1, 1384] | +| AccSum | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | +| AccRevSum | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | +| AntiDiagSum | [0, 1, 0, 0, 0, 11, 0, -192, 0] | EulerTan Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 8, 80, 480, 28560, 15232] | -| RowGcd | [1, 1, 2, 1, 4, 1, 2, 1, 8] | -| RowMax | [0, 1, 2, 3, 8, 20, 96, 336, 2176] | -| Middle | [0, 1, 2, 0, 0, -20, -40, 0, 0] | -| Central | [0, 2, 0, -40, 0] | -| LeftSide | [0, 1, 0, -2, 0, 16, 0, -272, 0] | -| RightSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [0, 2, 4, -10, -56, 362, 2764, -24610, -250736] | -| NegHalf | [0, -2, -4, 10, 56, -362, -2764, 24610, 250736] | -| Bin | [0, 1, 4, 7, -16, -159, -188, 4383, 26560] | -| InvBin | [0, -1, -4, -7, 16, 159, 188, -4383, -26560] | -| TransSqrs | [0, 0, 2, 12, 28, 0, -114, 476, 3480] | -| TransNat0 | [0, 0, 2, 6, 4, -20, 6, 434, 8] | -| TransNat1 | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 8, 80, 480, 28560, 15232] | +| RowGcd | [1, 1, 2, 1, 4, 1, 2, 1, 8] | +| RowMax | [0, 1, 2, 3, 8, 20, 96, 336, 2176] | +| ColMiddle | [0, 1, 2, 0, 0, -20, -40, 0, 0] | +| ColECenter | [0, 2, 0, -40, 0] | +| ColOCenter | [1, 0, -20, 0] | +| ColLeft | [0, 1, 0, -2, 0, 16, 0, -272, 0] | +| ColRight | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 12, 28, 0, -114, 476, 3480] | +| TransNat0 | [0, 0, 2, 6, 4, -20, 6, 434, 8] | +| TransNat1 | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | EulerTan Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ EulerTan Polynomial values as columns | PolyCol7 | [0, 1, 14, 145, 1316, 11041, 87794, 671665, 4998056] | | PolyCol8 | [0, 1, 16, 190, 1984, 19216, 176896, 1569520, 13548544] | -# EulerTanRev -True +# EulerTan:Rev +[] -EulerTanRev Triangle view +EulerTan:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [0] | @@ -122,47 +118,43 @@ EulerTanRev Triangle view | Row7 | [0, 7, 0, -70, 0, 336, 0, -272] | | Row8 | [0, 8, 0, -112, 0, 896, 0, -2176, 0] | -EulerTanRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 0, 1, 0, 2, 0, 0, 3, 0, -2, 0, 4, 0, -8, 0, 0, 5, 0, -20, 0, 16, 0, 6, 0, -40, 0, 96, 0, 0, 7, 0, -70, 0, 336, 0, -272, 0, 8, 0, -112, 0, 896, 0, -2176, 0] | -| TriRev | [0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0] | -| TriAcc | [0, 0, 1, 0, 2, 2, 0, 3, 3, 1, 0, 4, 4, -4, -4, 0, 5, 5, -15, -15, 1, 0, 6, 6, -34, -34, 62, 62, 0, 7, 7, -63, -63, 273, 273, 1, 0, 8, 8, -104, -104, 792, 792, -1384, -1384] | -| TriRevAcc | [0, 1, 0, 2, 2, 0, 1, 3, 3, 0, -4, -4, 4, 4, 0, 1, -15, -15, 5, 5, 0, 62, 62, -34, -34, 6, 6, 0, 1, 273, 273, -63, -63, 7, 7, 0, -1384, -1384, 792, 792, -104, -104, 8, 8, 0] | -| TriAccRev | [0, 1, 1, 0, 2, 2, -2, -2, 1, 1, 0, -8, -8, -4, -4, 16, 16, -4, -4, 1, 1, 0, 96, 96, 56, 56, 62, 62, -272, -272, 64, 64, -6, -6, 1, 1, 0, -2176, -2176, -1280, -1280, -1392, -1392, -1384, -1384] | -| TriDiag | [0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 4, 0, 0, 5, 0, -2, 0, 6, 0, -8] | - -EulerTanRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | -| EvenSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | -| AltSum | [0, -1, -2, -1, 4, -1, -62, -1, 1384] | -| AccSum | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | -| AccRevSum | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | -| RevAccSum | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | -| DiagSum | [0, 0, 1, 2, 3, 4, 3, -2] | - -EulerTanRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 8, 80, 480, 28560, 15232] | -| RowGcd | [1, 1, 2, 1, 4, 1, 2, 1, 8] | -| RowMax | [0, 1, 2, 3, 8, 20, 96, 336, 2176] | -| Middle | [0, 0, 2, 3, 0, 0, -40, -70, 0] | -| Central | [0, 2, 0, -40, 0] | -| LeftSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [0, 1, 0, -2, 0, 16, 0, -272, 0] | -| PosHalf | [0, 1, 4, 10, 16, 16, 64, 400, 256] | -| NegHalf | [0, 1, -4, 10, -16, 16, -64, 400, -256] | -| Bin | [0, 1, 4, 7, -16, -159, -188, 4383, 26560] | -| InvBin | [0, 1, -4, 7, 16, -159, 188, 4383, -26560] | -| TransSqrs | [0, 1, 2, -15, -68, 225, 2046, -5551, -85224] | -| TransNat0 | [0, 1, 2, -3, -20, 25, 366, -427, -11080] | -| TransNat1 | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | - -EulerTanRev Diagonals as rows +EulerTan:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [0, 1], [0, 2, 0], [0, 3, 0, -2], [0, 4, 0, -8, 0], [0, 5, 0, -20, 0, 16], [0, 6, 0, -40, 0, 96, 0], [0, 7, 0, -70, 0, 336, 0, -272], [0, 8, 0, -112, 0, 896, 0, -2176, 0]] | +| RevTabl | [[0], [1, 0], [0, 2, 0], [-2, 0, 3, 0], [0, -8, 0, 4, 0], [16, 0, -20, 0, 5, 0], [0, 96, 0, -40, 0, 6, 0], [-272, 0, 336, 0, -70, 0, 7, 0], [0, -2176, 0, 896, 0, -112, 0, 8, 0]] | +| AntiDiag | [[0], [0], [0, 1], [0, 2], [0, 3, 0], [0, 4, 0], [0, 5, 0, -2], [0, 6, 0, -8], [0, 7, 0, -20, 0]] | +| AccTabl | [[0], [0, 1], [0, 2, 2], [0, 3, 3, 1], [0, 4, 4, -4, -4], [0, 5, 5, -15, -15, 1], [0, 6, 6, -34, -34, 62, 62], [0, 7, 7, -63, -63, 273, 273, 1], [0, 8, 8, -104, -104, 792, 792, -1384, -1384]] | +| RevAccTabl | [[0], [1, 0], [2, 2, 0], [1, 3, 3, 0], [-4, -4, 4, 4, 0], [1, -15, -15, 5, 5, 0], [62, 62, -34, -34, 6, 6, 0], [1, 273, 273, -63, -63, 7, 7, 0], [-1384, -1384, 792, 792, -104, -104, 8, 8, 0]] | +| AccRevTabl | [[0], [1, 1], [0, 2, 2], [-2, -2, 1, 1], [0, -8, -8, -4, -4], [16, 16, -4, -4, 1, 1], [0, 96, 96, 56, 56, 62, 62], [-272, -272, 64, 64, -6, -6, 1, 1], [0, -2176, -2176, -1280, -1280, -1392, -1392, -1384, -1384]] | + +EulerTan:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | +| EvenSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 2, 1, -4, 1, 62, 1, -1384] | +| AltSum | [0, -1, -2, -1, 4, -1, -62, -1, 1384] | +| AccSum | [0, 1, 4, 7, 0, -19, 68, 435, -1376] | +| AccRevSum | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | +| AntiDiagSum | [0, 0, 1, 2, 3, 4, 3, -2, -13] | + +EulerTan:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 8, 80, 480, 28560, 15232] | +| RowGcd | [1, 1, 2, 1, 4, 1, 2, 1, 8] | +| RowMax | [0, 1, 2, 3, 8, 20, 96, 336, 2176] | +| ColMiddle | [0, 0, 2, 3, 0, 0, -40, -70, 0] | +| ColECenter | [0, 2, 0, -40, 0] | +| ColOCenter | [0, 3, 0, -70] | +| ColLeft | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [0, 1, 0, -2, 0, 16, 0, -272, 0] | +| TransSqrs | [0, 1, 2, -15, -68, 225, 2046, -5551, -85224] | +| TransNat0 | [0, 1, 2, -3, -20, 25, 366, -427, -11080] | +| TransNat1 | [0, 2, 4, -2, -24, 26, 428, -426, -12464] | + +EulerTan:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [0, 1, 0, -2, 0, 16, 0, -272, 0]| @@ -175,7 +167,7 @@ EulerTanRev Diagonals as rows | DiagRow7 | [0, 8, 0, -240, 0, 12672, 0, -933504, 0]| | DiagRow8 | [0, 9, 0, -330, 0, 20592, 0, -1750320, 0]| -EulerTanRev Diagonals as columns +EulerTan:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ EulerTanRev Diagonals as columns | DiagCol7 | [-272, -2176, -9792, -32640, -89760, -215424, -466752, -933504, -1750320] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -EulerTanRev Polynomial values as rows +EulerTan:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -201,7 +193,7 @@ EulerTanRev Polynomial values as rows | PolyRow7 | [0, 1, -24610, -515085, -4116836, -20208715, -73544934, -218380505, -559451080] | | PolyRow8 | [0, -1384, -250736, -4544184, -34741216, -167213960, -602197584, -1777008856, -4534099904] | -EulerTanRev Polynomial values as columns +EulerTan:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | diff --git a/data/md/Eulerian.md b/data/md/Eulerian.md index 8a6f60a..faa79a1 100644 --- a/data/md/Eulerian.md +++ b/data/md/Eulerian.md @@ -1,5 +1,5 @@ # Eulerian -['A008292', 'A123125', 'A173018'] +['A173018', 'A008292', 'A123125'] Eulerian Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Eulerian Triangle view | Row7 | [1, 120, 1191, 2416, 1191, 120, 1, 0] | | Row8 | [1, 247, 4293, 15619, 15619, 4293, 247, 1, 0] | -Eulerian Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 5, 6, 6, 1, 12, 23, 24, 24, 1, 27, 93, 119, 120, 120, 1, 58, 360, 662, 719, 720, 720, 1, 121, 1312, 3728, 4919, 5039, 5040, 5040, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 40320] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 6, 6, 5, 1, 24, 24, 23, 12, 1, 120, 120, 119, 93, 27, 1, 720, 720, 719, 662, 360, 58, 1, 5040, 5040, 5039, 4919, 3728, 1312, 121, 1, 40320, 40320, 40319, 40072, 35779, 20160, 4541, 248, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 12, 23, 24, 0, 1, 27, 93, 119, 120, 0, 1, 58, 360, 662, 719, 720, 0, 1, 121, 1312, 3728, 4919, 5039, 5040, 0, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 11, 1, 1, 26, 11, 0, 1, 57, 66, 1] | +Eulerian Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 4, 1, 0], [1, 11, 11, 1, 0], [1, 26, 66, 26, 1, 0], [1, 57, 302, 302, 57, 1, 0], [1, 120, 1191, 2416, 1191, 120, 1, 0], [1, 247, 4293, 15619, 15619, 4293, 247, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 4, 1], [0, 1, 11, 11, 1], [0, 1, 26, 66, 26, 1], [0, 1, 57, 302, 302, 57, 1], [0, 1, 120, 1191, 2416, 1191, 120, 1], [0, 1, 247, 4293, 15619, 15619, 4293, 247, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 4, 0], [1, 11, 1], [1, 26, 11, 0], [1, 57, 66, 1], [1, 120, 302, 26, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 5, 6, 6], [1, 12, 23, 24, 24], [1, 27, 93, 119, 120, 120], [1, 58, 360, 662, 719, 720, 720], [1, 121, 1312, 3728, 4919, 5039, 5040, 5040], [1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 40320]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [6, 6, 5, 1], [24, 24, 23, 12, 1], [120, 120, 119, 93, 27, 1], [720, 720, 719, 662, 360, 58, 1], [5040, 5040, 5039, 4919, 3728, 1312, 121, 1], [40320, 40320, 40319, 40072, 35779, 20160, 4541, 248, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 5, 6], [0, 1, 12, 23, 24], [0, 1, 27, 93, 119, 120], [0, 1, 58, 360, 662, 719, 720], [0, 1, 121, 1312, 3728, 4919, 5039, 5040], [0, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320]] | Eulerian Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 1, 1, 2, 12, 68, 360, 2384, 20160] | -| OddSum | [0, 0, 1, 4, 12, 52, 360, 2656, 20160] | -| AltSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| AccSum | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | -| AccRevSum | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | -| RevAccSum | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | -| DiagSum | [1, 1, 1, 2, 5, 13, 38, 125] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 1, 1, 2, 12, 68, 360, 2384, 20160] | +| OddSum | [0, 0, 1, 4, 12, 52, 360, 2656, 20160] | +| AltSum | [1, 1, 0, -2, 0, 16, 0, -272, 0] | +| AccSum | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | +| AccRevSum | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | +| AntiDiagSum | [1, 1, 1, 2, 5, 13, 38, 125, 449] | Eulerian Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 4, 11, 858, 17214, 14387280, 16561934649] | -| RowGcd | [1, 1, 1, 4, 11, 2, 1, 1, 1] | -| RowMax | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | -| Middle | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | -| Central | [1, 1, 11, 302, 15619] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670] | -| NegHalf | [1, -2, 2, 6, -30, -42, 882, -954, -39870] | -| Bin | [1, 1, 3, 16, 115, 1056, 11774, 154624, 2337507] | -| InvBin | [1, -1, -1, 8, 19, -276, -1002, 21216, 103395] | -| TransSqrs | [0, 0, 1, 8, 64, 540, 4920, 48720, 524160] | -| TransNat0 | [0, 0, 1, 6, 36, 240, 1800, 15120, 141120] | -| TransNat1 | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 11, 858, 17214, 14387280, 16561934649] | +| RowGcd | [1, 1, 1, 4, 11, 2, 1, 1, 1] | +| RowMax | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | +| ColMiddle | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | +| ColECenter | [1, 1, 11, 302, 15619] | +| ColOCenter | [1, 4, 66, 2416] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 8, 64, 540, 4920, 48720, 524160] | +| TransNat0 | [0, 0, 1, 6, 36, 240, 1800, 15120, 141120] | +| TransNat1 | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | Eulerian Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Eulerian Polynomial values as columns | PolyCol7 | [1, 1, 8, 78, 960, 14736, 272448, 5881968, 145105920] | | PolyCol8 | [1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825] | -# EulerianRev -True +# Eulerian:Rev +[] -EulerianRev Triangle view +Eulerian:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ EulerianRev Triangle view | Row7 | [0, 1, 120, 1191, 2416, 1191, 120, 1] | | Row8 | [0, 1, 247, 4293, 15619, 15619, 4293, 247, 1] | -EulerianRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 12, 23, 24, 0, 1, 27, 93, 119, 120, 0, 1, 58, 360, 662, 719, 720, 0, 1, 121, 1312, 3728, 4919, 5039, 5040, 0, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 6, 5, 1, 0, 24, 23, 12, 1, 0, 120, 119, 93, 27, 1, 0, 720, 719, 662, 360, 58, 1, 0, 5040, 5039, 4919, 3728, 1312, 121, 1, 0, 40320, 40319, 40072, 35779, 20160, 4541, 248, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 5, 6, 6, 1, 12, 23, 24, 24, 1, 27, 93, 119, 120, 120, 1, 58, 360, 662, 719, 720, 720, 1, 121, 1312, 3728, 4919, 5039, 5040, 5040, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 40320] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 4, 0, 1, 11, 1, 0, 1, 26, 11] | - -EulerianRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 0, 1, 4, 12, 52, 360, 2656, 20160] | -| OddSum | [0, 1, 1, 2, 12, 68, 360, 2384, 20160] | -| AltSum | [1, -1, 0, 2, 0, -16, 0, 272, 0] | -| AccSum | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | -| AccRevSum | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | -| RevAccSum | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | -| DiagSum | [1, 0, 1, 1, 2, 5, 13, 38] | - -EulerianRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 4, 11, 858, 17214, 14387280, 16561934649] | -| RowGcd | [1, 1, 1, 4, 11, 2, 1, 1, 1] | -| RowMax | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | -| Middle | [1, 0, 1, 1, 11, 26, 302, 1191, 15619] | -| Central | [1, 1, 11, 302, 15619] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| NegHalf | [1, 1, -1, -3, 15, 21, -441, 477, 19935] | -| Bin | [1, 1, 3, 16, 115, 1056, 11774, 154624, 2337507] | -| InvBin | [1, 1, -1, -8, 19, 276, -1002, -21216, 103395] | -| TransSqrs | [0, 1, 5, 26, 160, 1140, 9240, 84000, 846720] | -| TransNat0 | [0, 1, 3, 12, 60, 360, 2520, 20160, 181440] | -| TransNat1 | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | - -EulerianRev Diagonals as rows +Eulerian:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 4, 1], [0, 1, 11, 11, 1], [0, 1, 26, 66, 26, 1], [0, 1, 57, 302, 302, 57, 1], [0, 1, 120, 1191, 2416, 1191, 120, 1], [0, 1, 247, 4293, 15619, 15619, 4293, 247, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 4, 1, 0], [1, 11, 11, 1, 0], [1, 26, 66, 26, 1, 0], [1, 57, 302, 302, 57, 1, 0], [1, 120, 1191, 2416, 1191, 120, 1, 0], [1, 247, 4293, 15619, 15619, 4293, 247, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 4], [0, 1, 11, 1], [0, 1, 26, 11], [0, 1, 57, 66, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 5, 6], [0, 1, 12, 23, 24], [0, 1, 27, 93, 119, 120], [0, 1, 58, 360, 662, 719, 720], [0, 1, 121, 1312, 3728, 4919, 5039, 5040], [0, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [6, 5, 1, 0], [24, 23, 12, 1, 0], [120, 119, 93, 27, 1, 0], [720, 719, 662, 360, 58, 1, 0], [5040, 5039, 4919, 3728, 1312, 121, 1, 0], [40320, 40319, 40072, 35779, 20160, 4541, 248, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 5, 6, 6], [1, 12, 23, 24, 24], [1, 27, 93, 119, 120, 120], [1, 58, 360, 662, 719, 720, 720], [1, 121, 1312, 3728, 4919, 5039, 5040, 5040], [1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 40320]] | + +Eulerian:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 0, 1, 4, 12, 52, 360, 2656, 20160] | +| OddSum | [0, 1, 1, 2, 12, 68, 360, 2384, 20160] | +| AltSum | [1, -1, 0, 2, 0, -16, 0, 272, 0] | +| AccSum | [1, 1, 3, 12, 60, 360, 2520, 20160, 181440] | +| AccRevSum | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | +| AntiDiagSum | [1, 0, 1, 1, 2, 5, 13, 38, 125] | + +Eulerian:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 11, 858, 17214, 14387280, 16561934649] | +| RowGcd | [1, 1, 1, 4, 11, 2, 1, 1, 1] | +| RowMax | [1, 1, 1, 4, 11, 66, 302, 2416, 15619] | +| ColMiddle | [1, 0, 1, 1, 11, 26, 302, 1191, 15619] | +| ColECenter | [1, 1, 11, 302, 15619] | +| ColOCenter | [0, 1, 26, 1191] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 26, 160, 1140, 9240, 84000, 846720] | +| TransNat0 | [0, 1, 3, 12, 60, 360, 2520, 20160, 181440] | +| TransNat1 | [1, 2, 5, 18, 84, 480, 3240, 25200, 221760] | + +Eulerian:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ EulerianRev Diagonals as rows | DiagRow7 | [0, 1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472]| | DiagRow8 | [0, 1, 1013, 152637, 10187685, 423281535, 12843262863, 311387598411, 6382798925475]| -EulerianRev Diagonals as columns +Eulerian:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ EulerianRev Diagonals as columns | DiagCol7 | [1, 247, 14608, 455192, 9738114, 162512286, 2275172004, 27971176092, 311387598411] | | DiagCol8 | [1, 502, 47840, 2203488, 66318474, 1505621508, 27971176092, 447538817472, 6382798925475] | -EulerianRev Polynomial values as rows +Eulerian:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ EulerianRev Polynomial values as rows | PolyRow7 | [0, 5040, 94586, 608016, 2424132, 7336880, 18532590, 41173776, 83094536] | | PolyRow8 | [0, 40320, 1091670, 8855040, 41967540, 145879680, 413749770, 1015741440, 2238150600] | -EulerianRev Polynomial values as columns +Eulerian:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,3 +206,107 @@ EulerianRev Polynomial values as columns | PolyCol7 | [1, 7, 56, 546, 6720, 103152, 1907136, 41173776, 1015741440] | | PolyCol8 | [1, 8, 72, 776, 10440, 174728, 3525192, 83094536, 2238150600] | +# Eulerian:Rev:Inv +[] + +Eulerian:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [0, -1, 1] | +| Row3 | [0, 3, -4, 1] | +| Row4 | [0, -23, 33, -11, 1] | +| Row5 | [0, 425, -620, 220, -26, 1] | +| Row6 | [0, -18129, 26525, -9520, 1180, -57, 1] | +| Row7 | [0, 1721419, -2519664, 905765, -113050, 5649, -120, 1] | +| Row8 | [0, -353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1] | + +Eulerian:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 3, -4, 1], [0, -23, 33, -11, 1], [0, 425, -620, 220, -26, 1], [0, -18129, 26525, -9520, 1180, -57, 1], [0, 1721419, -2519664, 905765, -113050, 5649, -120, 1], [0, -353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -4, 3, 0], [1, -11, 33, -23, 0], [1, -26, 220, -620, 425, 0], [1, -57, 1180, -9520, 26525, -18129, 0], [1, -120, 5649, -113050, 905765, -2519664, 1721419, 0], [1, -247, 25347, -1166221, 23248085, -186123259, 517670461, -353654167, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 3, 1], [0, -23, -4], [0, 425, 33, 1], [0, -18129, -620, -11], [0, 1721419, 26525, 220, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 3, -1, 0], [0, -23, 10, -1, 0], [0, 425, -195, 25, -1, 0], [0, -18129, 8396, -1124, 56, -1, 0], [0, 1721419, -798245, 107520, -5530, 119, -1, 0], [0, -353654167, 164016294, -22106965, 1141120, -25101, 246, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 3, 0], [0, -1, 10, -23, 0], [0, -1, 25, -195, 425, 0], [0, -1, 56, -1124, 8396, -18129, 0], [0, -1, 119, -5530, 107520, -798245, 1721419, 0], [0, -1, 246, -25101, 1141120, -22106965, 164016294, -353654167, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -3, 0, 0], [1, -10, 23, 0, 0], [1, -25, 195, -425, 0, 0], [1, -56, 1124, -8396, 18129, 0, 0], [1, -119, 5530, -107520, 798245, -1721419, 0, 0], [1, -246, 25101, -1141120, 22106965, -164016294, 353654167, 0, 0]] | + +Eulerian:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -4, 34, -646, 27706, -2632834, 540943894] | +| OddSum | [0, 1, -1, 4, -34, 646, -27706, 2632834, -540943894] | +| AltSum | [1, -1, 2, -8, 68, -1292, 55412, -5265668, 1081887788] | +| AccSum | [1, 1, -1, 2, -14, 254, -10802, 1025282, -210628574] | +| AccRevSum | [1, 2, 1, -2, 14, -254, 10802, -1025282, 210628574] | +| AntiDiagSum | [1, 0, 1, -1, 4, -27, 459, -18760, 1748165] | + +Eulerian:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 12, 759, 7536100, 1026364915892400, 261891765713181433200, 7458125553629805584118669563264684517615] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 4, 33, 620, 26525, 2519664, 517670461] | +| ColMiddle | [1, 0, -1, 3, 33, -620, -9520, 905765, 23248085] | +| ColECenter | [1, -1, 33, -9520, 23248085] | +| ColOCenter | [0, 3, -620, 905765] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -4, 26, -466, 19782, -1877198, 385632634] | +| TransNat0 | [0, 1, 1, -2, 14, -254, 10802, -1025282, 210628574] | +| TransNat1 | [1, 2, 1, -2, 14, -254, 10802, -1025282, 210628574] | + +Eulerian:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [0, -1, -4, -11, -26, -57, -120, -247, -502]| +| DiagRow2 | [0, 3, 33, 220, 1180, 5649, 25347, 109386, 460686]| +| DiagRow3 | [0, -23, -620, -9520, -113050, -1166221, -11059468, -99446730, -863536410]| +| DiagRow4 | [0, 425, 26525, 905765, 23248085, 507795498, 10043953290, 186311714820, 3310534307520]| +| DiagRow5 | [0, -18129, -2519664, -186123259, -10119247684, -461102826618, -18816103842228, -714246416566848, -25816517930312538]| +| DiagRow6 | [0, 1721419, 517670461, 81009042744, 9188556369384, 863812603482906, 72133429144708470, 5569899127407415086, 407739804931874000586]| +| DiagRow7 | [0, -353654167, -225309742552, -73558129319520, -17213473445481060, -3311511577447284882, -562517197939195769556, -87969630451614365600718, -13000182161456131108373478]| +| DiagRow8 | [0, 153923102577, 204586385695065, 137800821538668585, 65989560146841989625, 25824118117025730219003, 8884259623968929185879179, 2804781888582977542701824505, 834753222582690830220723149085]| + +Eulerian:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, -1, 3, -23, 425, -18129, 1721419, -353654167, 153923102577] | +| DiagCol2 | [1, -4, 33, -620, 26525, -2519664, 517670461, -225309742552, 204586385695065] | +| DiagCol3 | [1, -11, 220, -9520, 905765, -186123259, 81009042744, -73558129319520, 137800821538668585] | +| DiagCol4 | [1, -26, 1180, -113050, 23248085, -10119247684, 9188556369384, -17213473445481060, 65989560146841989625] | +| DiagCol5 | [1, -57, 5649, -1166221, 507795498, -461102826618, 863812603482906, -3311511577447284882, 25824118117025730219003] | +| DiagCol6 | [1, -120, 25347, -11059468, 10043953290, -18816103842228, 72133429144708470, -562517197939195769556, 8884259623968929185879179] | +| DiagCol7 | [1, -247, 109386, -99446730, 186311714820, -714246416566848, 5569899127407415086, -87969630451614365600718, 2804781888582977542701824505] | +| DiagCol8 | [1, -502, 460686, -863536410, 3310534307520, -25816517930312538, 407739804931874000586, -13000182161456131108373478, 834753222582690830220723149085] | + +Eulerian:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [0, 0, 2, 6, 12, 20, 30, 42, 56] | +| PolyRow3 | [0, 0, -2, 0, 12, 40, 90, 168, 280] | +| PolyRow4 | [0, 0, 14, 12, -12, -40, -30, 84, 392] | +| PolyRow5 | [0, 0, -254, -228, 228, 1000, 1830, 2436, 2632] | +| PolyRow6 | [0, 0, 10802, 9756, -9588, -42520, -77490, -99708, -94024] | +| PolyRow7 | [0, 0, -1025282, -926700, 908652, 4036120, 7360890, 9479148, 8958040] | +| PolyRow8 | [0, 0, 210628574, 190391172, -186638412, -829163560, -1512303630, -1947629796, -1840661368] | + +Eulerian:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 2, 2, -2, 14, -254, 10802, -1025282, 210628574] | +| PolyCol3 | [1, 3, 6, 0, 12, -228, 9756, -926700, 190391172] | +| PolyCol4 | [1, 4, 12, 12, -12, 228, -9588, 908652, -186638412] | +| PolyCol5 | [1, 5, 20, 40, -40, 1000, -42520, 4036120, -829163560] | +| PolyCol6 | [1, 6, 30, 90, -30, 1830, -77490, 7360890, -1512303630] | +| PolyCol7 | [1, 7, 42, 168, 84, 2436, -99708, 9479148, -1947629796] | +| PolyCol8 | [1, 8, 56, 280, 392, 2632, -94024, 8958040, -1840661368] | + diff --git a/data/md/Eulerian2.md b/data/md/Eulerian2.md index 8e6fbf4..d6c2834 100644 --- a/data/md/Eulerian2.md +++ b/data/md/Eulerian2.md @@ -1,5 +1,5 @@ # Eulerian2 -['A008517', 'A112007', 'A163936', 'A340556'] +['A340556', 'A008517', 'A112007', 'A163936'] Eulerian2 Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Eulerian2 Triangle view | Row7 | [0, 1, 240, 5610, 32120, 58140, 33984, 5040] | | Row8 | [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320] | -Eulerian2 Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320] | -| TriRev | [1, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 3, 0, 1, 9, 15, 0, 1, 23, 81, 105, 0, 1, 53, 381, 825, 945, 0, 1, 115, 1567, 5967, 9675, 10395, 0, 1, 241, 5851, 37971, 96111, 130095, 135135, 0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025] | -| TriRevAcc | [1, 1, 0, 3, 1, 0, 15, 9, 1, 0, 105, 81, 23, 1, 0, 945, 825, 381, 53, 1, 0, 10395, 9675, 5967, 1567, 115, 1, 0, 135135, 130095, 96111, 37971, 5851, 241, 1, 0, 2027025, 1986705, 1645569, 860265, 216245, 20445, 495, 1, 0] | -| TriAccRev | [1, 1, 1, 2, 3, 3, 6, 14, 15, 15, 24, 82, 104, 105, 105, 120, 564, 892, 944, 945, 945, 720, 4428, 8828, 10280, 10394, 10395, 10395, 5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135, 40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 8, 0, 1, 22, 6, 0, 1, 52, 58] | +Eulerian2 Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 1, 8, 6], [0, 1, 22, 58, 24], [0, 1, 52, 328, 444, 120], [0, 1, 114, 1452, 4400, 3708, 720], [0, 1, 240, 5610, 32120, 58140, 33984, 5040], [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [6, 8, 1, 0], [24, 58, 22, 1, 0], [120, 444, 328, 52, 1, 0], [720, 3708, 4400, 1452, 114, 1, 0], [5040, 33984, 58140, 32120, 5610, 240, 1, 0], [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 8], [0, 1, 22, 6], [0, 1, 52, 58], [0, 1, 114, 328, 24]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 9, 15], [0, 1, 23, 81, 105], [0, 1, 53, 381, 825, 945], [0, 1, 115, 1567, 5967, 9675, 10395], [0, 1, 241, 5851, 37971, 96111, 130095, 135135], [0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [15, 9, 1, 0], [105, 81, 23, 1, 0], [945, 825, 381, 53, 1, 0], [10395, 9675, 5967, 1567, 115, 1, 0], [135135, 130095, 96111, 37971, 5851, 241, 1, 0], [2027025, 1986705, 1645569, 860265, 216245, 20445, 495, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [6, 14, 15, 15], [24, 82, 104, 105, 105], [120, 564, 892, 944, 945, 945], [720, 4428, 8828, 10280, 10394, 10395, 10395], [5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135], [40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025]] | Eulerian2 Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| EvenSum | [1, 0, 2, 8, 46, 496, 5234, 66344, 1021918] | -| OddSum | [0, 1, 1, 7, 59, 449, 5161, 68791, 1005107] | -| AltSum | [1, -1, 1, 1, -13, 47, 73, -2447, 16811] | -| AccSum | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | -| AccRevSum | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | -| RevAccSum | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | -| DiagSum | [1, 0, 1, 1, 3, 9, 29, 111] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| EvenSum | [1, 0, 2, 8, 46, 496, 5234, 66344, 1021918] | +| OddSum | [0, 1, 1, 7, 59, 449, 5161, 68791, 1005107] | +| AltSum | [1, -1, 1, 1, -13, 47, 73, -2447, 16811] | +| AccSum | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | +| AccRevSum | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | +| AntiDiagSum | [1, 0, 1, 1, 3, 9, 29, 111, 467] | Eulerian2 Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600] | -| RowGcd | [1, 1, 2, 2, 2, 4, 2, 2, 2] | -| RowMax | [1, 1, 2, 8, 58, 444, 4400, 58140, 785304] | -| Middle | [1, 0, 1, 1, 22, 52, 1452, 5610, 195800] | -| Central | [1, 1, 22, 1452, 195800] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912] | -| NegHalf | [1, 1, 0, -6, -12, 144, 1080, -5184, -127008] | -| Bin | [1, 1, 4, 33, 392, 6145, 119724, 2789465, 75660080] | -| InvBin | [1, 1, 0, -15, -80, 665, 17136, 55209, -4162752] | -| TransSqrs | [0, 1, 9, 87, 995, 13265, 202545, 3489255, 66981915] | -| TransNat0 | [0, 1, 5, 35, 315, 3465, 45045, 675675, 11486475] | -| TransNat1 | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600] | +| RowGcd | [1, 1, 2, 2, 2, 4, 2, 2, 2] | +| RowMax | [1, 1, 2, 8, 58, 444, 4400, 58140, 785304] | +| ColMiddle | [1, 0, 1, 1, 22, 52, 1452, 5610, 195800] | +| ColECenter | [1, 1, 22, 1452, 195800] | +| ColOCenter | [0, 1, 52, 5610] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 9, 87, 995, 13265, 202545, 3489255, 66981915] | +| TransNat0 | [0, 1, 5, 35, 315, 3465, 45045, 675675, 11486475] | +| TransNat1 | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | Eulerian2 Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Eulerian2 Polynomial values as columns | PolyCol7 | [1, 7, 105, 2457, 78603, 3197943, 158095665, 9205055433, 617068198467] | | PolyCol8 | [1, 8, 136, 3592, 129416, 5922056, 329020552, 21517930504, 1619649615752] | -# Eulerian2Rev -True +# Eulerian2:Rev +[] -Eulerian2Rev Triangle view +Eulerian2:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ Eulerian2Rev Triangle view | Row7 | [5040, 33984, 58140, 32120, 5610, 240, 1, 0] | | Row8 | [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0] | -Eulerian2Rev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320] | -| TriAcc | [1, 1, 1, 2, 3, 3, 6, 14, 15, 15, 24, 82, 104, 105, 105, 120, 564, 892, 944, 945, 945, 720, 4428, 8828, 10280, 10394, 10395, 10395, 5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135, 40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025] | -| TriRevAcc | [1, 1, 1, 3, 3, 2, 15, 15, 14, 6, 105, 105, 104, 82, 24, 945, 945, 944, 892, 564, 120, 10395, 10395, 10394, 10280, 8828, 4428, 720, 135135, 135135, 135134, 134894, 129284, 97164, 39024, 5040, 2027025, 2027025, 2027024, 2026530, 2006580, 1810780, 1166760, 381456, 40320] | -| TriAccRev | [1, 0, 1, 0, 1, 3, 0, 1, 9, 15, 0, 1, 23, 81, 105, 0, 1, 53, 381, 825, 945, 0, 1, 115, 1567, 5967, 9675, 10395, 0, 1, 241, 5851, 37971, 96111, 130095, 135135, 0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025] | -| TriDiag | [1, 1, 2, 0, 6, 1, 24, 8, 0, 120, 58, 1, 720, 444, 22, 0, 5040, 3708, 328, 1] | - -Eulerian2Rev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| EvenSum | [1, 1, 2, 7, 46, 449, 5234, 68791, 1021918] | -| OddSum | [0, 0, 1, 8, 59, 496, 5161, 66344, 1005107] | -| AltSum | [1, 1, 1, -1, -13, -47, 73, 2447, 16811] | -| AccSum | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | -| AccRevSum | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | -| RevAccSum | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | -| DiagSum | [1, 1, 2, 7, 32, 179, 1186, 9077] | - -Eulerian2Rev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600] | -| RowGcd | [1, 1, 2, 2, 2, 4, 2, 2, 2] | -| RowMax | [1, 1, 2, 8, 58, 444, 4400, 58140, 785304] | -| Middle | [1, 1, 1, 8, 22, 328, 1452, 32120, 195800] | -| Central | [1, 1, 22, 1452, 195800] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802] | -| NegHalf | [1, -2, 6, -18, 6, 846, -13338, 139374, -717498] | -| Bin | [1, 1, 4, 33, 392, 6145, 119724, 2789465, 75660080] | -| InvBin | [1, -1, 0, 15, -80, -665, 17136, -55209, -4162752] | -| TransSqrs | [0, 0, 1, 12, 155, 2240, 36225, 651420, 12927915] | -| TransNat0 | [0, 0, 1, 10, 105, 1260, 17325, 270270, 4729725] | -| TransNat1 | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | - -Eulerian2Rev Diagonals as rows +Eulerian2:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [6, 8, 1, 0], [24, 58, 22, 1, 0], [120, 444, 328, 52, 1, 0], [720, 3708, 4400, 1452, 114, 1, 0], [5040, 33984, 58140, 32120, 5610, 240, 1, 0], [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 8, 6], [0, 1, 22, 58, 24], [0, 1, 52, 328, 444, 120], [0, 1, 114, 1452, 4400, 3708, 720], [0, 1, 240, 5610, 32120, 58140, 33984, 5040], [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320]] | +| AntiDiag | [[1], [1], [2, 0], [6, 1], [24, 8, 0], [120, 58, 1], [720, 444, 22, 0], [5040, 3708, 328, 1], [40320, 33984, 4400, 52, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [6, 14, 15, 15], [24, 82, 104, 105, 105], [120, 564, 892, 944, 945, 945], [720, 4428, 8828, 10280, 10394, 10395, 10395], [5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135], [40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [15, 15, 14, 6], [105, 105, 104, 82, 24], [945, 945, 944, 892, 564, 120], [10395, 10395, 10394, 10280, 8828, 4428, 720], [135135, 135135, 135134, 134894, 129284, 97164, 39024, 5040], [2027025, 2027025, 2027024, 2026530, 2006580, 1810780, 1166760, 381456, 40320]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 9, 15], [0, 1, 23, 81, 105], [0, 1, 53, 381, 825, 945], [0, 1, 115, 1567, 5967, 9675, 10395], [0, 1, 241, 5851, 37971, 96111, 130095, 135135], [0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025]] | + +Eulerian2:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| EvenSum | [1, 1, 2, 7, 46, 449, 5234, 68791, 1021918] | +| OddSum | [0, 0, 1, 8, 59, 496, 5161, 66344, 1005107] | +| AltSum | [1, 1, 1, -1, -13, -47, 73, 2447, 16811] | +| AccSum | [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500] | +| AccRevSum | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | +| AntiDiagSum | [1, 1, 2, 7, 32, 179, 1186, 9077, 78756] | + +Eulerian2:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600] | +| RowGcd | [1, 1, 2, 2, 2, 4, 2, 2, 2] | +| RowMax | [1, 1, 2, 8, 58, 444, 4400, 58140, 785304] | +| ColMiddle | [1, 1, 1, 8, 22, 328, 1452, 32120, 195800] | +| ColECenter | [1, 1, 22, 1452, 195800] | +| ColOCenter | [1, 8, 328, 32120] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 12, 155, 2240, 36225, 651420, 12927915] | +| TransNat0 | [0, 0, 1, 10, 105, 1260, 17325, 270270, 4729725] | +| TransNat1 | [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750] | + +Eulerian2:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ Eulerian2Rev Diagonals as rows | DiagRow7 | [5040, 341136, 11026296, 238904904, 4002695088, 56041398784, 687720046384, 7634832149392, 78391384831312]| | DiagRow8 | [40320, 3733920, 162186912, 4642163952, 101180433024, 1818188642304, 28299910066112, 394365587815520, 5036317938475648]| -Eulerian2Rev Diagonals as columns +Eulerian2:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -188,7 +180,7 @@ Eulerian2Rev Diagonals as columns | DiagCol7 | [0, 1, 1004, 218848, 25243904, 2051482776, 134323420224, 7634832149392, 394365587815520] | | DiagCol8 | [0, 1, 2026, 695038, 114876376, 12669817776, 1084676512416, 78391384831312, 5036317938475648] | -Eulerian2Rev Polynomial values as rows +Eulerian2:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ Eulerian2Rev Polynomial values as rows | PolyRow7 | [5040, 135135, 660032, 2010951, 4812912, 9915335, 18423360, 31729887, 51548336] | | PolyRow8 | [40320, 2027025, 12818912, 46589967, 127780416, 294396725, 601682400, 1126164627, 1970081792] | -Eulerian2Rev Polynomial values as columns +Eulerian2:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | diff --git a/data/md/EulerianB.md b/data/md/EulerianB.md index acf6dff..3b8f0ac 100644 --- a/data/md/EulerianB.md +++ b/data/md/EulerianB.md @@ -14,45 +14,41 @@ EulerianB Triangle view | Row7 | [1, 2179, 60657, 259723, 259723, 60657, 2179, 1] | | Row8 | [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -EulerianB Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 230, 76, 1, 1, 237, 1682, 1682, 237, 1, 1, 722, 10543, 23548, 10543, 722, 1, 1, 2179, 60657, 259723, 259723, 60657, 2179, 1, 1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -| TriRev | [1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 230, 76, 1, 1, 237, 1682, 1682, 237, 1, 1, 722, 10543, 23548, 10543, 722, 1, 1, 2179, 60657, 259723, 259723, 60657, 2179, 1, 1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -| TriAcc | [1, 1, 2, 1, 7, 8, 1, 24, 47, 48, 1, 77, 307, 383, 384, 1, 238, 1920, 3602, 3839, 3840, 1, 723, 11266, 34814, 45357, 46079, 46080, 1, 2180, 62837, 322560, 582283, 642940, 645119, 645120, 1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 7, 1, 48, 47, 24, 1, 384, 383, 307, 77, 1, 3840, 3839, 3602, 1920, 238, 1, 46080, 46079, 45357, 34814, 11266, 723, 1, 645120, 645119, 642940, 582283, 322560, 62837, 2180, 1, 10321920, 10321919, 10315367, 9983755, 7498467, 2823453, 338165, 6553, 1] | -| TriAccRev | [1, 1, 2, 1, 7, 8, 1, 24, 47, 48, 1, 77, 307, 383, 384, 1, 238, 1920, 3602, 3839, 3840, 1, 723, 11266, 34814, 45357, 46079, 46080, 1, 2180, 62837, 322560, 582283, 642940, 645119, 645120, 1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920] | -| TriDiag | [1, 1, 1, 1, 1, 6, 1, 23, 1, 1, 76, 23, 1, 237, 230, 1, 1, 722, 1682, 76] | +EulerianB Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 6, 1], [1, 23, 23, 1], [1, 76, 230, 76, 1], [1, 237, 1682, 1682, 237, 1], [1, 722, 10543, 23548, 10543, 722, 1], [1, 2179, 60657, 259723, 259723, 60657, 2179, 1], [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1]] | +| RevTabl | [[1], [1, 1], [1, 6, 1], [1, 23, 23, 1], [1, 76, 230, 76, 1], [1, 237, 1682, 1682, 237, 1], [1, 722, 10543, 23548, 10543, 722, 1], [1, 2179, 60657, 259723, 259723, 60657, 2179, 1], [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 6], [1, 23, 1], [1, 76, 23], [1, 237, 230, 1], [1, 722, 1682, 76], [1, 2179, 10543, 1682, 1]] | +| AccTabl | [[1], [1, 2], [1, 7, 8], [1, 24, 47, 48], [1, 77, 307, 383, 384], [1, 238, 1920, 3602, 3839, 3840], [1, 723, 11266, 34814, 45357, 46079, 46080], [1, 2180, 62837, 322560, 582283, 642940, 645119, 645120], [1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 7, 1], [48, 47, 24, 1], [384, 383, 307, 77, 1], [3840, 3839, 3602, 1920, 238, 1], [46080, 46079, 45357, 34814, 11266, 723, 1], [645120, 645119, 642940, 582283, 322560, 62837, 2180, 1], [10321920, 10321919, 10315367, 9983755, 7498467, 2823453, 338165, 6553, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 7, 8], [1, 24, 47, 48], [1, 77, 307, 383, 384], [1, 238, 1920, 3602, 3839, 3840], [1, 723, 11266, 34814, 45357, 46079, 46080], [1, 2180, 62837, 322560, 582283, 642940, 645119, 645120], [1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920]] | EulerianB Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 2, 24, 232, 1920, 21088, 322560, 5338240] | -| OddSum | [0, 1, 6, 24, 152, 1920, 24992, 322560, 4983680] | -| AltSum | [1, 0, -4, 0, 80, 0, -3904, 0, 354560] | -| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| RevAccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| DiagSum | [1, 1, 2, 7, 25, 100, 469, 2481] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 2, 24, 232, 1920, 21088, 322560, 5338240] | +| OddSum | [0, 1, 6, 24, 152, 1920, 24992, 322560, 4983680] | +| AltSum | [1, 0, -4, 0, 80, 0, -3904, 0, 354560] | +| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AntiDiagSum | [1, 1, 2, 7, 25, 100, 469, 2481, 14406] | EulerianB Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 23, 8740, 398634, 89624229604, 34328005245969, 3371304106638422136] | -| RowGcd | [1, 1, 6, 23, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | -| Middle | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | -| Central | [1, 6, 230, 23548, 4675014] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377] | -| NegHalf | [1, -1, -7, 39, 177, -3441, -2007, 514359, -2913183] | -| Bin | [1, 2, 14, 140, 1990, 36012, 795916, 20758712, 624278342] | -| InvBin | [1, 0, -10, 0, 774, 0, -163332, 0, 67364166] | -| TransSqrs | [0, 1, 10, 124, 1696, 25920, 441600, 8332800, 172892160] | -| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | -| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 23, 8740, 398634, 89624229604, 34328005245969, 3371304106638422136] | +| RowGcd | [1, 1, 6, 23, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | +| ColMiddle | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | +| ColECenter | [1, 6, 230, 23548, 4675014] | +| ColOCenter | [1, 23, 1682, 259723] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 10, 124, 1696, 25920, 441600, 8332800, 172892160] | +| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | +| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | EulerianB Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ EulerianB Polynomial values as columns | PolyCol7 | [1, 8, 92, 1632, 40272, 1246848, 46164672, 1992310272, 98279131392] | | PolyCol8 | [1, 9, 113, 2169, 58337, 1974249, 79841873, 3761620569, 202567930817] | -# EulerianBInv -True +# EulerianB:Inv +[] -EulerianBInv Triangle view +EulerianB:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ EulerianBInv Triangle view | Row7 | [-1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1] | | Row8 | [11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1] | -EulerianBInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 5, -6, 1, -93, 115, -23, 1, 5993, -7436, 1518, -76, 1, -1272089, 1578757, -322762, 16330, -237, 1, 857402029, -1064110290, 217560951, -11012540, 160571, -722, 1, -1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1, 11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1] | -| TriRev | [1, 1, -1, 1, -6, 5, 1, -23, 115, -93, 1, -76, 1518, -7436, 5993, 1, -237, 16330, -322762, 1578757, -1272089, 1, -722, 160571, -11012540, 217560951, -1064110290, 857402029, 1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525, 1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057] | -| TriAcc | [1, -1, 0, 5, -1, 0, -93, 22, -1, 0, 5993, -1443, 75, -1, 0, -1272089, 306668, -16094, 236, -1, 0, 857402029, -206708261, 10852690, -159850, 721, -1, 0, -1792650585525, 432184866882, -22691017255, 334257820, -1510403, 2178, -1, 0, 11464255554367057, -2763883774564039, 145112312902789, -2137630911395, 9659553491, -13938645, 6551, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, 5, 0, -1, 22, -93, 0, -1, 75, -1443, 5993, 0, -1, 236, -16094, 306668, -1272089, 0, -1, 721, -159850, 10852690, -206708261, 857402029, 0, -1, 2178, -1510403, 334257820, -22691017255, 432184866882, -1792650585525, 0, -1, 6551, -13938645, 9659553491, -2137630911395, 145112312902789, -2763883774564039, 11464255554367057] | -| TriAccRev | [1, 1, 0, 1, -5, 0, 1, -22, 93, 0, 1, -75, 1443, -5993, 0, 1, -236, 16094, -306668, 1272089, 0, 1, -721, 159850, -10852690, 206708261, -857402029, 0, 1, -2178, 1510403, -334257820, 22691017255, -432184866882, 1792650585525, 0, 1, -6551, 13938645, -9659553491, 2137630911395, -145112312902789, 2763883774564039, -11464255554367057, 0] | -| TriDiag | [1, -1, 5, 1, -93, -6, 5993, 115, 1, -1272089, -7436, -23, 857402029, 1578757, 1518, 1, -1792650585525, -1064110290, -322762, -76] | - -EulerianBInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 6, -116, 7512, -1595088, 1075123552, -2247862240064, 14375398946243968] | -| OddSum | [0, 1, -6, 116, -7512, 1595088, -1075123552, 2247862240064, -14375398946243968] | -| AltSum | [1, -2, 12, -232, 15024, -3190176, 2150247104, -4495724480128, 28750797892487936] | -| AccSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | -| AccRevSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | -| RevAccSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | -| DiagSum | [1, -1, 6, -99, 6109, -1279548, 858982305, -1793715018653] | - -EulerianBInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 30, 10695, 4494102756, 1254352845580753602413730, 383972367241177062317562280537759735980, 715883465950829482992912712277956264095039430421605464700575, 4235300274399047501873201769211652027955886145279771055932028482670928385151774050533992] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 6, 115, 7436, 1578757, 1064110290, 2224835452407, 14228139328931096] | -| Middle | [1, -1, -6, 115, 1518, -322762, -11012540, 23025275075, 2147290464886] | -| Central | [1, -6, 1518, -11012540, 2147290464886] | -| LeftSide | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, 9, -329, 42321, -17963985, 24215716313, -101260109984153, 1295145618525878177] | -| NegHalf | [1, 3, 33, 1251, 161601, 68614851, 92494978401, 386775868795299, 4946973437912246913] | -| Bin | [1, 0, -6, 184, -14946, 3556192, -2481692012, 5022968503216, -29005196474209354] | -| InvBin | [1, 2, 18, 508, 45150, 12557980, 10728141732, 27736560614136, 215138109927100854] | -| TransSqrs | [0, 1, -2, 32, -2032, 430912, -290428224, 607224836096, -3883289161550592] | -| TransNat0 | [0, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | -| TransNat1 | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | - -EulerianBInv Diagonals as rows +EulerianB:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [5, -6, 1], [-93, 115, -23, 1], [5993, -7436, 1518, -76, 1], [-1272089, 1578757, -322762, 16330, -237, 1], [857402029, -1064110290, 217560951, -11012540, 160571, -722, 1], [-1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1], [11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1]] | +| RevTabl | [[1], [1, -1], [1, -6, 5], [1, -23, 115, -93], [1, -76, 1518, -7436, 5993], [1, -237, 16330, -322762, 1578757, -1272089], [1, -722, 160571, -11012540, 217560951, -1064110290, 857402029], [1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525], [1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057]] | +| AntiDiag | [[1], [-1], [5, 1], [-93, -6], [5993, 115, 1], [-1272089, -7436, -23], [857402029, 1578757, 1518, 1], [-1792650585525, -1064110290, -322762, -76], [11464255554367057, 2224835452407, 217560951, 16330, 1]] | +| AccTabl | [[1], [-1, 0], [5, -1, 0], [-93, 22, -1, 0], [5993, -1443, 75, -1, 0], [-1272089, 306668, -16094, 236, -1, 0], [857402029, -206708261, 10852690, -159850, 721, -1, 0], [-1792650585525, 432184866882, -22691017255, 334257820, -1510403, 2178, -1, 0], [11464255554367057, -2763883774564039, 145112312902789, -2137630911395, 9659553491, -13938645, 6551, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 5], [0, -1, 22, -93], [0, -1, 75, -1443, 5993], [0, -1, 236, -16094, 306668, -1272089], [0, -1, 721, -159850, 10852690, -206708261, 857402029], [0, -1, 2178, -1510403, 334257820, -22691017255, 432184866882, -1792650585525], [0, -1, 6551, -13938645, 9659553491, -2137630911395, 145112312902789, -2763883774564039, 11464255554367057]] | +| AccRevTabl | [[1], [1, 0], [1, -5, 0], [1, -22, 93, 0], [1, -75, 1443, -5993, 0], [1, -236, 16094, -306668, 1272089, 0], [1, -721, 159850, -10852690, 206708261, -857402029, 0], [1, -2178, 1510403, -334257820, 22691017255, -432184866882, 1792650585525, 0], [1, -6551, 13938645, -9659553491, 2137630911395, -145112312902789, 2763883774564039, -11464255554367057, 0]] | + +EulerianB:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 6, -116, 7512, -1595088, 1075123552, -2247862240064, 14375398946243968] | +| OddSum | [0, 1, -6, 116, -7512, 1595088, -1075123552, 2247862240064, -14375398946243968] | +| AltSum | [1, -2, 12, -232, 15024, -3190176, 2150247104, -4495724480128, 28750797892487936] | +| AccSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | +| AccRevSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| AntiDiagSum | [1, -1, 6, -99, 6109, -1279548, 858982305, -1793715018653, 11466480607396746] | + +EulerianB:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 30, 10695, 4494102756, 1254352845580753602413730, 383972367241177062317562280537759735980, 715883465950829482992912712277956264095039430421605464700575, 4235300274399047501873201769211652027955886145279771055932028482670928385151774050533992] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 6, 115, 7436, 1578757, 1064110290, 2224835452407, 14228139328931096] | +| ColMiddle | [1, -1, -6, 115, 1518, -322762, -11012540, 23025275075, 2147290464886] | +| ColECenter | [1, -6, 1518, -11012540, 2147290464886] | +| ColOCenter | [-1, 115, -322762, 23025275075] | +| ColLeft | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, -2, 32, -2032, 430912, -290428224, 607224836096, -3883289161550592] | +| TransNat0 | [0, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| TransNat1 | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | + +EulerianB:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ EulerianBInv Diagonals as rows | DiagRow7 | [-1792650585525, -14228139328931096, -56434463826320585284, -167312402773377971746920, -430241010049958628814059458, -1027422064996274234324991004232, -2358032188165387759999729133419316, -5296141644441170086929626440059219768, -11754596285418648169111713017129203693711]| | DiagRow8 | [11464255554367057, 276025608122908733321, 3305339964838291288943901, 29503676524653884693236314085, 228063754011124000036378055403359, 1635706386940760528736923482495252231, 11269427287836849598854819603136973269515, 75960197810240124116687805473107599893593955, 505872176876172128739413633340997793945445390118]| -EulerianBInv Diagonals as columns +EulerianB:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | @@ -188,7 +180,7 @@ EulerianBInv Diagonals as columns | DiagCol7 | [1, -6552, 127141156, -7446608913000, 1313126451395556490, -696066950858505682059208, 1108172771490732215668663853876, -5296141644441170086929626440059219768, 75960197810240124116687805473107599893593955] | | DiagCol8 | [1, -19673, 1152338433, -203202094563809, 107714121748314607559, -171486171954267914996574751, 819560884455816722716701613957119, -11754596285418648169111713017129203693711, 505872176876172128739413633340997793945445390118] | -EulerianBInv Polynomial values as rows +EulerianB:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ EulerianBInv Polynomial values as rows | PolyRow7 | [-1792650585525, 0, 1016394955037, 1382823986304, 1217877991767, 632626664960, -269205454455, -1391064478848, -2643394214461] | | PolyRow8 | [11464255554367057, 0, -6499990316797327, -8843356107415808, -7788503050691919, -4045737620558848, 1721607027968785, 8896055081595648, 16904881967491217] | -EulerianBInv Polynomial values as columns +EulerianB:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | @@ -214,10 +206,10 @@ EulerianBInv Polynomial values as columns | PolyCol7 | [1, 6, 12, -72, 4656, -987168, 665329344, -1391064478848, 8896055081595648] | | PolyCol8 | [1, 7, 21, -133, 8841, -1875825, 1264302557, -2643394214461, 16904881967491217] | -# EulerianBRev -True +# EulerianB:Rev +[] -EulerianBRev Triangle view +EulerianB:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ EulerianBRev Triangle view | Row7 | [1, 2179, 60657, 259723, 259723, 60657, 2179, 1] | | Row8 | [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -EulerianBRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 230, 76, 1, 1, 237, 1682, 1682, 237, 1, 1, 722, 10543, 23548, 10543, 722, 1, 1, 2179, 60657, 259723, 259723, 60657, 2179, 1, 1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -| TriRev | [1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 230, 76, 1, 1, 237, 1682, 1682, 237, 1, 1, 722, 10543, 23548, 10543, 722, 1, 1, 2179, 60657, 259723, 259723, 60657, 2179, 1, 1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1] | -| TriAcc | [1, 1, 2, 1, 7, 8, 1, 24, 47, 48, 1, 77, 307, 383, 384, 1, 238, 1920, 3602, 3839, 3840, 1, 723, 11266, 34814, 45357, 46079, 46080, 1, 2180, 62837, 322560, 582283, 642940, 645119, 645120, 1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 7, 1, 48, 47, 24, 1, 384, 383, 307, 77, 1, 3840, 3839, 3602, 1920, 238, 1, 46080, 46079, 45357, 34814, 11266, 723, 1, 645120, 645119, 642940, 582283, 322560, 62837, 2180, 1, 10321920, 10321919, 10315367, 9983755, 7498467, 2823453, 338165, 6553, 1] | -| TriAccRev | [1, 1, 2, 1, 7, 8, 1, 24, 47, 48, 1, 77, 307, 383, 384, 1, 238, 1920, 3602, 3839, 3840, 1, 723, 11266, 34814, 45357, 46079, 46080, 1, 2180, 62837, 322560, 582283, 642940, 645119, 645120, 1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920] | -| TriDiag | [1, 1, 1, 1, 1, 6, 1, 23, 1, 1, 76, 23, 1, 237, 230, 1, 1, 722, 1682, 76] | - -EulerianBRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 2, 24, 232, 1920, 21088, 322560, 5338240] | -| OddSum | [0, 1, 6, 24, 152, 1920, 24992, 322560, 4983680] | -| AltSum | [1, 0, -4, 0, 80, 0, -3904, 0, 354560] | -| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| RevAccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| DiagSum | [1, 1, 2, 7, 25, 100, 469, 2481] | - -EulerianBRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 23, 8740, 398634, 89624229604, 34328005245969, 3371304106638422136] | -| RowGcd | [1, 1, 6, 23, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | -| Middle | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | -| Central | [1, 6, 230, 23548, 4675014] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 17, 147, 1697, 24483, 423857, 8560947, 197613377] | -| NegHalf | [1, -1, -7, 39, 177, -3441, -2007, 514359, -2913183] | -| Bin | [1, 2, 14, 140, 1990, 36012, 795916, 20758712, 624278342] | -| InvBin | [1, 0, -10, 0, 774, 0, -163332, 0, 67364166] | -| TransSqrs | [0, 1, 10, 124, 1696, 25920, 441600, 8332800, 172892160] | -| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | -| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | - -EulerianBRev Diagonals as rows +EulerianB:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 6, 1], [1, 23, 23, 1], [1, 76, 230, 76, 1], [1, 237, 1682, 1682, 237, 1], [1, 722, 10543, 23548, 10543, 722, 1], [1, 2179, 60657, 259723, 259723, 60657, 2179, 1], [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1]] | +| RevTabl | [[1], [1, 1], [1, 6, 1], [1, 23, 23, 1], [1, 76, 230, 76, 1], [1, 237, 1682, 1682, 237, 1], [1, 722, 10543, 23548, 10543, 722, 1], [1, 2179, 60657, 259723, 259723, 60657, 2179, 1], [1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 6], [1, 23, 1], [1, 76, 23], [1, 237, 230, 1], [1, 722, 1682, 76], [1, 2179, 10543, 1682, 1]] | +| AccTabl | [[1], [1, 2], [1, 7, 8], [1, 24, 47, 48], [1, 77, 307, 383, 384], [1, 238, 1920, 3602, 3839, 3840], [1, 723, 11266, 34814, 45357, 46079, 46080], [1, 2180, 62837, 322560, 582283, 642940, 645119, 645120], [1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 7, 1], [48, 47, 24, 1], [384, 383, 307, 77, 1], [3840, 3839, 3602, 1920, 238, 1], [46080, 46079, 45357, 34814, 11266, 723, 1], [645120, 645119, 642940, 582283, 322560, 62837, 2180, 1], [10321920, 10321919, 10315367, 9983755, 7498467, 2823453, 338165, 6553, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 7, 8], [1, 24, 47, 48], [1, 77, 307, 383, 384], [1, 238, 1920, 3602, 3839, 3840], [1, 723, 11266, 34814, 45357, 46079, 46080], [1, 2180, 62837, 322560, 582283, 642940, 645119, 645120], [1, 6553, 338165, 2823453, 7498467, 9983755, 10315367, 10321919, 10321920]] | + +EulerianB:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 2, 24, 232, 1920, 21088, 322560, 5338240] | +| OddSum | [0, 1, 6, 24, 152, 1920, 24992, 322560, 4983680] | +| AltSum | [1, 0, -4, 0, 80, 0, -3904, 0, 354560] | +| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AntiDiagSum | [1, 1, 2, 7, 25, 100, 469, 2481, 14406] | + +EulerianB:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 23, 8740, 398634, 89624229604, 34328005245969, 3371304106638422136] | +| RowGcd | [1, 1, 6, 23, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | +| ColMiddle | [1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014] | +| ColECenter | [1, 6, 230, 23548, 4675014] | +| ColOCenter | [1, 23, 1682, 259723] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 10, 124, 1696, 25920, 441600, 8332800, 172892160] | +| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | +| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | + +EulerianB:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ EulerianBRev Diagonals as rows | DiagRow7 | [1, 6552, 1756340, 178300904, 10836823098, 477020564424, 16818059163492, 504541774904760, 13397724585164019]| | DiagRow8 | [1, 19673, 9116141, 1403080725, 121383780207, 7310750472183, 342917527152507, 13397724585164019, 455522635895576646]| -EulerianBRev Diagonals as columns +EulerianB:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ EulerianBRev Diagonals as columns | DiagCol7 | [1, 6552, 1756340, 178300904, 10836823098, 477020564424, 16818059163492, 504541774904760, 13397724585164019] | | DiagCol8 | [1, 19673, 9116141, 1403080725, 121383780207, 7310750472183, 342917527152507, 13397724585164019, 455522635895576646] | -EulerianBRev Polynomial values as rows +EulerianB:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ EulerianBRev Polynomial values as rows | PolyRow7 | [1, 645120, 8560947, 44932864, 155144925, 419997696, 968510095, 1992310272, 3761620569] | | PolyRow8 | [1, 10321920, 197613377, 1308788992, 5371825185, 16701095936, 43240726945, 98279131392, 202567930817] | -EulerianBRev Polynomial values as columns +EulerianB:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ EulerianBRev Polynomial values as columns | PolyCol7 | [1, 8, 92, 1632, 40272, 1246848, 46164672, 1992310272, 98279131392] | | PolyCol8 | [1, 9, 113, 2169, 58337, 1974249, 79841873, 3761620569, 202567930817] | -# EulerianBInvRev -True +# EulerianB:Inv:Rev +[] -EulerianBInvRev Triangle view +EulerianB:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ EulerianBInvRev Triangle view | Row7 | [1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525] | | Row8 | [1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057] | -EulerianBInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -6, 5, 1, -23, 115, -93, 1, -76, 1518, -7436, 5993, 1, -237, 16330, -322762, 1578757, -1272089, 1, -722, 160571, -11012540, 217560951, -1064110290, 857402029, 1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525, 1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057] | -| TriRev | [1, -1, 1, 5, -6, 1, -93, 115, -23, 1, 5993, -7436, 1518, -76, 1, -1272089, 1578757, -322762, 16330, -237, 1, 857402029, -1064110290, 217560951, -11012540, 160571, -722, 1, -1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1, 11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1] | -| TriAcc | [1, 1, 0, 1, -5, 0, 1, -22, 93, 0, 1, -75, 1443, -5993, 0, 1, -236, 16094, -306668, 1272089, 0, 1, -721, 159850, -10852690, 206708261, -857402029, 0, 1, -2178, 1510403, -334257820, 22691017255, -432184866882, 1792650585525, 0, 1, -6551, 13938645, -9659553491, 2137630911395, -145112312902789, 2763883774564039, -11464255554367057, 0] | -| TriRevAcc | [1, 0, 1, 0, -5, 1, 0, 93, -22, 1, 0, -5993, 1443, -75, 1, 0, 1272089, -306668, 16094, -236, 1, 0, -857402029, 206708261, -10852690, 159850, -721, 1, 0, 1792650585525, -432184866882, 22691017255, -334257820, 1510403, -2178, 1, 0, -11464255554367057, 2763883774564039, -145112312902789, 2137630911395, -9659553491, 13938645, -6551, 1] | -| TriAccRev | [1, -1, 0, 5, -1, 0, -93, 22, -1, 0, 5993, -1443, 75, -1, 0, -1272089, 306668, -16094, 236, -1, 0, 857402029, -206708261, 10852690, -159850, 721, -1, 0, -1792650585525, 432184866882, -22691017255, 334257820, -1510403, 2178, -1, 0, 11464255554367057, -2763883774564039, 145112312902789, -2137630911395, 9659553491, -13938645, 6551, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, -6, 1, -23, 5, 1, -76, 115, 1, -237, 1518, -93, 1, -722, 16330, -7436] | - -EulerianBInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 6, 116, 7512, 1595088, 1075123552, 2247862240064, 14375398946243968] | -| OddSum | [0, -1, -6, -116, -7512, -1595088, -1075123552, -2247862240064, -14375398946243968] | -| AltSum | [1, 2, 12, 232, 15024, 3190176, 2150247104, 4495724480128, 28750797892487936] | -| AccSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | -| AccRevSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | -| RevAccSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | -| DiagSum | [1, 1, 0, -5, -17, 40, 1189, 8173] | - -EulerianBInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 30, 10695, 4494102756, 1254352845580753602413730, 383972367241177062317562280537759735980, 715883465950829482992912712277956264095039430421605464700575, 4235300274399047501873201769211652027955886145279771055932028482670928385151774050533992] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 6, 115, 7436, 1578757, 1064110290, 2224835452407, 14228139328931096] | -| Middle | [1, 1, -6, -23, 1518, 16330, -11012540, -335768223, 2147290464886] | -| Central | [1, -6, 1518, -11012540, 2147290464886] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | -| PosHalf | [1, 1, -3, 53, -3399, 721257, -486128971, 1016394955037, -6499990316797327] | -| NegHalf | [1, -3, 21, -423, 27561, -5855115, 3946559037, -8251448061231, 52769185205128017] | -| Bin | [1, 0, -6, 184, -14946, 3556192, -2481692012, 5022968503216, -29005196474209354] | -| InvBin | [1, -2, 18, -508, 45150, -12557980, 10728141732, -27736560614136, 215138109927100854] | -| TransSqrs | [0, -1, 14, -400, 34960, -9381888, 7646219712, -18752310972160, 137610408557102336] | -| TransNat0 | [0, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | -| TransNat1 | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | - -EulerianBInvRev Diagonals as rows +EulerianB:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -6, 5], [1, -23, 115, -93], [1, -76, 1518, -7436, 5993], [1, -237, 16330, -322762, 1578757, -1272089], [1, -722, 160571, -11012540, 217560951, -1064110290, 857402029], [1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525], [1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057]] | +| RevTabl | [[1], [-1, 1], [5, -6, 1], [-93, 115, -23, 1], [5993, -7436, 1518, -76, 1], [-1272089, 1578757, -322762, 16330, -237, 1], [857402029, -1064110290, 217560951, -11012540, 160571, -722, 1], [-1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1], [11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -6], [1, -23, 5], [1, -76, 115], [1, -237, 1518, -93], [1, -722, 16330, -7436], [1, -2179, 160571, -322762, 5993]] | +| AccTabl | [[1], [1, 0], [1, -5, 0], [1, -22, 93, 0], [1, -75, 1443, -5993, 0], [1, -236, 16094, -306668, 1272089, 0], [1, -721, 159850, -10852690, 206708261, -857402029, 0], [1, -2178, 1510403, -334257820, 22691017255, -432184866882, 1792650585525, 0], [1, -6551, 13938645, -9659553491, 2137630911395, -145112312902789, 2763883774564039, -11464255554367057, 0]] | +| RevAccTabl | [[1], [0, 1], [0, -5, 1], [0, 93, -22, 1], [0, -5993, 1443, -75, 1], [0, 1272089, -306668, 16094, -236, 1], [0, -857402029, 206708261, -10852690, 159850, -721, 1], [0, 1792650585525, -432184866882, 22691017255, -334257820, 1510403, -2178, 1], [0, -11464255554367057, 2763883774564039, -145112312902789, 2137630911395, -9659553491, 13938645, -6551, 1]] | +| AccRevTabl | [[1], [-1, 0], [5, -1, 0], [-93, 22, -1, 0], [5993, -1443, 75, -1, 0], [-1272089, 306668, -16094, 236, -1, 0], [857402029, -206708261, 10852690, -159850, 721, -1, 0], [-1792650585525, 432184866882, -22691017255, 334257820, -1510403, 2178, -1, 0], [11464255554367057, -2763883774564039, 145112312902789, -2137630911395, 9659553491, -13938645, 6551, -1, 0]] | + +EulerianB:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 6, 116, 7512, 1595088, 1075123552, 2247862240064, 14375398946243968] | +| OddSum | [0, -1, -6, -116, -7512, -1595088, -1075123552, -2247862240064, -14375398946243968] | +| AltSum | [1, 2, 12, 232, 15024, 3190176, 2150247104, 4495724480128, 28750797892487936] | +| AccSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| AccRevSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | +| AntiDiagSum | [1, 1, 0, -5, -17, 40, 1189, 8173, -158376] | + +EulerianB:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 30, 10695, 4494102756, 1254352845580753602413730, 383972367241177062317562280537759735980, 715883465950829482992912712277956264095039430421605464700575, 4235300274399047501873201769211652027955886145279771055932028482670928385151774050533992] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 6, 115, 7436, 1578757, 1064110290, 2224835452407, 14228139328931096] | +| ColMiddle | [1, 1, -6, -23, 1518, 16330, -11012540, -335768223, 2147290464886] | +| ColECenter | [1, -6, 1518, -11012540, 2147290464886] | +| ColOCenter | [1, -23, 16330, -335768223] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | +| TransSqrs | [0, -1, 14, -400, 34960, -9381888, 7646219712, -18752310972160, 137610408557102336] | +| TransNat0 | [0, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | +| TransNat1 | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | + +EulerianB:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057]| @@ -391,7 +375,7 @@ EulerianBInvRev Diagonals as rows | DiagRow7 | [1, -6552, 127141156, -7446608913000, 1313126451395556490, -696066950858505682059208, 1108172771490732215668663853876, -5296141644441170086929626440059219768, 75960197810240124116687805473107599893593955]| | DiagRow8 | [1, -19673, 1152338433, -203202094563809, 107714121748314607559, -171486171954267914996574751, 819560884455816722716701613957119, -11754596285418648169111713017129203693711, 505872176876172128739413633340997793945445390118]| -EulerianBInvRev Diagonals as columns +EulerianB:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ EulerianBInvRev Diagonals as columns | DiagCol7 | [-1792650585525, -14228139328931096, -56434463826320585284, -167312402773377971746920, -430241010049958628814059458, -1027422064996274234324991004232, -2358032188165387759999729133419316, -5296141644441170086929626440059219768, -11754596285418648169111713017129203693711] | | DiagCol8 | [11464255554367057, 276025608122908733321, 3305339964838291288943901, 29503676524653884693236314085, 228063754011124000036378055403359, 1635706386940760528736923482495252231, 11269427287836849598854819603136973269515, 75960197810240124116687805473107599893593955, 505872176876172128739413633340997793945445390118] | -EulerianBInvRev Polynomial values as rows +EulerianB:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ EulerianBInvRev Polynomial values as rows | PolyRow7 | [1, 0, -101260109984153, -2407300630438016, -20717781080093307, -106694911324511744, -401532858032073845, -1222165105408418688, -3191044729160380631] | | PolyRow8 | [1, 0, 1295145618525878177, 46185090060715280128, 529972630365069384801, 3411645689491842262016, 15407158532713346931265, 54711438139920644481792, 163257480439413162352193] | -EulerianBInvRev Polynomial values as columns +EulerianB:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ EulerianBInvRev Polynomial values as columns | PolyCol7 | [1, -6, 204, -26424, 11912496, -17699313120, 83506584070848, -1222165105408418688, 54711438139920644481792] | | PolyCol8 | [1, -7, 273, -40439, 20836641, -35381434599, 190779533013041, -3191044729160380631, 163257480439413162352193] | +# EulerianB:Rev:Inv +[] + +EulerianB:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [5, -6, 1] | +| Row3 | [-93, 115, -23, 1] | +| Row4 | [5993, -7436, 1518, -76, 1] | +| Row5 | [-1272089, 1578757, -322762, 16330, -237, 1] | +| Row6 | [857402029, -1064110290, 217560951, -11012540, 160571, -722, 1] | +| Row7 | [-1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1] | +| Row8 | [11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1] | + +EulerianB:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [5, -6, 1], [-93, 115, -23, 1], [5993, -7436, 1518, -76, 1], [-1272089, 1578757, -322762, 16330, -237, 1], [857402029, -1064110290, 217560951, -11012540, 160571, -722, 1], [-1792650585525, 2224835452407, -454875884137, 23025275075, -335768223, 1512581, -2179, 1], [11464255554367057, -14228139328931096, 2908996087466828, -147249943814184, 2147290464886, -9673492136, 13945196, -6552, 1]] | +| RevTabl | [[1], [1, -1], [1, -6, 5], [1, -23, 115, -93], [1, -76, 1518, -7436, 5993], [1, -237, 16330, -322762, 1578757, -1272089], [1, -722, 160571, -11012540, 217560951, -1064110290, 857402029], [1, -2179, 1512581, -335768223, 23025275075, -454875884137, 2224835452407, -1792650585525], [1, -6552, 13945196, -9673492136, 2147290464886, -147249943814184, 2908996087466828, -14228139328931096, 11464255554367057]] | +| AntiDiag | [[1], [-1], [5, 1], [-93, -6], [5993, 115, 1], [-1272089, -7436, -23], [857402029, 1578757, 1518, 1], [-1792650585525, -1064110290, -322762, -76], [11464255554367057, 2224835452407, 217560951, 16330, 1]] | +| AccTabl | [[1], [-1, 0], [5, -1, 0], [-93, 22, -1, 0], [5993, -1443, 75, -1, 0], [-1272089, 306668, -16094, 236, -1, 0], [857402029, -206708261, 10852690, -159850, 721, -1, 0], [-1792650585525, 432184866882, -22691017255, 334257820, -1510403, 2178, -1, 0], [11464255554367057, -2763883774564039, 145112312902789, -2137630911395, 9659553491, -13938645, 6551, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 5], [0, -1, 22, -93], [0, -1, 75, -1443, 5993], [0, -1, 236, -16094, 306668, -1272089], [0, -1, 721, -159850, 10852690, -206708261, 857402029], [0, -1, 2178, -1510403, 334257820, -22691017255, 432184866882, -1792650585525], [0, -1, 6551, -13938645, 9659553491, -2137630911395, 145112312902789, -2763883774564039, 11464255554367057]] | +| AccRevTabl | [[1], [1, 0], [1, -5, 0], [1, -22, 93, 0], [1, -75, 1443, -5993, 0], [1, -236, 16094, -306668, 1272089, 0], [1, -721, 159850, -10852690, 206708261, -857402029, 0], [1, -2178, 1510403, -334257820, 22691017255, -432184866882, 1792650585525, 0], [1, -6551, 13938645, -9659553491, 2137630911395, -145112312902789, 2763883774564039, -11464255554367057, 0]] | + +EulerianB:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 6, -116, 7512, -1595088, 1075123552, -2247862240064, 14375398946243968] | +| OddSum | [0, 1, -6, 116, -7512, 1595088, -1075123552, 2247862240064, -14375398946243968] | +| AltSum | [1, -2, 12, -232, 15024, -3190176, 2150247104, -4495724480128, 28750797892487936] | +| AccSum | [1, -1, 4, -72, 4624, -981280, 661387328, -1382823986304, 8843356107415808] | +| AccRevSum | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| AntiDiagSum | [1, -1, 6, -99, 6109, -1279548, 858982305, -1793715018653, 11466480607396746] | + +EulerianB:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 30, 10695, 4494102756, 1254352845580753602413730, 383972367241177062317562280537759735980, 715883465950829482992912712277956264095039430421605464700575, 4235300274399047501873201769211652027955886145279771055932028482670928385151774050533992] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 6, 115, 7436, 1578757, 1064110290, 2224835452407, 14228139328931096] | +| ColMiddle | [1, -1, -6, 115, 1518, -322762, -11012540, 23025275075, 2147290464886] | +| ColECenter | [1, -6, 1518, -11012540, 2147290464886] | +| ColOCenter | [-1, 115, -322762, 23025275075] | +| ColLeft | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, -2, 32, -2032, 430912, -290428224, 607224836096, -3883289161550592] | +| TransNat0 | [0, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| TransNat1 | [1, 1, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | + +EulerianB:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -6, -23, -76, -237, -722, -2179, -6552, -19673]| +| DiagRow2 | [5, 115, 1518, 16330, 160571, 1512581, 13945196, 127141156, 1152338433]| +| DiagRow3 | [-93, -7436, -322762, -11012540, -335768223, -9673492136, -270538484020, -7446608913000, -203202094563809]| +| DiagRow4 | [5993, 1578757, 217560951, 23025275075, 2147290464886, 187665608020478, 15845312257310658, 1313126451395556490, 107714121748314607559]| +| DiagRow5 | [-1272089, -1064110290, -454875884137, -147249943814184, -41657391444153086, -10991486289326969076, -2794144125548953124662, -696066950858505682059208, -171486171954267914996574751]| +| DiagRow6 | [857402029, 2224835452407, 2908996087466828, 2856645864675796564, 2439853795947184617546, 1938226040820344383545710, 1481130305191078491484585996, 1108172771490732215668663853876, 819560884455816722716701613957119]| +| DiagRow7 | [-1792650585525, -14228139328931096, -56434463826320585284, -167312402773377971746920, -430241010049958628814059458, -1027422064996274234324991004232, -2358032188165387759999729133419316, -5296141644441170086929626440059219768, -11754596285418648169111713017129203693711]| +| DiagRow8 | [11464255554367057, 276025608122908733321, 3305339964838291288943901, 29503676524653884693236314085, 228063754011124000036378055403359, 1635706386940760528736923482495252231, 11269427287836849598854819603136973269515, 75960197810240124116687805473107599893593955, 505872176876172128739413633340997793945445390118]| + +EulerianB:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | +| DiagCol1 | [1, -6, 115, -7436, 1578757, -1064110290, 2224835452407, -14228139328931096, 276025608122908733321] | +| DiagCol2 | [1, -23, 1518, -322762, 217560951, -454875884137, 2908996087466828, -56434463826320585284, 3305339964838291288943901] | +| DiagCol3 | [1, -76, 16330, -11012540, 23025275075, -147249943814184, 2856645864675796564, -167312402773377971746920, 29503676524653884693236314085] | +| DiagCol4 | [1, -237, 160571, -335768223, 2147290464886, -41657391444153086, 2439853795947184617546, -430241010049958628814059458, 228063754011124000036378055403359] | +| DiagCol5 | [1, -722, 1512581, -9673492136, 187665608020478, -10991486289326969076, 1938226040820344383545710, -1027422064996274234324991004232, 1635706386940760528736923482495252231] | +| DiagCol6 | [1, -2179, 13945196, -270538484020, 15845312257310658, -2794144125548953124662, 1481130305191078491484585996, -2358032188165387759999729133419316, 11269427287836849598854819603136973269515] | +| DiagCol7 | [1, -6552, 127141156, -7446608913000, 1313126451395556490, -696066950858505682059208, 1108172771490732215668663853876, -5296141644441170086929626440059219768, 75960197810240124116687805473107599893593955] | +| DiagCol8 | [1, -19673, 1152338433, -203202094563809, 107714121748314607559, -171486171954267914996574751, 819560884455816722716701613957119, -11754596285418648169111713017129203693711, 505872176876172128739413633340997793945445390118] | + +EulerianB:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [5, 0, -3, -4, -3, 0, 5, 12, 21] | +| PolyRow3 | [-93, 0, 53, 72, 63, 32, -15, -72, -133] | +| PolyRow4 | [5993, 0, -3399, -4624, -4071, -2112, 905, 4656, 8841] | +| PolyRow5 | [-1272089, 0, 721257, 981280, 864219, 448896, -191075, -987168, -1875825] | +| PolyRow6 | [857402029, 0, -486128971, -661387328, -582495531, -302576896, 128758285, 665329344, 1264302557] | +| PolyRow7 | [-1792650585525, 0, 1016394955037, 1382823986304, 1217877991767, 632626664960, -269205454455, -1391064478848, -2643394214461] | +| PolyRow8 | [11464255554367057, 0, -6499990316797327, -8843356107415808, -7788503050691919, -4045737620558848, 1721607027968785, 8896055081595648, 16904881967491217] | + +EulerianB:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 5, -93, 5993, -1272089, 857402029, -1792650585525, 11464255554367057] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, -3, 53, -3399, 721257, -486128971, 1016394955037, -6499990316797327] | +| PolyCol3 | [1, 2, -4, 72, -4624, 981280, -661387328, 1382823986304, -8843356107415808] | +| PolyCol4 | [1, 3, -3, 63, -4071, 864219, -582495531, 1217877991767, -7788503050691919] | +| PolyCol5 | [1, 4, 0, 32, -2112, 448896, -302576896, 632626664960, -4045737620558848] | +| PolyCol6 | [1, 5, 5, -15, 905, -191075, 128758285, -269205454455, 1721607027968785] | +| PolyCol7 | [1, 6, 12, -72, 4656, -987168, 665329344, -1391064478848, 8896055081595648] | +| PolyCol8 | [1, 7, 21, -133, 8841, -1875825, 1264302557, -2643394214461, 16904881967491217] | + diff --git a/data/md/FallingFact.md b/data/md/FallingFact.md index c5e8f3c..e3e6bf6 100644 --- a/data/md/FallingFact.md +++ b/data/md/FallingFact.md @@ -14,45 +14,41 @@ FallingFact Triangle view | Row7 | [1, 7, 42, 210, 840, 2520, 5040, 5040] | | Row8 | [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -FallingFact Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -| TriRev | [1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, 120, 120, 60, 20, 5, 1, 720, 720, 360, 120, 30, 6, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601] | -| TriRevAcc | [1, 2, 1, 5, 3, 1, 16, 10, 4, 1, 65, 41, 17, 5, 1, 326, 206, 86, 26, 6, 1, 1957, 1237, 517, 157, 37, 7, 1, 13700, 8660, 3620, 1100, 260, 50, 8, 1, 109601, 69281, 28961, 8801, 2081, 401, 65, 9, 1] | -| TriAccRev | [1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 6, 1, 5, 12, 6, 1, 6, 20, 24] | +FallingFact Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 2], [1, 3, 6, 6], [1, 4, 12, 24, 24], [1, 5, 20, 60, 120, 120], [1, 6, 30, 120, 360, 720, 720], [1, 7, 42, 210, 840, 2520, 5040, 5040], [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320]] | +| RevTabl | [[1], [1, 1], [2, 2, 1], [6, 6, 3, 1], [24, 24, 12, 4, 1], [120, 120, 60, 20, 5, 1], [720, 720, 360, 120, 30, 6, 1], [5040, 5040, 2520, 840, 210, 42, 7, 1], [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 2], [1, 4, 6], [1, 5, 12, 6], [1, 6, 20, 24], [1, 7, 30, 60, 24]] | +| AccTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 10, 16], [1, 5, 17, 41, 65], [1, 6, 26, 86, 206, 326], [1, 7, 37, 157, 517, 1237, 1957], [1, 8, 50, 260, 1100, 3620, 8660, 13700], [1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601]] | +| RevAccTabl | [[1], [2, 1], [5, 3, 1], [16, 10, 4, 1], [65, 41, 17, 5, 1], [326, 206, 86, 26, 6, 1], [1957, 1237, 517, 157, 37, 7, 1], [13700, 8660, 3620, 1100, 260, 50, 8, 1], [109601, 69281, 28961, 8801, 2081, 401, 65, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 4, 5], [6, 12, 15, 16], [24, 48, 60, 64, 65], [120, 240, 300, 320, 325, 326], [720, 1440, 1800, 1920, 1950, 1956, 1957], [5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700], [40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601]] | FallingFact Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 1, 3, 7, 37, 141, 1111, 5923, 62217] | -| OddSum | [0, 1, 2, 9, 28, 185, 846, 7777, 47384] | -| AltSum | [1, 0, 1, -2, 9, -44, 265, -1854, 14833] | -| AccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| AccRevSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| RevAccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| DiagSum | [1, 1, 2, 3, 6, 11, 24, 51] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 1, 3, 7, 37, 141, 1111, 5923, 62217] | +| OddSum | [0, 1, 2, 9, 28, 185, 846, 7777, 47384] | +| AltSum | [1, 0, 1, -2, 9, -44, 265, -1854, 14833] | +| AccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | +| AccRevSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | +| AntiDiagSum | [1, 1, 2, 3, 6, 11, 24, 51, 122] | FallingFact Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 1, 2, 3, 12, 20, 120, 210, 1680] | -| Central | [1, 2, 12, 120, 1680] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 10, 38, 168, 872, 5296, 37200, 297856] | -| NegHalf | [1, -1, 2, -2, 8, 8, 112, 656, 5504] | -| Bin | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| InvBin | [1, 0, -1, -4, -15, -56, -185, -204, 6209] | -| TransSqrs | [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056] | -| TransNat0 | [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208] | -| TransNat1 | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 1, 2, 3, 12, 20, 120, 210, 1680] | +| ColECenter | [1, 2, 12, 120, 1680] | +| ColOCenter | [1, 3, 20, 210] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056] | +| TransNat0 | [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208] | +| TransNat1 | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | FallingFact Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ FallingFact Polynomial values as columns | PolyCol7 | [1, 8, 113, 2374, 66473, 2326556, 97715353, 4788052298, 268130928689] | | PolyCol8 | [1, 9, 145, 3481, 111393, 4455721, 213874609, 11976978105, 766526598721] | -# FallingFactRev -True +# FallingFact:Rev +[] -FallingFactRev Triangle view +FallingFact:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ FallingFactRev Triangle view | Row7 | [5040, 5040, 2520, 840, 210, 42, 7, 1] | | Row8 | [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -FallingFactRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, 120, 120, 60, 20, 5, 1, 720, 720, 360, 120, 30, 6, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -| TriAcc | [1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601] | -| TriRevAcc | [1, 2, 1, 5, 4, 2, 16, 15, 12, 6, 65, 64, 60, 48, 24, 326, 325, 320, 300, 240, 120, 1957, 1956, 1950, 1920, 1800, 1440, 720, 13700, 13699, 13692, 13650, 13440, 12600, 10080, 5040, 109601, 109600, 109592, 109536, 109200, 107520, 100800, 80640, 40320] | -| TriAccRev | [1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601] | -| TriDiag | [1, 1, 2, 1, 6, 2, 24, 6, 1, 120, 24, 3, 720, 120, 12, 1, 5040, 720, 60, 4] | - -FallingFactRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 1, 3, 9, 37, 185, 1111, 7777, 62217] | -| OddSum | [0, 1, 2, 7, 28, 141, 846, 5923, 47384] | -| AltSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| AccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| AccRevSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| RevAccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| DiagSum | [1, 1, 3, 8, 31, 147, 853, 5824] | - -FallingFactRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 1, 2, 6, 12, 60, 120, 840, 1680] | -| Central | [1, 2, 12, 120, 1680] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969] | -| NegHalf | [1, -1, 5, -29, 233, -2329, 27949, -391285, 6260561] | -| Bin | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| InvBin | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | -| TransSqrs | [0, 1, 6, 27, 124, 645, 3906, 27391, 219192] | -| TransNat0 | [0, 1, 4, 15, 64, 325, 1956, 13699, 109600] | -| TransNat1 | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | - -FallingFactRev Diagonals as rows +FallingFact:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 2, 1], [6, 6, 3, 1], [24, 24, 12, 4, 1], [120, 120, 60, 20, 5, 1], [720, 720, 360, 120, 30, 6, 1], [5040, 5040, 2520, 840, 210, 42, 7, 1], [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 2], [1, 3, 6, 6], [1, 4, 12, 24, 24], [1, 5, 20, 60, 120, 120], [1, 6, 30, 120, 360, 720, 720], [1, 7, 42, 210, 840, 2520, 5040, 5040], [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 2], [24, 6, 1], [120, 24, 3], [720, 120, 12, 1], [5040, 720, 60, 4], [40320, 5040, 360, 20, 1]] | +| AccTabl | [[1], [1, 2], [2, 4, 5], [6, 12, 15, 16], [24, 48, 60, 64, 65], [120, 240, 300, 320, 325, 326], [720, 1440, 1800, 1920, 1950, 1956, 1957], [5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700], [40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 2], [16, 15, 12, 6], [65, 64, 60, 48, 24], [326, 325, 320, 300, 240, 120], [1957, 1956, 1950, 1920, 1800, 1440, 720], [13700, 13699, 13692, 13650, 13440, 12600, 10080, 5040], [109601, 109600, 109592, 109536, 109200, 107520, 100800, 80640, 40320]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 10, 16], [1, 5, 17, 41, 65], [1, 6, 26, 86, 206, 326], [1, 7, 37, 157, 517, 1237, 1957], [1, 8, 50, 260, 1100, 3620, 8660, 13700], [1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601]] | + +FallingFact:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 1, 3, 9, 37, 185, 1111, 7777, 62217] | +| OddSum | [0, 1, 2, 7, 28, 141, 846, 5923, 47384] | +| AltSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| AccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | +| AccRevSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | +| AntiDiagSum | [1, 1, 3, 8, 31, 147, 853, 5824, 45741] | + +FallingFact:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 1, 2, 6, 12, 60, 120, 840, 1680] | +| ColECenter | [1, 2, 12, 120, 1680] | +| ColOCenter | [1, 6, 60, 840] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 27, 124, 645, 3906, 27391, 219192] | +| TransNat0 | [0, 1, 4, 15, 64, 325, 1956, 13699, 109600] | +| TransNat1 | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | + +FallingFact:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ FallingFactRev Diagonals as rows | DiagRow7 | [5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400]| | DiagRow8 | [40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200, 518918400]| -FallingFactRev Diagonals as columns +FallingFact:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -188,7 +180,7 @@ FallingFactRev Diagonals as columns | DiagCol7 | [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200] | | DiagCol8 | [1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400] | -FallingFactRev Polynomial values as rows +FallingFact:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ FallingFactRev Polynomial values as rows | PolyRow7 | [5040, 13700, 37200, 100026, 261104, 648240, 1512720, 3309110, 6805296] | | PolyRow8 | [40320, 109601, 297856, 806769, 2154368, 5576545, 13781376, 32237681, 71219584] | -FallingFactRev Polynomial values as columns +FallingFact:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -214,3 +206,107 @@ FallingFactRev Polynomial values as columns | PolyCol7 | [1, 8, 65, 538, 4553, 39572, 355081, 3309110, 32237681] | | PolyCol8 | [1, 9, 82, 758, 7128, 68408, 672592, 6805296, 71219584] | +# FallingFact:Rev:Inv +[] + +FallingFact:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [0, -2, 1] | +| Row3 | [0, 0, -3, 1] | +| Row4 | [0, 0, 0, -4, 1] | +| Row5 | [0, 0, 0, 0, -5, 1] | +| Row6 | [0, 0, 0, 0, 0, -6, 1] | +| Row7 | [0, 0, 0, 0, 0, 0, -7, 1] | +| Row8 | [0, 0, 0, 0, 0, 0, 0, -8, 1] | + +FallingFact:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -2, 1], [0, 0, -3, 1], [0, 0, 0, -4, 1], [0, 0, 0, 0, -5, 1], [0, 0, 0, 0, 0, -6, 1], [0, 0, 0, 0, 0, 0, -7, 1], [0, 0, 0, 0, 0, 0, 0, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 0], [1, -3, 0, 0], [1, -4, 0, 0, 0], [1, -5, 0, 0, 0, 0], [1, -6, 0, 0, 0, 0, 0], [1, -7, 0, 0, 0, 0, 0, 0], [1, -8, 0, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [0, 1], [0, -2], [0, 0, 1], [0, 0, -3], [0, 0, 0, 1], [0, 0, 0, -4], [0, 0, 0, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -2, -1], [0, 0, -3, -2], [0, 0, 0, -4, -3], [0, 0, 0, 0, -5, -4], [0, 0, 0, 0, 0, -6, -5], [0, 0, 0, 0, 0, 0, -7, -6], [0, 0, 0, 0, 0, 0, 0, -8, -7]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 0], [-2, -3, 0, 0], [-3, -4, 0, 0, 0], [-4, -5, 0, 0, 0, 0], [-5, -6, 0, 0, 0, 0, 0], [-6, -7, 0, 0, 0, 0, 0, 0], [-7, -8, 0, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, -1, -1], [1, -2, -2, -2], [1, -3, -3, -3, -3], [1, -4, -4, -4, -4, -4], [1, -5, -5, -5, -5, -5, -5], [1, -6, -6, -6, -6, -6, -6, -6], [1, -7, -7, -7, -7, -7, -7, -7, -7]] | + +FallingFact:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| EvenSum | [1, -1, 1, -3, 1, -5, 1, -7, 1] | +| OddSum | [0, 1, -2, 1, -4, 1, -6, 1, -8] | +| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| AccSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | +| AccRevSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | +| AntiDiagSum | [1, -1, 1, -2, 1, -3, 1, -4, 1] | + +FallingFact:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColMiddle | [1, -1, -2, 0, 0, 0, 0, 0, 0] | +| ColECenter | [1, -2, 0, 0, 0] | +| ColOCenter | [-1, 0, 0, 0] | +| ColLeft | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -3, -20, -55, -114, -203, -328] | +| TransNat0 | [0, 1, 0, -3, -8, -15, -24, -35, -48] | +| TransNat1 | [1, 1, -1, -5, -11, -19, -29, -41, -55] | + +FallingFact:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -2, -3, -4, -5, -6, -7, -8, -9]| +| DiagRow2 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow3 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow4 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow5 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow6 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| + +FallingFact:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, -2, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol2 | [1, -3, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol3 | [1, -4, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol4 | [1, -5, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol5 | [1, -6, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol6 | [1, -7, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol7 | [1, -8, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol8 | [1, -9, 0, 0, 0, 0, 0, 0, 0] | + +FallingFact:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [0, -1, 0, 3, 8, 15, 24, 35, 48] | +| PolyRow3 | [0, -2, -4, 0, 16, 50, 108, 196, 320] | +| PolyRow4 | [0, -3, -16, -27, 0, 125, 432, 1029, 2048] | +| PolyRow5 | [0, -4, -48, -162, -256, 0, 1296, 4802, 12288] | +| PolyRow6 | [0, -5, -128, -729, -2048, -3125, 0, 16807, 65536] | +| PolyRow7 | [0, -6, -320, -2916, -12288, -31250, -46656, 0, 262144] | +| PolyRow8 | [0, -7, -768, -10935, -65536, -234375, -559872, -823543, 0] | + +FallingFact:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| PolyCol2 | [1, 1, 0, -4, -16, -48, -128, -320, -768] | +| PolyCol3 | [1, 2, 3, 0, -27, -162, -729, -2916, -10935] | +| PolyCol4 | [1, 3, 8, 16, 0, -256, -2048, -12288, -65536] | +| PolyCol5 | [1, 4, 15, 50, 125, 0, -3125, -31250, -234375] | +| PolyCol6 | [1, 5, 24, 108, 432, 1296, 0, -46656, -559872] | +| PolyCol7 | [1, 6, 35, 196, 1029, 4802, 16807, 0, -823543] | +| PolyCol8 | [1, 7, 48, 320, 2048, 12288, 65536, 262144, 0] | + diff --git a/data/md/Fibonacci.md b/data/md/Fibonacci.md index 1f2fb44..2636397 100644 --- a/data/md/Fibonacci.md +++ b/data/md/Fibonacci.md @@ -1,5 +1,5 @@ # Fibonacci -['A105809', 'A228074', 'A354267'] +['A354267', 'A105809', 'A228074'] Fibonacci Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Fibonacci Triangle view | Row7 | [8, 13, 20, 26, 25, 16, 6, 1] | | Row8 | [13, 21, 33, 46, 51, 41, 22, 7, 1] | -Fibonacci Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1] | -| TriRev | [1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 2, 1, 4, 7, 7, 5, 3, 1, 5, 11, 14, 12, 8, 5, 1, 6, 16, 25, 26, 20, 13, 8, 1, 7, 22, 41, 51, 46, 33, 21, 13] | -| TriAcc | [1, 0, 1, 1, 2, 3, 1, 3, 5, 6, 2, 5, 9, 12, 13, 3, 8, 15, 22, 26, 27, 5, 13, 25, 39, 50, 55, 56, 8, 21, 41, 67, 92, 108, 114, 115, 13, 34, 67, 113, 164, 205, 227, 234, 235] | -| TriRevAcc | [1, 1, 0, 3, 2, 1, 6, 5, 3, 1, 13, 12, 9, 5, 2, 27, 26, 22, 15, 8, 3, 56, 55, 50, 39, 25, 13, 5, 115, 114, 108, 92, 67, 41, 21, 8, 235, 234, 227, 205, 164, 113, 67, 34, 13] | -| TriAccRev | [1, 1, 1, 1, 2, 3, 1, 3, 5, 6, 1, 4, 8, 11, 13, 1, 5, 12, 19, 24, 27, 1, 6, 17, 31, 43, 51, 56, 1, 7, 23, 48, 74, 94, 107, 115, 1, 8, 30, 71, 122, 168, 201, 222, 235] | -| TriDiag | [1, 0, 1, 1, 1, 1, 2, 2, 1, 3, 3, 2, 5, 5, 4, 1, 8, 8, 7, 3] | +Fibonacci Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 1, 1], [1, 2, 2, 1], [2, 3, 4, 3, 1], [3, 5, 7, 7, 4, 1], [5, 8, 12, 14, 11, 5, 1], [8, 13, 20, 26, 25, 16, 6, 1], [13, 21, 33, 46, 51, 41, 22, 7, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 1], [1, 2, 2, 1], [1, 3, 4, 3, 2], [1, 4, 7, 7, 5, 3], [1, 5, 11, 14, 12, 8, 5], [1, 6, 16, 25, 26, 20, 13, 8], [1, 7, 22, 41, 51, 46, 33, 21, 13]] | +| AntiDiag | [[1], [0], [1, 1], [1, 1], [2, 2, 1], [3, 3, 2], [5, 5, 4, 1], [8, 8, 7, 3], [13, 13, 12, 7, 1]] | +| AccTabl | [[1], [0, 1], [1, 2, 3], [1, 3, 5, 6], [2, 5, 9, 12, 13], [3, 8, 15, 22, 26, 27], [5, 13, 25, 39, 50, 55, 56], [8, 21, 41, 67, 92, 108, 114, 115], [13, 34, 67, 113, 164, 205, 227, 234, 235]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 1], [6, 5, 3, 1], [13, 12, 9, 5, 2], [27, 26, 22, 15, 8, 3], [56, 55, 50, 39, 25, 13, 5], [115, 114, 108, 92, 67, 41, 21, 8], [235, 234, 227, 205, 164, 113, 67, 34, 13]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 3], [1, 3, 5, 6], [1, 4, 8, 11, 13], [1, 5, 12, 19, 24, 27], [1, 6, 17, 31, 43, 51, 56], [1, 7, 23, 48, 74, 94, 107, 115], [1, 8, 30, 71, 122, 168, 201, 222, 235]] | Fibonacci Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 6, 13, 27, 56, 115, 235] | -| EvenSum | [1, 0, 2, 3, 7, 14, 29, 59, 120] | -| OddSum | [0, 1, 1, 3, 6, 13, 27, 56, 115] | -| AltSum | [1, -1, 1, 0, 1, 1, 2, 3, 5] | -| AccSum | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | -| AccRevSum | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | -| RevAccSum | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | -| DiagSum | [1, 0, 2, 2, 5, 8, 15, 26] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 6, 13, 27, 56, 115, 235] | +| EvenSum | [1, 0, 2, 3, 7, 14, 29, 59, 120] | +| OddSum | [0, 1, 1, 3, 6, 13, 27, 56, 115] | +| AltSum | [1, -1, 1, 0, 1, 1, 2, 3, 5] | +| AccSum | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | +| AccRevSum | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | +| AntiDiagSum | [1, 0, 2, 2, 5, 8, 15, 26, 46] | Fibonacci Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 12, 420, 9240, 15600, 96282186] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 4, 7, 14, 26, 51] | -| Middle | [1, 0, 1, 2, 4, 7, 14, 26, 51] | -| Central | [1, 1, 4, 14, 51] | -| LeftSide | [1, 0, 1, 1, 2, 3, 5, 8, 13] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 7, 21, 79, 269, 935, 3189, 10847] | -| NegHalf | [1, 1, 3, -3, 19, -51, 179, -563, 1843] | -| Bin | [1, 1, 4, 14, 51, 189, 709, 2683, 10220] | -| InvBin | [1, 1, 0, 0, 3, 3, -7, -7, 28] | -| TransSqrs | [0, 1, 5, 19, 62, 185, 519, 1392, 3607] | -| TransNat0 | [0, 1, 3, 9, 24, 61, 149, 354, 823] | -| TransNat1 | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 12, 420, 9240, 15600, 96282186] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 4, 7, 14, 26, 51] | +| ColMiddle | [1, 0, 1, 2, 4, 7, 14, 26, 51] | +| ColECenter | [1, 1, 4, 14, 51] | +| ColOCenter | [0, 2, 7, 26] | +| ColLeft | [1, 0, 1, 1, 2, 3, 5, 8, 13] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 19, 62, 185, 519, 1392, 3607] | +| TransNat0 | [0, 1, 3, 9, 24, 61, 149, 354, 823] | +| TransNat1 | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | Fibonacci Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Fibonacci Polynomial values as columns | PolyCol7 | [1, 7, 57, 456, 3649, 29193, 233546, 1868371, 14946973] | | PolyCol8 | [1, 8, 73, 657, 5914, 53227, 479045, 4311408, 38802677] | -# FibonacciRev -True +# Fibonacci:Rev +[] -FibonacciRev Triangle view +Fibonacci:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ FibonacciRev Triangle view | Row7 | [1, 6, 16, 25, 26, 20, 13, 8] | | Row8 | [1, 7, 22, 41, 51, 46, 33, 21, 13] | -FibonacciRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 2, 1, 4, 7, 7, 5, 3, 1, 5, 11, 14, 12, 8, 5, 1, 6, 16, 25, 26, 20, 13, 8, 1, 7, 22, 41, 51, 46, 33, 21, 13] | -| TriRev | [1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1] | -| TriAcc | [1, 1, 1, 1, 2, 3, 1, 3, 5, 6, 1, 4, 8, 11, 13, 1, 5, 12, 19, 24, 27, 1, 6, 17, 31, 43, 51, 56, 1, 7, 23, 48, 74, 94, 107, 115, 1, 8, 30, 71, 122, 168, 201, 222, 235] | -| TriRevAcc | [1, 1, 1, 3, 2, 1, 6, 5, 3, 1, 13, 11, 8, 4, 1, 27, 24, 19, 12, 5, 1, 56, 51, 43, 31, 17, 6, 1, 115, 107, 94, 74, 48, 23, 7, 1, 235, 222, 201, 168, 122, 71, 30, 8, 1] | -| TriAccRev | [1, 0, 1, 1, 2, 3, 1, 3, 5, 6, 2, 5, 9, 12, 13, 3, 8, 15, 22, 26, 27, 5, 13, 25, 39, 50, 55, 56, 8, 21, 41, 67, 92, 108, 114, 115, 13, 34, 67, 113, 164, 205, 227, 234, 235] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 4, 1, 1, 5, 7, 3] | - -FibonacciRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 6, 13, 27, 56, 115, 235] | -| EvenSum | [1, 1, 2, 3, 7, 13, 29, 56, 120] | -| OddSum | [0, 0, 1, 3, 6, 14, 27, 59, 115] | -| AltSum | [1, 1, 1, 0, 1, -1, 2, -3, 5] | -| AccSum | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | -| AccRevSum | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | -| RevAccSum | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | -| DiagSum | [1, 1, 1, 2, 4, 6, 10, 16] | - -FibonacciRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 12, 420, 9240, 15600, 96282186] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 4, 7, 14, 26, 51] | -| Middle | [1, 1, 1, 2, 4, 7, 14, 25, 51] | -| Central | [1, 1, 4, 14, 51] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 1, 2, 3, 5, 8, 13] | -| PosHalf | [1, 2, 7, 21, 64, 193, 581, 1746, 5243] | -| NegHalf | [1, -2, 3, -3, 4, -3, 5, -2, 7] | -| Bin | [1, 1, 4, 14, 51, 189, 709, 2683, 10220] | -| InvBin | [1, -1, 0, 0, 3, -3, -7, 7, 28] | -| TransSqrs | [0, 0, 5, 19, 78, 250, 747, 2071, 5479] | -| TransNat0 | [0, 0, 3, 9, 28, 74, 187, 451, 1057] | -| TransNat1 | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | - -FibonacciRev Diagonals as rows +Fibonacci:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 1], [1, 2, 2, 1], [1, 3, 4, 3, 2], [1, 4, 7, 7, 5, 3], [1, 5, 11, 14, 12, 8, 5], [1, 6, 16, 25, 26, 20, 13, 8], [1, 7, 22, 41, 51, 46, 33, 21, 13]] | +| RevTabl | [[1], [0, 1], [1, 1, 1], [1, 2, 2, 1], [2, 3, 4, 3, 1], [3, 5, 7, 7, 4, 1], [5, 8, 12, 14, 11, 5, 1], [8, 13, 20, 26, 25, 16, 6, 1], [13, 21, 33, 46, 51, 41, 22, 7, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 2, 1], [1, 3, 2], [1, 4, 4, 1], [1, 5, 7, 3], [1, 6, 11, 7, 2]] | +| AccTabl | [[1], [1, 1], [1, 2, 3], [1, 3, 5, 6], [1, 4, 8, 11, 13], [1, 5, 12, 19, 24, 27], [1, 6, 17, 31, 43, 51, 56], [1, 7, 23, 48, 74, 94, 107, 115], [1, 8, 30, 71, 122, 168, 201, 222, 235]] | +| RevAccTabl | [[1], [1, 1], [3, 2, 1], [6, 5, 3, 1], [13, 11, 8, 4, 1], [27, 24, 19, 12, 5, 1], [56, 51, 43, 31, 17, 6, 1], [115, 107, 94, 74, 48, 23, 7, 1], [235, 222, 201, 168, 122, 71, 30, 8, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 2, 3], [1, 3, 5, 6], [2, 5, 9, 12, 13], [3, 8, 15, 22, 26, 27], [5, 13, 25, 39, 50, 55, 56], [8, 21, 41, 67, 92, 108, 114, 115], [13, 34, 67, 113, 164, 205, 227, 234, 235]] | + +Fibonacci:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 6, 13, 27, 56, 115, 235] | +| EvenSum | [1, 1, 2, 3, 7, 13, 29, 56, 120] | +| OddSum | [0, 0, 1, 3, 6, 14, 27, 59, 115] | +| AltSum | [1, 1, 1, 0, 1, -1, 2, -3, 5] | +| AccSum | [1, 2, 6, 15, 37, 88, 205, 469, 1058] | +| AccRevSum | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | +| AntiDiagSum | [1, 1, 1, 2, 4, 6, 10, 16, 27] | + +Fibonacci:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 12, 420, 9240, 15600, 96282186] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 4, 7, 14, 26, 51] | +| ColMiddle | [1, 1, 1, 2, 4, 7, 14, 25, 51] | +| ColECenter | [1, 1, 4, 14, 51] | +| ColOCenter | [1, 2, 7, 25] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 1, 2, 3, 5, 8, 13] | +| TransSqrs | [0, 0, 5, 19, 78, 250, 747, 2071, 5479] | +| TransNat0 | [0, 0, 3, 9, 28, 74, 187, 451, 1057] | +| TransNat1 | [1, 1, 6, 15, 41, 101, 243, 566, 1292] | + +Fibonacci:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 1, 2, 3, 5, 8, 13]| @@ -175,7 +167,7 @@ FibonacciRev Diagonals as rows | DiagRow7 | [1, 7, 29, 92, 247, 591, 1300, 2683, 5270]| | DiagRow8 | [1, 8, 37, 129, 376, 967, 2267, 4950, 10220]| -FibonacciRev Diagonals as columns +Fibonacci:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -188,7 +180,7 @@ FibonacciRev Diagonals as columns | DiagCol7 | [8, 21, 54, 133, 309, 674, 1383, 2683, 4950] | | DiagCol8 | [13, 34, 88, 221, 530, 1204, 2587, 5270, 10220] | -FibonacciRev Polynomial values as rows +Fibonacci:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ FibonacciRev Polynomial values as rows | PolyRow7 | [1, 115, 3189, 34777, 213337, 910431, 3041245, 8525749, 20960817] | | PolyRow8 | [1, 235, 10847, 171913, 1394365, 7415711, 29686795, 97029997, 272533433] | -FibonacciRev Polynomial values as columns +Fibonacci:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Fubini.md b/data/md/Fubini.md index 96b4ef9..53e4784 100644 --- a/data/md/Fubini.md +++ b/data/md/Fubini.md @@ -1,5 +1,5 @@ # Fubini -['A019538', 'A090582', 'A131689', 'A278075'] +['A131689', 'A019538', 'A090582', 'A278075'] Fubini Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Fubini Triangle view | Row7 | [0, 1, 126, 1806, 8400, 16800, 15120, 5040] | | Row8 | [0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320] | -Fubini Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320] | -| TriRev | [1, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 14, 1, 0, 120, 240, 150, 30, 1, 0, 720, 1800, 1560, 540, 62, 1, 0, 5040, 15120, 16800, 8400, 1806, 126, 1, 0, 40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 3, 0, 1, 7, 13, 0, 1, 15, 51, 75, 0, 1, 31, 181, 421, 541, 0, 1, 63, 603, 2163, 3963, 4683, 0, 1, 127, 1933, 10333, 27133, 42253, 47293, 0, 1, 255, 6051, 46875, 172875, 364395, 505515, 545835] | -| TriRevAcc | [1, 1, 0, 3, 1, 0, 13, 7, 1, 0, 75, 51, 15, 1, 0, 541, 421, 181, 31, 1, 0, 4683, 3963, 2163, 603, 63, 1, 0, 47293, 42253, 27133, 10333, 1933, 127, 1, 0, 545835, 505515, 364395, 172875, 46875, 6051, 255, 1, 0] | -| TriAccRev | [1, 1, 1, 2, 3, 3, 6, 12, 13, 13, 24, 60, 74, 75, 75, 120, 360, 510, 540, 541, 541, 720, 2520, 4080, 4620, 4682, 4683, 4683, 5040, 20160, 36960, 45360, 47166, 47292, 47293, 47293, 40320, 181440, 372960, 498960, 539784, 545580, 545834, 545835, 545835] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 6, 0, 1, 14, 6, 0, 1, 30, 36] | +Fubini Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 1, 6, 6], [0, 1, 14, 36, 24], [0, 1, 30, 150, 240, 120], [0, 1, 62, 540, 1560, 1800, 720], [0, 1, 126, 1806, 8400, 16800, 15120, 5040], [0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [6, 6, 1, 0], [24, 36, 14, 1, 0], [120, 240, 150, 30, 1, 0], [720, 1800, 1560, 540, 62, 1, 0], [5040, 15120, 16800, 8400, 1806, 126, 1, 0], [40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 6], [0, 1, 14, 6], [0, 1, 30, 36], [0, 1, 62, 150, 24]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 7, 13], [0, 1, 15, 51, 75], [0, 1, 31, 181, 421, 541], [0, 1, 63, 603, 2163, 3963, 4683], [0, 1, 127, 1933, 10333, 27133, 42253, 47293], [0, 1, 255, 6051, 46875, 172875, 364395, 505515, 545835]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [13, 7, 1, 0], [75, 51, 15, 1, 0], [541, 421, 181, 31, 1, 0], [4683, 3963, 2163, 603, 63, 1, 0], [47293, 42253, 27133, 10333, 1933, 127, 1, 0], [545835, 505515, 364395, 172875, 46875, 6051, 255, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [6, 12, 13, 13], [24, 60, 74, 75, 75], [120, 360, 510, 540, 541, 541], [720, 2520, 4080, 4620, 4682, 4683, 4683], [5040, 20160, 36960, 45360, 47166, 47292, 47293, 47293], [40320, 181440, 372960, 498960, 539784, 545580, 545834, 545835, 545835]] | Fubini Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| EvenSum | [1, 0, 2, 6, 38, 270, 2342, 23646, 272918] | -| OddSum | [0, 1, 1, 7, 37, 271, 2341, 23647, 272917] | -| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| AccSum | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | -| AccRevSum | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | -| RevAccSum | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | -| DiagSum | [1, 0, 1, 1, 3, 7, 21, 67] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| EvenSum | [1, 0, 2, 6, 38, 270, 2342, 23646, 272918] | +| OddSum | [0, 1, 1, 7, 37, 271, 2341, 23647, 272917] | +| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| AccSum | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | +| AccRevSum | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | +| AntiDiagSum | [1, 0, 1, 1, 3, 7, 21, 67, 237] | Fubini Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 504, 1200, 4352400, 6501600, 31719676464000] | -| RowGcd | [1, 1, 2, 6, 2, 30, 2, 42, 2] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 16800, 191520] | -| Middle | [1, 0, 1, 1, 14, 30, 540, 1806, 40824] | -| Central | [1, 1, 14, 540, 40824] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680] | -| NegHalf | [1, 1, 0, -2, 0, 16, 0, -272, 0] | -| Bin | [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216] | -| InvBin | [1, 1, 0, -9, -40, 125, 3444, 18571, -241872] | -| TransSqrs | [0, 1, 9, 79, 765, 8311, 100989, 1362439, 20246445] | -| TransNat0 | [0, 1, 5, 31, 233, 2071, 21305, 249271, 3270713] | -| TransNat1 | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 504, 1200, 4352400, 6501600, 31719676464000] | +| RowGcd | [1, 1, 2, 6, 2, 30, 2, 42, 2] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 16800, 191520] | +| ColMiddle | [1, 0, 1, 1, 14, 30, 540, 1806, 40824] | +| ColECenter | [1, 1, 14, 540, 40824] | +| ColOCenter | [0, 1, 30, 1806] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 9, 79, 765, 8311, 100989, 1362439, 20246445] | +| TransNat0 | [0, 1, 5, 31, 233, 2071, 21305, 249271, 3270713] | +| TransNat1 | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | Fubini Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Fubini Polynomial values as columns | PolyCol7 | [1, 7, 105, 2359, 70665, 2646007, 118893705, 6232661239, 373405001865] | | PolyCol8 | [1, 8, 136, 3464, 117640, 4993928, 254396296, 15119104904, 1026912225160] | -# FubiniRev -True +# Fubini:Rev +[] -FubiniRev Triangle view +Fubini:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ FubiniRev Triangle view | Row7 | [5040, 15120, 16800, 8400, 1806, 126, 1, 0] | | Row8 | [40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 0] | -FubiniRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 14, 1, 0, 120, 240, 150, 30, 1, 0, 720, 1800, 1560, 540, 62, 1, 0, 5040, 15120, 16800, 8400, 1806, 126, 1, 0, 40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320] | -| TriAcc | [1, 1, 1, 2, 3, 3, 6, 12, 13, 13, 24, 60, 74, 75, 75, 120, 360, 510, 540, 541, 541, 720, 2520, 4080, 4620, 4682, 4683, 4683, 5040, 20160, 36960, 45360, 47166, 47292, 47293, 47293, 40320, 181440, 372960, 498960, 539784, 545580, 545834, 545835, 545835] | -| TriRevAcc | [1, 1, 1, 3, 3, 2, 13, 13, 12, 6, 75, 75, 74, 60, 24, 541, 541, 540, 510, 360, 120, 4683, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320] | -| TriAccRev | [1, 0, 1, 0, 1, 3, 0, 1, 7, 13, 0, 1, 15, 51, 75, 0, 1, 31, 181, 421, 541, 0, 1, 63, 603, 2163, 3963, 4683, 0, 1, 127, 1933, 10333, 27133, 42253, 47293, 0, 1, 255, 6051, 46875, 172875, 364395, 505515, 545835] | -| TriDiag | [1, 1, 2, 0, 6, 1, 24, 6, 0, 120, 36, 1, 720, 240, 14, 0, 5040, 1800, 150, 1] | - -FubiniRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| EvenSum | [1, 1, 2, 7, 38, 271, 2342, 23647, 272918] | -| OddSum | [0, 0, 1, 6, 37, 270, 2341, 23646, 272917] | -| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| AccSum | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | -| AccRevSum | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | -| RevAccSum | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | -| DiagSum | [1, 1, 2, 7, 30, 157, 974, 6991] | - -FubiniRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 504, 1200, 4352400, 6501600, 31719676464000] | -| RowGcd | [1, 1, 2, 6, 2, 30, 2, 42, 2] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 16800, 191520] | -| Middle | [1, 1, 1, 6, 14, 150, 540, 8400, 40824] | -| Central | [1, 1, 14, 540, 40824] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 10, 74, 730, 9002, 133210, 2299754, 45375130] | -| NegHalf | [1, -2, 6, -26, 150, -1082, 9366, -94586, 1091670] | -| Bin | [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216] | -| InvBin | [1, -1, 0, 9, -40, -125, 3444, -18571, -241872] | -| TransSqrs | [0, 0, 1, 10, 101, 1126, 13917, 190002, 2848477] | -| TransNat0 | [0, 0, 1, 8, 67, 634, 6793, 81780, 1095967] | -| TransNat1 | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | - -FubiniRev Diagonals as rows +Fubini:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [6, 6, 1, 0], [24, 36, 14, 1, 0], [120, 240, 150, 30, 1, 0], [720, 1800, 1560, 540, 62, 1, 0], [5040, 15120, 16800, 8400, 1806, 126, 1, 0], [40320, 141120, 191520, 126000, 40824, 5796, 254, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 6, 6], [0, 1, 14, 36, 24], [0, 1, 30, 150, 240, 120], [0, 1, 62, 540, 1560, 1800, 720], [0, 1, 126, 1806, 8400, 16800, 15120, 5040], [0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320]] | +| AntiDiag | [[1], [1], [2, 0], [6, 1], [24, 6, 0], [120, 36, 1], [720, 240, 14, 0], [5040, 1800, 150, 1], [40320, 15120, 1560, 30, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [6, 12, 13, 13], [24, 60, 74, 75, 75], [120, 360, 510, 540, 541, 541], [720, 2520, 4080, 4620, 4682, 4683, 4683], [5040, 20160, 36960, 45360, 47166, 47292, 47293, 47293], [40320, 181440, 372960, 498960, 539784, 545580, 545834, 545835, 545835]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [13, 13, 12, 6], [75, 75, 74, 60, 24], [541, 541, 540, 510, 360, 120], [4683, 4683, 4682, 4620, 4080, 2520, 720], [47293, 47293, 47292, 47166, 45360, 36960, 20160, 5040], [545835, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 7, 13], [0, 1, 15, 51, 75], [0, 1, 31, 181, 421, 541], [0, 1, 63, 603, 2163, 3963, 4683], [0, 1, 127, 1933, 10333, 27133, 42253, 47293], [0, 1, 255, 6051, 46875, 172875, 364395, 505515, 545835]] | + +Fubini:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| EvenSum | [1, 1, 2, 7, 38, 271, 2342, 23647, 272918] | +| OddSum | [0, 0, 1, 6, 37, 270, 2341, 23646, 272917] | +| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AccSum | [1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548] | +| AccRevSum | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | +| AntiDiagSum | [1, 1, 2, 7, 30, 157, 974, 6991, 57030] | + +Fubini:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 504, 1200, 4352400, 6501600, 31719676464000] | +| RowGcd | [1, 1, 2, 6, 2, 30, 2, 42, 2] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 16800, 191520] | +| ColMiddle | [1, 1, 1, 6, 14, 150, 540, 8400, 40824] | +| ColECenter | [1, 1, 14, 540, 40824] | +| ColOCenter | [1, 6, 150, 8400] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 10, 101, 1126, 13917, 190002, 2848477] | +| TransNat0 | [0, 0, 1, 8, 67, 634, 6793, 81780, 1095967] | +| TransNat1 | [1, 1, 4, 21, 142, 1175, 11476, 129073, 1641802] | + +Fubini:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ FubiniRev Diagonals as rows | DiagRow7 | [5040, 141120, 2328480, 29635200, 322494480, 3162075840, 28805736960, 248619571200, 2060056318320]| | DiagRow8 | [40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960]| -FubiniRev Diagonals as columns +Fubini:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -188,7 +180,7 @@ FubiniRev Diagonals as columns | DiagCol7 | [0, 1, 510, 55980, 3498000, 165528000, 6711344640, 248619571200, 8734434508800] | | DiagCol8 | [0, 1, 1022, 171006, 14676024, 901020120, 45674188560, 2060056318320, 86355926616960] | -FubiniRev Polynomial values as rows +Fubini:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ FubiniRev Polynomial values as rows | PolyRow7 | [5040, 47293, 202672, 606033, 1467376, 3088765, 5881968, 10386817, 17290288] | | PolyRow8 | [40320, 545835, 2951680, 10491885, 29175936, 68958295, 145105920, 279768825, 503758720] | -FubiniRev Polynomial values as columns +Fubini:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | diff --git a/data/md/FussCatalan.md b/data/md/FussCatalan.md index ab36cac..79abe8a 100644 --- a/data/md/FussCatalan.md +++ b/data/md/FussCatalan.md @@ -1,5 +1,5 @@ # FussCatalan -['A030237', 'A054445', 'A355173'] +['A355173', 'A030237', 'A054445'] FussCatalan Triangle view | Row | Seq | @@ -14,45 +14,41 @@ FussCatalan Triangle view | Row7 | [0, 1, 7, 27, 75, 165, 297, 429] | | Row8 | [0, 1, 8, 35, 110, 275, 572, 1001, 1430] | -FussCatalan Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430] | -| TriRev | [1, 1, 0, 2, 1, 0, 5, 3, 1, 0, 14, 9, 4, 1, 0, 42, 28, 14, 5, 1, 0, 132, 90, 48, 20, 6, 1, 0, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1001, 572, 275, 110, 35, 8, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 3, 0, 1, 4, 9, 0, 1, 5, 14, 28, 0, 1, 6, 20, 48, 90, 0, 1, 7, 27, 75, 165, 297, 0, 1, 8, 35, 110, 275, 572, 1001, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432] | -| TriRevAcc | [1, 1, 0, 3, 1, 0, 9, 4, 1, 0, 28, 14, 5, 1, 0, 90, 48, 20, 6, 1, 0, 297, 165, 75, 27, 7, 1, 0, 1001, 572, 275, 110, 35, 8, 1, 0, 3432, 2002, 1001, 429, 154, 44, 9, 1, 0] | -| TriAccRev | [1, 1, 1, 2, 3, 3, 5, 8, 9, 9, 14, 23, 27, 28, 28, 42, 70, 84, 89, 90, 90, 132, 222, 270, 290, 296, 297, 297, 429, 726, 891, 966, 993, 1000, 1001, 1001, 1430, 2431, 3003, 3278, 3388, 3423, 3431, 3432, 3432] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 5, 0, 1, 5, 9] | +FussCatalan Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 5], [0, 1, 4, 9, 14], [0, 1, 5, 14, 28, 42], [0, 1, 6, 20, 48, 90, 132], [0, 1, 7, 27, 75, 165, 297, 429], [0, 1, 8, 35, 110, 275, 572, 1001, 1430]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [5, 3, 1, 0], [14, 9, 4, 1, 0], [42, 28, 14, 5, 1, 0], [132, 90, 48, 20, 6, 1, 0], [429, 297, 165, 75, 27, 7, 1, 0], [1430, 1001, 572, 275, 110, 35, 8, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 3], [0, 1, 4, 5], [0, 1, 5, 9], [0, 1, 6, 14, 14]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 4, 9], [0, 1, 5, 14, 28], [0, 1, 6, 20, 48, 90], [0, 1, 7, 27, 75, 165, 297], [0, 1, 8, 35, 110, 275, 572, 1001], [0, 1, 9, 44, 154, 429, 1001, 2002, 3432]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [9, 4, 1, 0], [28, 14, 5, 1, 0], [90, 48, 20, 6, 1, 0], [297, 165, 75, 27, 7, 1, 0], [1001, 572, 275, 110, 35, 8, 1, 0], [3432, 2002, 1001, 429, 154, 44, 9, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [5, 8, 9, 9], [14, 23, 27, 28, 28], [42, 70, 84, 89, 90, 90], [132, 222, 270, 290, 296, 297, 297], [429, 726, 891, 966, 993, 1000, 1001, 1001], [1430, 2431, 3003, 3278, 3388, 3423, 3431, 3432, 3432]] | FussCatalan Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 9, 28, 90, 297, 1001, 3432] | -| EvenSum | [1, 0, 2, 3, 18, 33, 186, 379, 2120] | -| OddSum | [0, 1, 1, 6, 10, 57, 111, 622, 1312] | -| AltSum | [1, -1, 1, -3, 8, -24, 75, -243, 808] | -| AccSum | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | -| AccRevSum | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | -| RevAccSum | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | -| DiagSum | [1, 0, 1, 1, 3, 4, 10, 15] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 9, 28, 90, 297, 1001, 3432] | +| EvenSum | [1, 0, 2, 3, 18, 33, 186, 379, 2120] | +| OddSum | [0, 1, 1, 6, 10, 57, 111, 622, 1312] | +| AltSum | [1, -1, 1, -3, 8, -24, 75, -243, 808] | +| AccSum | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | +| AccRevSum | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | +| AntiDiagSum | [1, 0, 1, 1, 3, 4, 10, 15, 35] | FussCatalan Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 15, 252, 420, 7920, 675675, 200200] | -| RowGcd | [1, 1, 2, 1, 1, 1, 2, 1, 1] | -| RowMax | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| Middle | [1, 0, 1, 1, 4, 5, 20, 27, 110] | -| Central | [1, 1, 4, 20, 110] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| PosHalf | [1, 1, 4, 15, 56, 210, 792, 3003, 11440] | -| NegHalf | [1, 1, 0, 3, 4, 18, 48, 167, 540] | -| Bin | [1, 1, 4, 17, 78, 377, 1888, 9697, 50746] | -| InvBin | [1, 1, 0, -1, -2, -3, -4, -5, -6] | -| TransSqrs | [0, 1, 9, 58, 322, 1645, 7975, 37310, 170144] | -| TransNat0 | [0, 1, 5, 22, 92, 375, 1507, 6006, 23816] | -| TransNat1 | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 15, 252, 420, 7920, 675675, 200200] | +| RowGcd | [1, 1, 2, 1, 1, 1, 2, 1, 1] | +| RowMax | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| ColMiddle | [1, 0, 1, 1, 4, 5, 20, 27, 110] | +| ColECenter | [1, 1, 4, 20, 110] | +| ColOCenter | [0, 1, 5, 27] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| TransSqrs | [0, 1, 9, 58, 322, 1645, 7975, 37310, 170144] | +| TransNat0 | [0, 1, 5, 22, 92, 375, 1507, 6006, 23816] | +| TransNat1 | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | FussCatalan Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ FussCatalan Polynomial values as columns | PolyCol7 | [1, 7, 105, 1869, 36904, 778176, 17164707, 391204541, 9140225640] | | PolyCol8 | [1, 8, 136, 2760, 62216, 1498440, 37759368, 983263176, 26250094600] | -# FussCatalanRev -True +# FussCatalan:Rev +[] -FussCatalanRev Triangle view +FussCatalan:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ FussCatalanRev Triangle view | Row7 | [429, 297, 165, 75, 27, 7, 1, 0] | | Row8 | [1430, 1001, 572, 275, 110, 35, 8, 1, 0] | -FussCatalanRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 1, 0, 5, 3, 1, 0, 14, 9, 4, 1, 0, 42, 28, 14, 5, 1, 0, 132, 90, 48, 20, 6, 1, 0, 429, 297, 165, 75, 27, 7, 1, 0, 1430, 1001, 572, 275, 110, 35, 8, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430] | -| TriAcc | [1, 1, 1, 2, 3, 3, 5, 8, 9, 9, 14, 23, 27, 28, 28, 42, 70, 84, 89, 90, 90, 132, 222, 270, 290, 296, 297, 297, 429, 726, 891, 966, 993, 1000, 1001, 1001, 1430, 2431, 3003, 3278, 3388, 3423, 3431, 3432, 3432] | -| TriRevAcc | [1, 1, 1, 3, 3, 2, 9, 9, 8, 5, 28, 28, 27, 23, 14, 90, 90, 89, 84, 70, 42, 297, 297, 296, 290, 270, 222, 132, 1001, 1001, 1000, 993, 966, 891, 726, 429, 3432, 3432, 3431, 3423, 3388, 3278, 3003, 2431, 1430] | -| TriAccRev | [1, 0, 1, 0, 1, 3, 0, 1, 4, 9, 0, 1, 5, 14, 28, 0, 1, 6, 20, 48, 90, 0, 1, 7, 27, 75, 165, 297, 0, 1, 8, 35, 110, 275, 572, 1001, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432] | -| TriDiag | [1, 1, 2, 0, 5, 1, 14, 3, 0, 42, 9, 1, 132, 28, 4, 0, 429, 90, 14, 1] | - -FussCatalanRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 9, 28, 90, 297, 1001, 3432] | -| EvenSum | [1, 1, 2, 6, 18, 57, 186, 622, 2120] | -| OddSum | [0, 0, 1, 3, 10, 33, 111, 379, 1312] | -| AltSum | [1, 1, 1, 3, 8, 24, 75, 243, 808] | -| AccSum | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | -| AccRevSum | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | -| RevAccSum | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | -| DiagSum | [1, 1, 2, 6, 17, 52, 164, 534] | - -FussCatalanRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 15, 252, 420, 7920, 675675, 200200] | -| RowGcd | [1, 1, 2, 1, 1, 1, 2, 1, 1] | -| RowMax | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| Middle | [1, 1, 1, 3, 4, 14, 20, 75, 110] | -| Central | [1, 1, 4, 20, 110] | -| LeftSide | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 10, 54, 314, 1926, 12282, 80646, 541690] | -| NegHalf | [1, -2, 6, -30, 166, -990, 6198, -40174, 267270] | -| Bin | [1, 1, 4, 17, 78, 377, 1888, 9697, 50746] | -| InvBin | [1, -1, 0, 1, -2, 3, -4, 5, -6] | -| TransSqrs | [0, 0, 1, 7, 34, 145, 583, 2275, 8736] | -| TransNat0 | [0, 0, 1, 5, 20, 75, 275, 1001, 3640] | -| TransNat1 | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | - -FussCatalanRev Diagonals as rows +FussCatalan:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [5, 3, 1, 0], [14, 9, 4, 1, 0], [42, 28, 14, 5, 1, 0], [132, 90, 48, 20, 6, 1, 0], [429, 297, 165, 75, 27, 7, 1, 0], [1430, 1001, 572, 275, 110, 35, 8, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 5], [0, 1, 4, 9, 14], [0, 1, 5, 14, 28, 42], [0, 1, 6, 20, 48, 90, 132], [0, 1, 7, 27, 75, 165, 297, 429], [0, 1, 8, 35, 110, 275, 572, 1001, 1430]] | +| AntiDiag | [[1], [1], [2, 0], [5, 1], [14, 3, 0], [42, 9, 1], [132, 28, 4, 0], [429, 90, 14, 1], [1430, 297, 48, 5, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [5, 8, 9, 9], [14, 23, 27, 28, 28], [42, 70, 84, 89, 90, 90], [132, 222, 270, 290, 296, 297, 297], [429, 726, 891, 966, 993, 1000, 1001, 1001], [1430, 2431, 3003, 3278, 3388, 3423, 3431, 3432, 3432]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [9, 9, 8, 5], [28, 28, 27, 23, 14], [90, 90, 89, 84, 70, 42], [297, 297, 296, 290, 270, 222, 132], [1001, 1001, 1000, 993, 966, 891, 726, 429], [3432, 3432, 3431, 3423, 3388, 3278, 3003, 2431, 1430]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 4, 9], [0, 1, 5, 14, 28], [0, 1, 6, 20, 48, 90], [0, 1, 7, 27, 75, 165, 297], [0, 1, 8, 35, 110, 275, 572, 1001], [0, 1, 9, 44, 154, 429, 1001, 2002, 3432]] | + +FussCatalan:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 9, 28, 90, 297, 1001, 3432] | +| EvenSum | [1, 1, 2, 6, 18, 57, 186, 622, 2120] | +| OddSum | [0, 0, 1, 3, 10, 33, 111, 379, 1312] | +| AltSum | [1, 1, 1, 3, 8, 24, 75, 243, 808] | +| AccSum | [1, 2, 8, 31, 120, 465, 1804, 7007, 27248] | +| AccRevSum | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | +| AntiDiagSum | [1, 1, 2, 6, 17, 52, 164, 534, 1780] | + +FussCatalan:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 15, 252, 420, 7920, 675675, 200200] | +| RowGcd | [1, 1, 2, 1, 1, 1, 2, 1, 1] | +| RowMax | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| ColMiddle | [1, 1, 1, 3, 4, 14, 20, 75, 110] | +| ColECenter | [1, 1, 4, 20, 110] | +| ColOCenter | [1, 3, 14, 75] | +| ColLeft | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 7, 34, 145, 583, 2275, 8736] | +| TransNat0 | [0, 0, 1, 5, 20, 75, 275, 1001, 3640] | +| TransNat1 | [1, 1, 4, 14, 48, 165, 572, 2002, 7072] | + +FussCatalan:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ FussCatalanRev Diagonals as rows | DiagRow7 | [429, 1001, 2002, 3640, 6188, 9996, 15504, 23256, 33915]| | DiagRow8 | [1430, 3432, 7072, 13260, 23256, 38760, 62016, 95931, 144210]| -FussCatalanRev Diagonals as columns +FussCatalan:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | @@ -188,7 +180,7 @@ FussCatalanRev Diagonals as columns | DiagCol7 | [0, 1, 9, 54, 273, 1260, 5508, 23256, 95931] | | DiagCol8 | [0, 1, 10, 65, 350, 1700, 7752, 33915, 144210] | -FussCatalanRev Polynomial values as rows +FussCatalan:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ FussCatalanRev Polynomial values as rows | PolyRow7 | [429, 1001, 3003, 9447, 27233, 69789, 160431, 336443, 653877] | | PolyRow8 | [1430, 3432, 11440, 42440, 145338, 436360, 1155332, 2747880, 5978590] | -FussCatalanRev Polynomial values as columns +FussCatalan:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | diff --git a/data/md/Gaussq2.md b/data/md/Gaussq2.md index 98c9a9a..5183b2f 100644 --- a/data/md/Gaussq2.md +++ b/data/md/Gaussq2.md @@ -14,45 +14,41 @@ Gaussq2 Triangle view | Row7 | [1, 127, 2667, 11811, 11811, 2667, 127, 1] | | Row8 | [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -Gaussq2 Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -| TriRev | [1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -| TriAcc | [1, 1, 2, 1, 4, 5, 1, 8, 15, 16, 1, 16, 51, 66, 67, 1, 32, 187, 342, 373, 374, 1, 64, 715, 2110, 2761, 2824, 2825, 1, 128, 2795, 14606, 26417, 29084, 29211, 29212, 1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199] | -| TriRevAcc | [1, 2, 1, 5, 4, 1, 16, 15, 8, 1, 67, 66, 51, 16, 1, 374, 373, 342, 187, 32, 1, 2825, 2824, 2761, 2110, 715, 64, 1, 29212, 29211, 29084, 26417, 14606, 2795, 128, 1, 417199, 417198, 416943, 406148, 308993, 108206, 11051, 256, 1] | -| TriAccRev | [1, 1, 2, 1, 4, 5, 1, 8, 15, 16, 1, 16, 51, 66, 67, 1, 32, 187, 342, 373, 374, 1, 64, 715, 2110, 2761, 2824, 2825, 1, 128, 2795, 14606, 26417, 29084, 29211, 29212, 1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 7, 1, 31, 35, 1, 1, 63, 155, 15] | +Gaussq2 Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 1], [1, 7, 7, 1], [1, 15, 35, 15, 1], [1, 31, 155, 155, 31, 1], [1, 63, 651, 1395, 651, 63, 1], [1, 127, 2667, 11811, 11811, 2667, 127, 1], [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 1], [1, 7, 7, 1], [1, 15, 35, 15, 1], [1, 31, 155, 155, 31, 1], [1, 63, 651, 1395, 651, 63, 1], [1, 127, 2667, 11811, 11811, 2667, 127, 1], [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 7, 1], [1, 15, 7], [1, 31, 35, 1], [1, 63, 155, 15], [1, 127, 651, 155, 1]] | +| AccTabl | [[1], [1, 2], [1, 4, 5], [1, 8, 15, 16], [1, 16, 51, 66, 67], [1, 32, 187, 342, 373, 374], [1, 64, 715, 2110, 2761, 2824, 2825], [1, 128, 2795, 14606, 26417, 29084, 29211, 29212], [1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 1], [16, 15, 8, 1], [67, 66, 51, 16, 1], [374, 373, 342, 187, 32, 1], [2825, 2824, 2761, 2110, 715, 64, 1], [29212, 29211, 29084, 26417, 14606, 2795, 128, 1], [417199, 417198, 416943, 406148, 308993, 108206, 11051, 256, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 5], [1, 8, 15, 16], [1, 16, 51, 66, 67], [1, 32, 187, 342, 373, 374], [1, 64, 715, 2110, 2761, 2824, 2825], [1, 128, 2795, 14606, 26417, 29084, 29211, 29212], [1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199]] | Gaussq2 Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 67, 374, 2825, 29212, 417199] | -| EvenSum | [1, 1, 2, 8, 37, 187, 1304, 14606, 222379] | -| OddSum | [0, 1, 3, 8, 30, 187, 1521, 14606, 194820] | -| AltSum | [1, 0, -1, 0, 7, 0, -217, 0, 27559] | -| AccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| AccRevSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| RevAccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| DiagSum | [1, 1, 2, 4, 9, 23, 68, 234] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 67, 374, 2825, 29212, 417199] | +| EvenSum | [1, 1, 2, 8, 37, 187, 1304, 14606, 222379] | +| OddSum | [0, 1, 3, 8, 30, 187, 1521, 14606, 194820] | +| AltSum | [1, 0, -1, 0, 7, 0, -217, 0, 27559] | +| AccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | +| AccRevSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | +| AntiDiagSum | [1, 1, 2, 4, 9, 23, 68, 234, 935] | Gaussq2 Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 7, 105, 155, 9765, 82677, 3011805] | -| RowGcd | [1, 1, 3, 7, 5, 31, 3, 127, 17] | -| RowMax | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | -| Middle | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | -| Central | [1, 3, 35, 1395, 200787] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 11, 51, 307, 2451, 26387, 387987, 7866259] | -| NegHalf | [1, -1, -1, 7, 7, -217, -217, 27559, 27559] | -| Bin | [1, 2, 8, 44, 332, 3412, 48188, 940564, 25545052] | -| InvBin | [1, 0, -4, 0, 92, 0, -9124, 0, 3774172] | -| TransSqrs | [0, 1, 7, 44, 306, 2567, 27249, 377366, 6960476] | -| TransNat0 | [0, 1, 5, 24, 134, 935, 8475, 102242, 1668796] | -| TransNat1 | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 7, 105, 155, 9765, 82677, 3011805] | +| RowGcd | [1, 1, 3, 7, 5, 31, 3, 127, 17] | +| RowMax | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | +| ColMiddle | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | +| ColECenter | [1, 3, 35, 1395, 200787] | +| ColOCenter | [1, 7, 155, 11811] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 44, 306, 2567, 27249, 377366, 6960476] | +| TransNat0 | [0, 1, 5, 24, 134, 935, 8475, 102242, 1668796] | +| TransNat1 | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | Gaussq2 Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Gaussq2 Polynomial values as columns | PolyCol7 | [1, 8, 71, 736, 9367, 152216, 3250367, 93130192, 3634617799] | | PolyCol8 | [1, 9, 89, 1017, 14137, 249273, 5749433, 177378489, 7437830329] | -# Gaussq2Inv -True +# Gaussq2:Inv +[] -Gaussq2Inv Triangle view +Gaussq2:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ Gaussq2Inv Triangle view | Row7 | [-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1] | | Row8 | [268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1] | -Gaussq2Inv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 2, -3, 1, -8, 14, -7, 1, 64, -120, 70, -15, 1, -1024, 1984, -1240, 310, -31, 1, 32768, -64512, 41664, -11160, 1302, -63, 1, -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1, 268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1] | -| TriRev | [1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456] | -| TriAcc | [1, -1, 0, 2, -1, 0, -8, 6, -1, 0, 64, -56, 14, -1, 0, -1024, 960, -280, 30, -1, 0, 32768, -31744, 9920, -1240, 62, -1, 0, -2097152, 2064384, -666624, 89280, -5208, 126, -1, 0, 268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, 2, 0, -1, 6, -8, 0, -1, 14, -56, 64, 0, -1, 30, -280, 960, -1024, 0, -1, 62, -1240, 9920, -31744, 32768, 0, -1, 126, -5208, 89280, -666624, 2064384, -2097152, 0, -1, 254, -21336, 755904, -12094464, 87392256, -266338304, 268435456] | -| TriAccRev | [1, 1, 0, 1, -2, 0, 1, -6, 8, 0, 1, -14, 56, -64, 0, 1, -30, 280, -960, 1024, 0, 1, -62, 1240, -9920, 31744, -32768, 0, 1, -126, 5208, -89280, 666624, -2064384, 2097152, 0, 1, -254, 21336, -755904, 12094464, -87392256, 266338304, -268435456, 0] | -| TriDiag | [1, -1, 2, 1, -8, -3, 64, 14, 1, -1024, -120, -7, 32768, 1984, 70, 1, -2097152, -64512, -1240, -15] | - -Gaussq2Inv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 3, -15, 135, -2295, 75735, -4922775, 635037975] | -| OddSum | [0, 1, -3, 15, -135, 2295, -75735, 4922775, -635037975] | -| AltSum | [1, -2, 6, -30, 270, -4590, 151470, -9845550, 1270075950] | -| AccSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | -| AccRevSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | -| RevAccSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | -| DiagSum | [1, -1, 3, -11, 79, -1151, 34823, -2162919] | - -Gaussq2Inv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 56, 6720, 158720, 319979520, 173386235904, 808475248558080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760] | -| Middle | [1, -1, -3, 14, 70, -1240, -11160, 755904, 12850368] | -| Central | [1, -3, 70, -11160, 12850368] | -| LeftSide | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, 3, -21, 315, -9765, 615195, -78129765, 19923090075] | -| NegHalf | [1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575] | -| Bin | [1, 0, -3, 14, -55, -558, 66609, -7056882, 1180047537] | -| InvBin | [1, 2, 9, 72, 1025, 26600, 1287909, 118455696, 20965995297] | -| TransSqrs | [0, 1, 1, -5, 41, -657, 20997, -1342341, 171707697] | -| TransNat0 | [0, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | -| TransNat1 | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | - -Gaussq2Inv Diagonals as rows +Gaussq2:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [2, -3, 1], [-8, 14, -7, 1], [64, -120, 70, -15, 1], [-1024, 1984, -1240, 310, -31, 1], [32768, -64512, 41664, -11160, 1302, -63, 1], [-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1], [268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 2], [1, -7, 14, -8], [1, -15, 70, -120, 64], [1, -31, 310, -1240, 1984, -1024], [1, -63, 1302, -11160, 41664, -64512, 32768], [1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152], [1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456]] | +| AntiDiag | [[1], [-1], [2, 1], [-8, -3], [64, 14, 1], [-1024, -120, -7], [32768, 1984, 70, 1], [-2097152, -64512, -1240, -15], [268435456, 4161536, 41664, 310, 1]] | +| AccTabl | [[1], [-1, 0], [2, -1, 0], [-8, 6, -1, 0], [64, -56, 14, -1, 0], [-1024, 960, -280, 30, -1, 0], [32768, -31744, 9920, -1240, 62, -1, 0], [-2097152, 2064384, -666624, 89280, -5208, 126, -1, 0], [268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 2], [0, -1, 6, -8], [0, -1, 14, -56, 64], [0, -1, 30, -280, 960, -1024], [0, -1, 62, -1240, 9920, -31744, 32768], [0, -1, 126, -5208, 89280, -666624, 2064384, -2097152], [0, -1, 254, -21336, 755904, -12094464, 87392256, -266338304, 268435456]] | +| AccRevTabl | [[1], [1, 0], [1, -2, 0], [1, -6, 8, 0], [1, -14, 56, -64, 0], [1, -30, 280, -960, 1024, 0], [1, -62, 1240, -9920, 31744, -32768, 0], [1, -126, 5208, -89280, 666624, -2064384, 2097152, 0], [1, -254, 21336, -755904, 12094464, -87392256, 266338304, -268435456, 0]] | + +Gaussq2:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 3, -15, 135, -2295, 75735, -4922775, 635037975] | +| OddSum | [0, 1, -3, 15, -135, 2295, -75735, 4922775, -635037975] | +| AltSum | [1, -2, 6, -30, 270, -4590, 151470, -9845550, 1270075950] | +| AccSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | +| AccRevSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | +| AntiDiagSum | [1, -1, 3, -11, 79, -1151, 34823, -2162919, 272638967] | + +Gaussq2:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 56, 6720, 158720, 319979520, 173386235904, 808475248558080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760] | +| ColMiddle | [1, -1, -3, 14, 70, -1240, -11160, 755904, 12850368] | +| ColECenter | [1, -3, 70, -11160, 12850368] | +| ColOCenter | [-1, 14, -1240, 755904] | +| ColLeft | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -5, 41, -657, 20997, -1342341, 171707697] | +| TransNat0 | [0, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | +| TransNat1 | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | + +Gaussq2:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ Gaussq2Inv Diagonals as rows | DiagRow7 | [-2097152, -534773760, -91089797120, -13312123207680, -1816661080408064, -239975068524871680, -31200568036305141760, -4024873276683363287040, -517188324145426528731136]| | DiagRow8 | [268435456, 137170518016, 46775146643456, 13678389311307776, 3734200281987022848, 986672080959861424128, 256581725434371582722048, 66200105490614595677585408, 17013427111087951089139449856]| -Gaussq2Inv Diagonals as columns +Gaussq2:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | @@ -188,7 +180,7 @@ Gaussq2Inv Diagonals as columns | DiagCol7 | [1, -255, 86870, -50781720, 55440096448, -117175326428160, 487508875567267840, -4024873276683363287040, 66200105490614595677585408] | | DiagCol8 | [1, -511, 348502, -407647768, 890302725312, -3763855289306112, 31321011405562937344, -517188324145426528731136, 17013427111087951089139449856] | -Gaussq2Inv Polynomial values as rows +Gaussq2:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ Gaussq2Inv Polynomial values as rows | PolyRow7 | [-2097152, 0, 0, -229970, 0, 630828, 1206400, 1154250, 0] | | PolyRow8 | [268435456, 0, 0, 28746250, 0, -77591844, -147180800, -139664250, 0] | -Gaussq2Inv Polynomial values as columns +Gaussq2:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | @@ -214,10 +206,10 @@ Gaussq2Inv Polynomial values as columns | PolyCol7 | [1, 6, 30, 90, -90, 810, -20250, 1154250, -139664250] | | PolyCol8 | [1, 7, 42, 168, 0, 0, 0, 0, 0] | -# Gaussq2Rev -True +# Gaussq2:Rev +[] -Gaussq2Rev Triangle view +Gaussq2:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ Gaussq2Rev Triangle view | Row7 | [1, 127, 2667, 11811, 11811, 2667, 127, 1] | | Row8 | [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -Gaussq2Rev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -| TriRev | [1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1] | -| TriAcc | [1, 1, 2, 1, 4, 5, 1, 8, 15, 16, 1, 16, 51, 66, 67, 1, 32, 187, 342, 373, 374, 1, 64, 715, 2110, 2761, 2824, 2825, 1, 128, 2795, 14606, 26417, 29084, 29211, 29212, 1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199] | -| TriRevAcc | [1, 2, 1, 5, 4, 1, 16, 15, 8, 1, 67, 66, 51, 16, 1, 374, 373, 342, 187, 32, 1, 2825, 2824, 2761, 2110, 715, 64, 1, 29212, 29211, 29084, 26417, 14606, 2795, 128, 1, 417199, 417198, 416943, 406148, 308993, 108206, 11051, 256, 1] | -| TriAccRev | [1, 1, 2, 1, 4, 5, 1, 8, 15, 16, 1, 16, 51, 66, 67, 1, 32, 187, 342, 373, 374, 1, 64, 715, 2110, 2761, 2824, 2825, 1, 128, 2795, 14606, 26417, 29084, 29211, 29212, 1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 7, 1, 31, 35, 1, 1, 63, 155, 15] | - -Gaussq2Rev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 67, 374, 2825, 29212, 417199] | -| EvenSum | [1, 1, 2, 8, 37, 187, 1304, 14606, 222379] | -| OddSum | [0, 1, 3, 8, 30, 187, 1521, 14606, 194820] | -| AltSum | [1, 0, -1, 0, 7, 0, -217, 0, 27559] | -| AccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| AccRevSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| RevAccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | -| DiagSum | [1, 1, 2, 4, 9, 23, 68, 234] | - -Gaussq2Rev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 7, 105, 155, 9765, 82677, 3011805] | -| RowGcd | [1, 1, 3, 7, 5, 31, 3, 127, 17] | -| RowMax | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | -| Middle | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | -| Central | [1, 3, 35, 1395, 200787] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 11, 51, 307, 2451, 26387, 387987, 7866259] | -| NegHalf | [1, -1, -1, 7, 7, -217, -217, 27559, 27559] | -| Bin | [1, 2, 8, 44, 332, 3412, 48188, 940564, 25545052] | -| InvBin | [1, 0, -4, 0, 92, 0, -9124, 0, 3774172] | -| TransSqrs | [0, 1, 7, 44, 306, 2567, 27249, 377366, 6960476] | -| TransNat0 | [0, 1, 5, 24, 134, 935, 8475, 102242, 1668796] | -| TransNat1 | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | - -Gaussq2Rev Diagonals as rows +Gaussq2:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 1], [1, 7, 7, 1], [1, 15, 35, 15, 1], [1, 31, 155, 155, 31, 1], [1, 63, 651, 1395, 651, 63, 1], [1, 127, 2667, 11811, 11811, 2667, 127, 1], [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 1], [1, 7, 7, 1], [1, 15, 35, 15, 1], [1, 31, 155, 155, 31, 1], [1, 63, 651, 1395, 651, 63, 1], [1, 127, 2667, 11811, 11811, 2667, 127, 1], [1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 7, 1], [1, 15, 7], [1, 31, 35, 1], [1, 63, 155, 15], [1, 127, 651, 155, 1]] | +| AccTabl | [[1], [1, 2], [1, 4, 5], [1, 8, 15, 16], [1, 16, 51, 66, 67], [1, 32, 187, 342, 373, 374], [1, 64, 715, 2110, 2761, 2824, 2825], [1, 128, 2795, 14606, 26417, 29084, 29211, 29212], [1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 1], [16, 15, 8, 1], [67, 66, 51, 16, 1], [374, 373, 342, 187, 32, 1], [2825, 2824, 2761, 2110, 715, 64, 1], [29212, 29211, 29084, 26417, 14606, 2795, 128, 1], [417199, 417198, 416943, 406148, 308993, 108206, 11051, 256, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 5], [1, 8, 15, 16], [1, 16, 51, 66, 67], [1, 32, 187, 342, 373, 374], [1, 64, 715, 2110, 2761, 2824, 2825], [1, 128, 2795, 14606, 26417, 29084, 29211, 29212], [1, 256, 11051, 108206, 308993, 406148, 416943, 417198, 417199]] | + +Gaussq2:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 67, 374, 2825, 29212, 417199] | +| EvenSum | [1, 1, 2, 8, 37, 187, 1304, 14606, 222379] | +| OddSum | [0, 1, 3, 8, 30, 187, 1521, 14606, 194820] | +| AltSum | [1, 0, -1, 0, 7, 0, -217, 0, 27559] | +| AccSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | +| AccRevSum | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | +| AntiDiagSum | [1, 1, 2, 4, 9, 23, 68, 234, 935] | + +Gaussq2:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 7, 105, 155, 9765, 82677, 3011805] | +| RowGcd | [1, 1, 3, 7, 5, 31, 3, 127, 17] | +| RowMax | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | +| ColMiddle | [1, 1, 3, 7, 35, 155, 1395, 11811, 200787] | +| ColECenter | [1, 3, 35, 1395, 200787] | +| ColOCenter | [1, 7, 155, 11811] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 44, 306, 2567, 27249, 377366, 6960476] | +| TransNat0 | [0, 1, 5, 24, 134, 935, 8475, 102242, 1668796] | +| TransNat1 | [1, 3, 10, 40, 201, 1309, 11300, 131454, 2085995] | + +Gaussq2:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ Gaussq2Rev Diagonals as rows | DiagRow7 | [1, 255, 43435, 6347715, 866251507, 114429029715, 14877590196755, 1919209135381395, 246614610741341843]| | DiagRow8 | [1, 511, 174251, 50955971, 13910980083, 3675639930963, 955841412523283, 246614610741341843, 63379954960524853651]| -Gaussq2Rev Diagonals as columns +Gaussq2:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ Gaussq2Rev Diagonals as columns | DiagCol7 | [1, 255, 43435, 6347715, 866251507, 114429029715, 14877590196755, 1919209135381395, 246614610741341843] | | DiagCol8 | [1, 511, 174251, 50955971, 13910980083, 3675639930963, 955841412523283, 246614610741341843, 63379954960524853651] | -Gaussq2Rev Polynomial values as rows +Gaussq2:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ Gaussq2Rev Polynomial values as rows | PolyRow7 | [1, 29212, 387987, 2042824, 7090285, 19322436, 44898847, 93130192, 177378489] | | PolyRow8 | [1, 417199, 7866259, 51027319, 205739629, 630501151, 1613787679, 3634617799, 7437830329] | -Gaussq2Rev Polynomial values as columns +Gaussq2:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ Gaussq2Rev Polynomial values as columns | PolyCol7 | [1, 8, 71, 736, 9367, 152216, 3250367, 93130192, 3634617799] | | PolyCol8 | [1, 9, 89, 1017, 14137, 249273, 5749433, 177378489, 7437830329] | -# Gaussq2InvRev -True +# Gaussq2:Inv:Rev +[] -Gaussq2InvRev Triangle view +Gaussq2:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ Gaussq2InvRev Triangle view | Row7 | [1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152] | | Row8 | [1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456] | -Gaussq2InvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456] | -| TriRev | [1, -1, 1, 2, -3, 1, -8, 14, -7, 1, 64, -120, 70, -15, 1, -1024, 1984, -1240, 310, -31, 1, 32768, -64512, 41664, -11160, 1302, -63, 1, -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1, 268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1] | -| TriAcc | [1, 1, 0, 1, -2, 0, 1, -6, 8, 0, 1, -14, 56, -64, 0, 1, -30, 280, -960, 1024, 0, 1, -62, 1240, -9920, 31744, -32768, 0, 1, -126, 5208, -89280, 666624, -2064384, 2097152, 0, 1, -254, 21336, -755904, 12094464, -87392256, 266338304, -268435456, 0] | -| TriRevAcc | [1, 0, 1, 0, -2, 1, 0, 8, -6, 1, 0, -64, 56, -14, 1, 0, 1024, -960, 280, -30, 1, 0, -32768, 31744, -9920, 1240, -62, 1, 0, 2097152, -2064384, 666624, -89280, 5208, -126, 1, 0, -268435456, 266338304, -87392256, 12094464, -755904, 21336, -254, 1] | -| TriAccRev | [1, -1, 0, 2, -1, 0, -8, 6, -1, 0, 64, -56, 14, -1, 0, -1024, 960, -280, 30, -1, 0, 32768, -31744, 9920, -1240, 62, -1, 0, -2097152, 2064384, -666624, 89280, -5208, 126, -1, 0, 268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, -3, 1, -7, 2, 1, -15, 14, 1, -31, 70, -8, 1, -63, 310, -120] | - -Gaussq2InvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 3, 15, 135, 2295, 75735, 4922775, 635037975] | -| OddSum | [0, -1, -3, -15, -135, -2295, -75735, -4922775, -635037975] | -| AltSum | [1, 2, 6, 30, 270, 4590, 151470, 9845550, 1270075950] | -| AccSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | -| AccRevSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | -| RevAccSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | -| DiagSum | [1, 1, 0, -2, -4, 0, 32, 128] | - -Gaussq2InvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 56, 6720, 158720, 319979520, 173386235904, 808475248558080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760] | -| Middle | [1, 1, -3, -7, 70, 310, -11160, -94488, 12850368] | -| Central | [1, -3, 70, -11160, 12850368] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | -| PosHalf | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| NegHalf | [1, -3, 12, -72, 720, -12960, 440640, -29082240, 3780691200] | -| Bin | [1, 0, -3, 14, -55, -558, 66609, -7056882, 1180047537] | -| InvBin | [1, -2, 9, -72, 1025, -26600, 1287909, -118455696, 20965995297] | -| TransSqrs | [0, -1, 5, -23, 209, -3807, 138177, -9955071, 1421783937] | -| TransNat0 | [0, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | -| TransNat1 | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | - -Gaussq2InvRev Diagonals as rows +Gaussq2:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -3, 2], [1, -7, 14, -8], [1, -15, 70, -120, 64], [1, -31, 310, -1240, 1984, -1024], [1, -63, 1302, -11160, 41664, -64512, 32768], [1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152], [1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456]] | +| RevTabl | [[1], [-1, 1], [2, -3, 1], [-8, 14, -7, 1], [64, -120, 70, -15, 1], [-1024, 1984, -1240, 310, -31, 1], [32768, -64512, 41664, -11160, 1302, -63, 1], [-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1], [268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -3], [1, -7, 2], [1, -15, 14], [1, -31, 70, -8], [1, -63, 310, -120], [1, -127, 1302, -1240, 64]] | +| AccTabl | [[1], [1, 0], [1, -2, 0], [1, -6, 8, 0], [1, -14, 56, -64, 0], [1, -30, 280, -960, 1024, 0], [1, -62, 1240, -9920, 31744, -32768, 0], [1, -126, 5208, -89280, 666624, -2064384, 2097152, 0], [1, -254, 21336, -755904, 12094464, -87392256, 266338304, -268435456, 0]] | +| RevAccTabl | [[1], [0, 1], [0, -2, 1], [0, 8, -6, 1], [0, -64, 56, -14, 1], [0, 1024, -960, 280, -30, 1], [0, -32768, 31744, -9920, 1240, -62, 1], [0, 2097152, -2064384, 666624, -89280, 5208, -126, 1], [0, -268435456, 266338304, -87392256, 12094464, -755904, 21336, -254, 1]] | +| AccRevTabl | [[1], [-1, 0], [2, -1, 0], [-8, 6, -1, 0], [64, -56, 14, -1, 0], [-1024, 960, -280, 30, -1, 0], [32768, -31744, 9920, -1240, 62, -1, 0], [-2097152, 2064384, -666624, 89280, -5208, 126, -1, 0], [268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 0]] | + +Gaussq2:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 3, 15, 135, 2295, 75735, 4922775, 635037975] | +| OddSum | [0, -1, -3, -15, -135, -2295, -75735, -4922775, -635037975] | +| AltSum | [1, 2, 6, 30, 270, 4590, 151470, 9845550, 1270075950] | +| AccSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | +| AccRevSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | +| AntiDiagSum | [1, 1, 0, -2, -4, 0, 32, 128, 0] | + +Gaussq2:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 56, 6720, 158720, 319979520, 173386235904, 808475248558080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760] | +| ColMiddle | [1, 1, -3, -7, 70, 310, -11160, -94488, 12850368] | +| ColECenter | [1, -3, 70, -11160, 12850368] | +| ColOCenter | [1, -7, 310, -94488] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | +| TransSqrs | [0, -1, 5, -23, 209, -3807, 138177, -9955071, 1421783937] | +| TransNat0 | [0, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | +| TransNat1 | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | + +Gaussq2:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456]| @@ -391,7 +375,7 @@ Gaussq2InvRev Diagonals as rows | DiagRow7 | [1, -255, 86870, -50781720, 55440096448, -117175326428160, 487508875567267840, -4024873276683363287040, 66200105490614595677585408]| | DiagRow8 | [1, -511, 348502, -407647768, 890302725312, -3763855289306112, 31321011405562937344, -517188324145426528731136, 17013427111087951089139449856]| -Gaussq2InvRev Diagonals as columns +Gaussq2:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ Gaussq2InvRev Diagonals as columns | DiagCol7 | [-2097152, -534773760, -91089797120, -13312123207680, -1816661080408064, -239975068524871680, -31200568036305141760, -4024873276683363287040, -517188324145426528731136] | | DiagCol8 | [268435456, 137170518016, 46775146643456, 13678389311307776, 3734200281987022848, 986672080959861424128, 256581725434371582722048, 66200105490614595677585408, 17013427111087951089139449856] | -Gaussq2InvRev Polynomial values as rows +Gaussq2:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ Gaussq2InvRev Polynomial values as rows | PolyRow7 | [1, 0, -78129765, -2157621950, -19923090075, -106889638284, -413184603425, -1281611575530, -3393566342775] | | PolyRow8 | [1, 0, 19923090075, 826369206850, 10180699028325, 68302478863476, 316912590826975, 1147042360099350, 3471618368658825] | -Gaussq2InvRev Polynomial values as columns +Gaussq2:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ Gaussq2InvRev Polynomial values as columns | PolyCol7 | [1, -6, 78, -2106, 115830, -12857130, 2867139990, -1281611575530, 1147042360099350] | | PolyCol8 | [1, -7, 105, -3255, 205065, -26043255, 6641030025, -3393566342775, 3471618368658825] | +# Gaussq2:Rev:Inv +[] + +Gaussq2:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [2, -3, 1] | +| Row3 | [-8, 14, -7, 1] | +| Row4 | [64, -120, 70, -15, 1] | +| Row5 | [-1024, 1984, -1240, 310, -31, 1] | +| Row6 | [32768, -64512, 41664, -11160, 1302, -63, 1] | +| Row7 | [-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1] | +| Row8 | [268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1] | + +Gaussq2:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [2, -3, 1], [-8, 14, -7, 1], [64, -120, 70, -15, 1], [-1024, 1984, -1240, 310, -31, 1], [32768, -64512, 41664, -11160, 1302, -63, 1], [-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1], [268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 2], [1, -7, 14, -8], [1, -15, 70, -120, 64], [1, -31, 310, -1240, 1984, -1024], [1, -63, 1302, -11160, 41664, -64512, 32768], [1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152], [1, -255, 21590, -777240, 12850368, -99486720, 353730560, -534773760, 268435456]] | +| AntiDiag | [[1], [-1], [2, 1], [-8, -3], [64, 14, 1], [-1024, -120, -7], [32768, 1984, 70, 1], [-2097152, -64512, -1240, -15], [268435456, 4161536, 41664, 310, 1]] | +| AccTabl | [[1], [-1, 0], [2, -1, 0], [-8, 6, -1, 0], [64, -56, 14, -1, 0], [-1024, 960, -280, 30, -1, 0], [32768, -31744, 9920, -1240, 62, -1, 0], [-2097152, 2064384, -666624, 89280, -5208, 126, -1, 0], [268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 2], [0, -1, 6, -8], [0, -1, 14, -56, 64], [0, -1, 30, -280, 960, -1024], [0, -1, 62, -1240, 9920, -31744, 32768], [0, -1, 126, -5208, 89280, -666624, 2064384, -2097152], [0, -1, 254, -21336, 755904, -12094464, 87392256, -266338304, 268435456]] | +| AccRevTabl | [[1], [1, 0], [1, -2, 0], [1, -6, 8, 0], [1, -14, 56, -64, 0], [1, -30, 280, -960, 1024, 0], [1, -62, 1240, -9920, 31744, -32768, 0], [1, -126, 5208, -89280, 666624, -2064384, 2097152, 0], [1, -254, 21336, -755904, 12094464, -87392256, 266338304, -268435456, 0]] | + +Gaussq2:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 3, -15, 135, -2295, 75735, -4922775, 635037975] | +| OddSum | [0, 1, -3, 15, -135, 2295, -75735, 4922775, -635037975] | +| AltSum | [1, -2, 6, -30, 270, -4590, 151470, -9845550, 1270075950] | +| AccSum | [1, -1, 1, -3, 21, -315, 9765, -615195, 78129765] | +| AccRevSum | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | +| AntiDiagSum | [1, -1, 3, -11, 79, -1151, 34823, -2162919, 272638967] | + +Gaussq2:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 56, 6720, 158720, 319979520, 173386235904, 808475248558080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760] | +| ColMiddle | [1, -1, -3, 14, 70, -1240, -11160, 755904, 12850368] | +| ColECenter | [1, -3, 70, -11160, 12850368] | +| ColOCenter | [-1, 14, -1240, 755904] | +| ColLeft | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -5, 41, -657, 20997, -1342341, 171707697] | +| TransNat0 | [0, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | +| TransNat1 | [1, 1, -1, 3, -21, 315, -9765, 615195, -78129765] | + +Gaussq2:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -3, -7, -15, -31, -63, -127, -255, -511]| +| DiagRow2 | [2, 14, 70, 310, 1302, 5334, 21590, 86870, 348502]| +| DiagRow3 | [-8, -120, -1240, -11160, -94488, -777240, -6304280, -50781720, -407647768]| +| DiagRow4 | [64, 1984, 41664, 755904, 12850368, 211823808, 3439615168, 55440096448, 890302725312]| +| DiagRow5 | [-1024, -64512, -2731008, -99486720, -3389180928, -111842970624, -3634008902656, -117175326428160, -3763855289306112]| +| DiagRow6 | [32768, 4161536, 353730560, 25822330880, 1761082966016, 116288284884992, 7558738517524480, 487508875567267840, 31321011405562937344]| +| DiagRow7 | [-2097152, -534773760, -91089797120, -13312123207680, -1816661080408064, -239975068524871680, -31200568036305141760, -4024873276683363287040, -517188324145426528731136]| +| DiagRow8 | [268435456, 137170518016, 46775146643456, 13678389311307776, 3734200281987022848, 986672080959861424128, 256581725434371582722048, 66200105490614595677585408, 17013427111087951089139449856]| + +Gaussq2:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | +| DiagCol1 | [1, -3, 14, -120, 1984, -64512, 4161536, -534773760, 137170518016] | +| DiagCol2 | [1, -7, 70, -1240, 41664, -2731008, 353730560, -91089797120, 46775146643456] | +| DiagCol3 | [1, -15, 310, -11160, 755904, -99486720, 25822330880, -13312123207680, 13678389311307776] | +| DiagCol4 | [1, -31, 1302, -94488, 12850368, -3389180928, 1761082966016, -1816661080408064, 3734200281987022848] | +| DiagCol5 | [1, -63, 5334, -777240, 211823808, -111842970624, 116288284884992, -239975068524871680, 986672080959861424128] | +| DiagCol6 | [1, -127, 21590, -6304280, 3439615168, -3634008902656, 7558738517524480, -31200568036305141760, 256581725434371582722048] | +| DiagCol7 | [1, -255, 86870, -50781720, 55440096448, -117175326428160, 487508875567267840, -4024873276683363287040, 66200105490614595677585408] | +| DiagCol8 | [1, -511, 348502, -407647768, 890302725312, -3763855289306112, 31321011405562937344, -517188324145426528731136, 17013427111087951089139449856] | + +Gaussq2:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [2, 0, 0, 2, 6, 12, 20, 30, 42] | +| PolyRow3 | [-8, 0, 0, -2, 0, 12, 40, 90, 168] | +| PolyRow4 | [64, 0, 0, 10, 0, -36, -80, -90, 0] | +| PolyRow5 | [-1024, 0, 0, -130, 0, 396, 800, 810, 0] | +| PolyRow6 | [32768, 0, 0, 3770, 0, -10692, -20800, -20250, 0] | +| PolyRow7 | [-2097152, 0, 0, -229970, 0, 630828, 1206400, 1154250, 0] | +| PolyRow8 | [268435456, 0, 0, 28746250, 0, -77591844, -147180800, -139664250, 0] | + +Gaussq2:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 2, -8, 64, -1024, 32768, -2097152, 268435456] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol3 | [1, 2, 2, -2, 10, -130, 3770, -229970, 28746250] | +| PolyCol4 | [1, 3, 6, 0, 0, 0, 0, 0, 0] | +| PolyCol5 | [1, 4, 12, 12, -36, 396, -10692, 630828, -77591844] | +| PolyCol6 | [1, 5, 20, 40, -80, 800, -20800, 1206400, -147180800] | +| PolyCol7 | [1, 6, 30, 90, -90, 810, -20250, 1154250, -139664250] | +| PolyCol8 | [1, 7, 42, 168, 0, 0, 0, 0, 0] | + diff --git a/data/md/Genocchi.md b/data/md/Genocchi.md index 09350b8..4671a15 100644 --- a/data/md/Genocchi.md +++ b/data/md/Genocchi.md @@ -14,45 +14,41 @@ Genocchi Triangle view | Row7 | [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569] | | Row8 | [5410688, 10623104, 15448416, 19716064, 23281432, 26031912, 27891050, 28820619, 28820619] | -Genocchi Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 3, 3, 8, 14, 17, 17, 56, 104, 138, 155, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569, 5410688, 10623104, 15448416, 19716064, 23281432, 26031912, 27891050, 28820619, 28820619] | -| TriRev | [1, 1, 1, 3, 3, 2, 17, 17, 14, 8, 155, 155, 138, 104, 56, 2073, 2073, 1918, 1608, 1160, 608, 38227, 38227, 36154, 32008, 25944, 18272, 9440, 929569, 929569, 891342, 814888, 702280, 557664, 387104, 198272, 28820619, 28820619, 27891050, 26031912, 23281432, 19716064, 15448416, 10623104, 5410688] | -| TriAcc | [1, 1, 2, 2, 5, 8, 8, 22, 39, 56, 56, 160, 298, 453, 608, 608, 1768, 3376, 5294, 7367, 9440, 9440, 27712, 53656, 85664, 121818, 160045, 198272, 198272, 585376, 1143040, 1845320, 2660208, 3551550, 4481119, 5410688, 5410688, 16033792, 31482208, 51198272, 74479704, 100511616, 128402666, 157223285, 186043904] | -| TriRevAcc | [1, 2, 1, 8, 5, 2, 56, 39, 22, 8, 608, 453, 298, 160, 56, 9440, 7367, 5294, 3376, 1768, 608, 198272, 160045, 121818, 85664, 53656, 27712, 9440, 5410688, 4481119, 3551550, 2660208, 1845320, 1143040, 585376, 198272, 186043904, 157223285, 128402666, 100511616, 74479704, 51198272, 31482208, 16033792, 5410688] | -| TriAccRev | [1, 1, 2, 3, 6, 8, 17, 34, 48, 56, 155, 310, 448, 552, 608, 2073, 4146, 6064, 7672, 8832, 9440, 38227, 76454, 112608, 144616, 170560, 188832, 198272, 929569, 1859138, 2750480, 3565368, 4267648, 4825312, 5212416, 5410688, 28820619, 57641238, 85532288, 111564200, 134845632, 154561696, 170010112, 180633216, 186043904] | -| TriDiag | [1, 1, 2, 1, 8, 3, 56, 14, 3, 608, 104, 17, 9440, 1160, 138, 17, 198272, 18272, 1608, 155] | +Genocchi Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 3, 3], [8, 14, 17, 17], [56, 104, 138, 155, 155], [608, 1160, 1608, 1918, 2073, 2073], [9440, 18272, 25944, 32008, 36154, 38227, 38227], [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569], [5410688, 10623104, 15448416, 19716064, 23281432, 26031912, 27891050, 28820619, 28820619]] | +| RevTabl | [[1], [1, 1], [3, 3, 2], [17, 17, 14, 8], [155, 155, 138, 104, 56], [2073, 2073, 1918, 1608, 1160, 608], [38227, 38227, 36154, 32008, 25944, 18272, 9440], [929569, 929569, 891342, 814888, 702280, 557664, 387104, 198272], [28820619, 28820619, 27891050, 26031912, 23281432, 19716064, 15448416, 10623104, 5410688]] | +| AntiDiag | [[1], [1], [2, 1], [8, 3], [56, 14, 3], [608, 104, 17], [9440, 1160, 138, 17], [198272, 18272, 1608, 155], [5410688, 387104, 25944, 1918, 155]] | +| AccTabl | [[1], [1, 2], [2, 5, 8], [8, 22, 39, 56], [56, 160, 298, 453, 608], [608, 1768, 3376, 5294, 7367, 9440], [9440, 27712, 53656, 85664, 121818, 160045, 198272], [198272, 585376, 1143040, 1845320, 2660208, 3551550, 4481119, 5410688], [5410688, 16033792, 31482208, 51198272, 74479704, 100511616, 128402666, 157223285, 186043904]] | +| RevAccTabl | [[1], [2, 1], [8, 5, 2], [56, 39, 22, 8], [608, 453, 298, 160, 56], [9440, 7367, 5294, 3376, 1768, 608], [198272, 160045, 121818, 85664, 53656, 27712, 9440], [5410688, 4481119, 3551550, 2660208, 1845320, 1143040, 585376, 198272], [186043904, 157223285, 128402666, 100511616, 74479704, 51198272, 31482208, 16033792, 5410688]] | +| AccRevTabl | [[1], [1, 2], [3, 6, 8], [17, 34, 48, 56], [155, 310, 448, 552, 608], [2073, 4146, 6064, 7672, 8832, 9440], [38227, 76454, 112608, 144616, 170560, 188832, 198272], [929569, 1859138, 2750480, 3565368, 4267648, 4825312, 5212416, 5410688], [28820619, 57641238, 85532288, 111564200, 134845632, 154561696, 170010112, 180633216, 186043904]] | Genocchi Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904] | -| EvenSum | [1, 1, 5, 25, 349, 4289, 109765, 2500393, 100852205] | -| OddSum | [0, 1, 3, 31, 259, 5151, 88507, 2910295, 85191699] | -| AltSum | [1, 0, 2, -6, 90, -862, 21258, -409902, 15660506] | -| AccSum | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | -| AccRevSum | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | -| RevAccSum | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | -| DiagSum | [1, 1, 3, 11, 73, 729, 10755, 218307] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904] | +| EvenSum | [1, 1, 5, 25, 349, 4289, 109765, 2500393, 100852205] | +| OddSum | [0, 1, 3, 31, 259, 5151, 88507, 2910295, 85191699] | +| AltSum | [1, 0, 2, -6, 90, -862, 21258, -409902, 15660506] | +| AccSum | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | +| AccRevSum | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | +| AntiDiagSum | [1, 1, 3, 11, 73, 729, 10755, 218307, 5825809] | Genocchi Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 952, 7785960, 11742600707040, 48330376367013905045280, 1394963925815260169445240796533120, 114573109682952121489233528400295906719269475200] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | -| Middle | [1, 1, 3, 14, 138, 1608, 32008, 702280, 23281432] | -| Central | [1, 3, 138, 32008, 23281432] | -| LeftSide | [1, 1, 2, 8, 56, 608, 9440, 198272, 5410688] | -| RightSide | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | -| PosHalf | [1, 3, 17, 171, 2745, 64771, 2119329, 92108379, 5143290377] | -| NegHalf | [1, -1, 5, -25, 461, -8161, 284885, -11096233, 630154589] | -| Bin | [1, 2, 11, 118, 2075, 54106, 1958291, 93874558, 5754873035] | -| InvBin | [1, 0, -1, 0, 3, 0, -17, 0, 155] | -| TransSqrs | [0, 1, 15, 235, 4531, 109847, 3320431, 123273403, 5533969803] | -| TransNat0 | [0, 1, 9, 99, 1465, 28787, 731297, 23409931, 923609001] | -| TransNat1 | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 952, 7785960, 11742600707040, 48330376367013905045280, 1394963925815260169445240796533120, 114573109682952121489233528400295906719269475200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | +| ColMiddle | [1, 1, 3, 14, 138, 1608, 32008, 702280, 23281432] | +| ColECenter | [1, 3, 138, 32008, 23281432] | +| ColOCenter | [1, 14, 1608, 702280] | +| ColLeft | [1, 1, 2, 8, 56, 608, 9440, 198272, 5410688] | +| ColRight | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | +| TransSqrs | [0, 1, 15, 235, 4531, 109847, 3320431, 123273403, 5533969803] | +| TransNat0 | [0, 1, 9, 99, 1465, 28787, 731297, 23409931, 923609001] | +| TransNat1 | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | Genocchi Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Genocchi Polynomial values as columns | PolyCol7 | [1, 8, 170, 6770, 432866, 40563578, 5239042610, 892111352906, 193662522824354] | | PolyCol8 | [1, 9, 218, 9912, 723960, 77513888, 11439891936, 2226072252288, 552241988561792] | -# GenocchiRev -True +# Genocchi:Rev +[] -GenocchiRev Triangle view +Genocchi:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ GenocchiRev Triangle view | Row7 | [929569, 929569, 891342, 814888, 702280, 557664, 387104, 198272] | | Row8 | [28820619, 28820619, 27891050, 26031912, 23281432, 19716064, 15448416, 10623104, 5410688] | -GenocchiRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 3, 3, 2, 17, 17, 14, 8, 155, 155, 138, 104, 56, 2073, 2073, 1918, 1608, 1160, 608, 38227, 38227, 36154, 32008, 25944, 18272, 9440, 929569, 929569, 891342, 814888, 702280, 557664, 387104, 198272, 28820619, 28820619, 27891050, 26031912, 23281432, 19716064, 15448416, 10623104, 5410688] | -| TriRev | [1, 1, 1, 2, 3, 3, 8, 14, 17, 17, 56, 104, 138, 155, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569, 5410688, 10623104, 15448416, 19716064, 23281432, 26031912, 27891050, 28820619, 28820619] | -| TriAcc | [1, 1, 2, 3, 6, 8, 17, 34, 48, 56, 155, 310, 448, 552, 608, 2073, 4146, 6064, 7672, 8832, 9440, 38227, 76454, 112608, 144616, 170560, 188832, 198272, 929569, 1859138, 2750480, 3565368, 4267648, 4825312, 5212416, 5410688, 28820619, 57641238, 85532288, 111564200, 134845632, 154561696, 170010112, 180633216, 186043904] | -| TriRevAcc | [1, 2, 1, 8, 6, 3, 56, 48, 34, 17, 608, 552, 448, 310, 155, 9440, 8832, 7672, 6064, 4146, 2073, 198272, 188832, 170560, 144616, 112608, 76454, 38227, 5410688, 5212416, 4825312, 4267648, 3565368, 2750480, 1859138, 929569, 186043904, 180633216, 170010112, 154561696, 134845632, 111564200, 85532288, 57641238, 28820619] | -| TriAccRev | [1, 1, 2, 2, 5, 8, 8, 22, 39, 56, 56, 160, 298, 453, 608, 608, 1768, 3376, 5294, 7367, 9440, 9440, 27712, 53656, 85664, 121818, 160045, 198272, 198272, 585376, 1143040, 1845320, 2660208, 3551550, 4481119, 5410688, 5410688, 16033792, 31482208, 51198272, 74479704, 100511616, 128402666, 157223285, 186043904] | -| TriDiag | [1, 1, 3, 1, 17, 3, 155, 17, 2, 2073, 155, 14, 38227, 2073, 138, 8, 929569, 38227, 1918, 104] | - -GenocchiRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904] | -| EvenSum | [1, 1, 5, 31, 349, 5151, 109765, 2910295, 100852205] | -| OddSum | [0, 1, 3, 25, 259, 4289, 88507, 2500393, 85191699] | -| AltSum | [1, 0, 2, 6, 90, 862, 21258, 409902, 15660506] | -| AccSum | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | -| AccRevSum | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | -| RevAccSum | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | -| DiagSum | [1, 1, 4, 20, 174, 2242, 40446, 969818] | - -GenocchiRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 952, 7785960, 11742600707040, 48330376367013905045280, 1394963925815260169445240796533120, 114573109682952121489233528400295906719269475200] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | -| Middle | [1, 1, 3, 17, 138, 1918, 32008, 814888, 23281432] | -| Central | [1, 3, 138, 32008, 23281432] | -| LeftSide | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | -| RightSide | [1, 1, 2, 8, 56, 608, 9440, 198272, 5410688] | -| PosHalf | [1, 3, 20, 240, 4536, 124208, 4654080, 228859776, 14303848064] | -| NegHalf | [1, -1, 8, -88, 1640, -43792, 1622336, -78940672, 4901777792] | -| Bin | [1, 2, 11, 118, 2075, 54106, 1958291, 93874558, 5754873035] | -| InvBin | [1, 0, -1, 0, 3, 0, -17, 0, 155] | -| TransSqrs | [0, 1, 11, 145, 2539, 57977, 1682659, 60658081, 2663035643] | -| TransNat0 | [0, 1, 7, 69, 967, 18413, 458335, 14464885, 564742231] | -| TransNat1 | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | - -GenocchiRev Diagonals as rows +Genocchi:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [3, 3, 2], [17, 17, 14, 8], [155, 155, 138, 104, 56], [2073, 2073, 1918, 1608, 1160, 608], [38227, 38227, 36154, 32008, 25944, 18272, 9440], [929569, 929569, 891342, 814888, 702280, 557664, 387104, 198272], [28820619, 28820619, 27891050, 26031912, 23281432, 19716064, 15448416, 10623104, 5410688]] | +| RevTabl | [[1], [1, 1], [2, 3, 3], [8, 14, 17, 17], [56, 104, 138, 155, 155], [608, 1160, 1608, 1918, 2073, 2073], [9440, 18272, 25944, 32008, 36154, 38227, 38227], [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569], [5410688, 10623104, 15448416, 19716064, 23281432, 26031912, 27891050, 28820619, 28820619]] | +| AntiDiag | [[1], [1], [3, 1], [17, 3], [155, 17, 2], [2073, 155, 14], [38227, 2073, 138, 8], [929569, 38227, 1918, 104], [28820619, 929569, 36154, 1608, 56]] | +| AccTabl | [[1], [1, 2], [3, 6, 8], [17, 34, 48, 56], [155, 310, 448, 552, 608], [2073, 4146, 6064, 7672, 8832, 9440], [38227, 76454, 112608, 144616, 170560, 188832, 198272], [929569, 1859138, 2750480, 3565368, 4267648, 4825312, 5212416, 5410688], [28820619, 57641238, 85532288, 111564200, 134845632, 154561696, 170010112, 180633216, 186043904]] | +| RevAccTabl | [[1], [2, 1], [8, 6, 3], [56, 48, 34, 17], [608, 552, 448, 310, 155], [9440, 8832, 7672, 6064, 4146, 2073], [198272, 188832, 170560, 144616, 112608, 76454, 38227], [5410688, 5212416, 4825312, 4267648, 3565368, 2750480, 1859138, 929569], [186043904, 180633216, 170010112, 154561696, 134845632, 111564200, 85532288, 57641238, 28820619]] | +| AccRevTabl | [[1], [1, 2], [2, 5, 8], [8, 22, 39, 56], [56, 160, 298, 453, 608], [608, 1768, 3376, 5294, 7367, 9440], [9440, 27712, 53656, 85664, 121818, 160045, 198272], [198272, 585376, 1143040, 1845320, 2660208, 3551550, 4481119, 5410688], [5410688, 16033792, 31482208, 51198272, 74479704, 100511616, 128402666, 157223285, 186043904]] | + +Genocchi:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 56, 608, 9440, 198272, 5410688, 186043904] | +| EvenSum | [1, 1, 5, 31, 349, 5151, 109765, 2910295, 100852205] | +| OddSum | [0, 1, 3, 25, 259, 4289, 88507, 2500393, 85191699] | +| AltSum | [1, 0, 2, 6, 90, 862, 21258, 409902, 15660506] | +| AccSum | [1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905] | +| AccRevSum | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | +| AntiDiagSum | [1, 1, 4, 20, 174, 2242, 40446, 969818, 29788006] | + +Genocchi:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 952, 7785960, 11742600707040, 48330376367013905045280, 1394963925815260169445240796533120, 114573109682952121489233528400295906719269475200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | +| ColMiddle | [1, 1, 3, 17, 138, 1918, 32008, 814888, 23281432] | +| ColECenter | [1, 3, 138, 32008, 23281432] | +| ColOCenter | [1, 17, 1918, 814888] | +| ColLeft | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | +| ColRight | [1, 1, 2, 8, 56, 608, 9440, 198272, 5410688] | +| TransSqrs | [0, 1, 11, 145, 2539, 57977, 1682659, 60658081, 2663035643] | +| TransNat0 | [0, 1, 7, 69, 967, 18413, 458335, 14464885, 564742231] | +| TransNat1 | [1, 3, 15, 125, 1575, 27853, 656607, 19875573, 750786135] | + +Genocchi:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 8, 56, 608, 9440, 198272, 5410688]| @@ -175,7 +167,7 @@ GenocchiRev Diagonals as rows | DiagRow7 | [929569, 28820619, 1080832286, 48614323016, 2594601005000, 162537878467808, 11829206389864736, 990793141677516416, 94695920709718106240]| | DiagRow8 | [28820619, 1109652905, 50833628826, 2749321197288, 173950709510808, 12756855185141472, 1074917736062704224, 103230184431408446592, 11215719940701859747200]| -GenocchiRev Diagonals as columns +Genocchi:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | @@ -188,7 +180,7 @@ GenocchiRev Diagonals as columns | DiagCol7 | [198272, 10623104, 536687232, 29642785408, 1852978233472, 132131747882112, 10739019716120192, 990793141677516416, 103230184431408446592] | | DiagCol8 | [5410688, 366677120, 22864454016, 1527608809600, 113590431554944, 9500302595955840, 894839340902463872, 94695920709718106240, 11215719940701859747200] | -GenocchiRev Polynomial values as rows +Genocchi:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ GenocchiRev Polynomial values as rows | PolyRow7 | [929569, 5410688, 92108379, 941959042, 5655960197, 23849846964, 79025356063, 220217368214, 538916219817] | | PolyRow8 | [28820619, 186043904, 5143290377, 77740110774, 620326195031, 3265149283964, 12972742238709, 42155406002642, 117861044130659] | -GenocchiRev Polynomial values as columns +Genocchi:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619] | diff --git a/data/md/Harmonic.md b/data/md/Harmonic.md index 6d566f8..c977e12 100644 --- a/data/md/Harmonic.md +++ b/data/md/Harmonic.md @@ -1,5 +1,5 @@ # Harmonic -['A109822', 'A358694'] +['A358694', 'A109822'] Harmonic Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Harmonic Triangle view | Row7 | [0, 5040, 4320, 2556, 932, 197, 22, 1] | | Row8 | [0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1] | -Harmonic Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 18, 7, 1, 0, 120, 96, 46, 11, 1, 0, 720, 600, 326, 101, 16, 1, 0, 5040, 4320, 2556, 932, 197, 22, 1, 0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1] | -| TriRev | [1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 7, 18, 24, 0, 1, 11, 46, 96, 120, 0, 1, 16, 101, 326, 600, 720, 0, 1, 22, 197, 932, 2556, 4320, 5040, 0, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 0] | -| TriAcc | [1, 0, 1, 0, 2, 3, 0, 6, 10, 11, 0, 24, 42, 49, 50, 0, 120, 216, 262, 273, 274, 0, 720, 1320, 1646, 1747, 1763, 1764, 0, 5040, 9360, 11916, 12848, 13045, 13067, 13068, 0, 40320, 75600, 97812, 106892, 109203, 109554, 109583, 109584] | -| TriRevAcc | [1, 1, 0, 3, 2, 0, 11, 10, 6, 0, 50, 49, 42, 24, 0, 274, 273, 262, 216, 120, 0, 1764, 1763, 1747, 1646, 1320, 720, 0, 13068, 13067, 13045, 12848, 11916, 9360, 5040, 0, 109584, 109583, 109554, 109203, 106892, 97812, 75600, 40320, 0] | -| TriAccRev | [1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 8, 26, 50, 50, 1, 12, 58, 154, 274, 274, 1, 17, 118, 444, 1044, 1764, 1764, 1, 23, 220, 1152, 3708, 8028, 13068, 13068, 1, 30, 381, 2692, 11772, 33984, 69264, 109584, 109584] | -| TriDiag | [1, 0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 4, 0, 120, 18, 1, 0, 720, 96, 7] | +Harmonic Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 1], [0, 6, 4, 1], [0, 24, 18, 7, 1], [0, 120, 96, 46, 11, 1], [0, 720, 600, 326, 101, 16, 1], [0, 5040, 4320, 2556, 932, 197, 22, 1], [0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1]] | +| RevTabl | [[1], [1, 0], [1, 2, 0], [1, 4, 6, 0], [1, 7, 18, 24, 0], [1, 11, 46, 96, 120, 0], [1, 16, 101, 326, 600, 720, 0], [1, 22, 197, 932, 2556, 4320, 5040, 0], [1, 29, 351, 2311, 9080, 22212, 35280, 40320, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 6, 1], [0, 24, 4], [0, 120, 18, 1], [0, 720, 96, 7], [0, 5040, 600, 46, 1]] | +| AccTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 10, 11], [0, 24, 42, 49, 50], [0, 120, 216, 262, 273, 274], [0, 720, 1320, 1646, 1747, 1763, 1764], [0, 5040, 9360, 11916, 12848, 13045, 13067, 13068], [0, 40320, 75600, 97812, 106892, 109203, 109554, 109583, 109584]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 0], [11, 10, 6, 0], [50, 49, 42, 24, 0], [274, 273, 262, 216, 120, 0], [1764, 1763, 1747, 1646, 1320, 720, 0], [13068, 13067, 13045, 12848, 11916, 9360, 5040, 0], [109584, 109583, 109554, 109203, 106892, 97812, 75600, 40320, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 11, 11], [1, 8, 26, 50, 50], [1, 12, 58, 154, 274, 274], [1, 17, 118, 444, 1044, 1764, 1764], [1, 23, 220, 1152, 3708, 8028, 13068, 13068], [1, 30, 381, 2692, 11772, 33984, 69264, 109584, 109584]] | Harmonic Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | -| EvenSum | [1, 0, 1, 4, 19, 107, 702, 5274, 44712] | -| OddSum | [0, 1, 2, 7, 31, 167, 1062, 7794, 64872] | -| AltSum | [1, -1, -1, -3, -12, -60, -360, -2520, -20160] | -| AccSum | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | -| AccRevSum | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | -| RevAccSum | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | -| DiagSum | [1, 0, 1, 2, 7, 28, 139, 823] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | +| EvenSum | [1, 0, 1, 4, 19, 107, 702, 5274, 44712] | +| OddSum | [0, 1, 2, 7, 31, 167, 1062, 7794, 64872] | +| AltSum | [1, -1, -1, -3, -12, -60, -360, -2520, -20160] | +| AccSum | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | +| AccRevSum | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | +| AntiDiagSum | [1, 0, 1, 2, 7, 28, 139, 823, 5687] | Harmonic Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 504, 121440, 59266800, 1084064113440, 103321840810930560] | -| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 0, 2, 6, 18, 96, 326, 2556, 9080] | -| Central | [1, 2, 18, 326, 9080] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 33, 279, 2895, 35685, 509985, 8294895] | -| NegHalf | [1, 1, -3, 17, -133, 1315, -15675, 218505, -3485685] | -| Bin | [1, 1, 5, 31, 233, 2076, 21452, 252372, 3329349] | -| InvBin | [1, 1, -3, 7, -15, 46, -420, 5384, -62811] | -| TransSqrs | [0, 1, 6, 31, 175, 1119, 8106, 66002, 598524] | -| TransNat0 | [0, 1, 4, 17, 85, 499, 3388, 26200, 227708] | -| TransNat1 | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 504, 121440, 59266800, 1084064113440, 103321840810930560] | +| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 0, 2, 6, 18, 96, 326, 2556, 9080] | +| ColECenter | [1, 2, 18, 326, 9080] | +| ColOCenter | [0, 6, 96, 2556] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 31, 175, 1119, 8106, 66002, 598524] | +| TransNat0 | [0, 1, 4, 17, 85, 499, 3388, 26200, 227708] | +| TransNat1 | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | Harmonic Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Harmonic Polynomial values as columns | PolyCol7 | [1, 7, 63, 581, 5852, 64540, 775320, 10084200, 141214080] | | PolyCol8 | [1, 8, 80, 816, 9024, 108480, 1411200, 19762560, 296478720] | -# HarmonicInv -True +# Harmonic:Inv +[] -HarmonicInv Triangle view +Harmonic:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ HarmonicInv Triangle view | Row7 | [0, 2, -94, 391, -415, 155, -22, 1] | | Row8 | [0, -2, 190, -1267, 2051, -1190, 287, -29, 1] | -HarmonicInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, -2, 10, -7, 1, 0, 2, -22, 31, -11, 1, 0, -2, 46, -115, 75, -16, 1, 0, 2, -94, 391, -415, 155, -22, 1, 0, -2, 190, -1267, 2051, -1190, 287, -29, 1] | -| TriRev | [1, 1, 0, 1, -2, 0, 1, -4, 2, 0, 1, -7, 10, -2, 0, 1, -11, 31, -22, 2, 0, 1, -16, 75, -115, 46, -2, 0, 1, -22, 155, -415, 391, -94, 2, 0, 1, -29, 287, -1190, 2051, -1267, 190, -2, 0] | -| TriAcc | [1, 0, 1, 0, -2, -1, 0, 2, -2, -1, 0, -2, 8, 1, 2, 0, 2, -20, 11, 0, 1, 0, -2, 44, -71, 4, -12, -11, 0, 2, -92, 299, -116, 39, 17, 18, 0, -2, 188, -1079, 972, -218, 69, 40, 41] | -| TriRevAcc | [1, 1, 0, -1, -2, 0, -1, -2, 2, 0, 2, 1, 8, -2, 0, 1, 0, 11, -20, 2, 0, -11, -12, 4, -71, 44, -2, 0, 18, 17, 39, -116, 299, -92, 2, 0, 41, 40, 69, -218, 972, -1079, 188, -2, 0] | -| TriAccRev | [1, 1, 1, 1, -1, -1, 1, -3, -1, -1, 1, -6, 4, 2, 2, 1, -10, 21, -1, 1, 1, 1, -15, 60, -55, -9, -11, -11, 1, -21, 134, -281, 110, 16, 18, 18, 1, -28, 259, -931, 1120, -147, 43, 41, 41] | -| TriDiag | [1, 0, 0, 1, 0, -2, 0, 2, 1, 0, -2, -4, 0, 2, 10, 1, 0, -2, -22, -7] | - -HarmonicInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -1, 2, 1, -11, 18, 41] | -| EvenSum | [1, 0, 1, -4, 11, -33, 122, -531, 2529] | -| OddSum | [0, 1, -2, 3, -9, 34, -133, 549, -2488] | -| AltSum | [1, -1, 3, -7, 20, -67, 255, -1080, 5017] | -| AccSum | [1, 1, -3, -1, 9, -6, -48, 167, 11] | -| AccRevSum | [1, 1, -3, -1, 9, -6, -48, 167, 11] | -| RevAccSum | [1, 2, -1, -4, 3, 13, -40, -5, 399] | -| DiagSum | [1, 0, 1, -2, 3, -6, 13, -31] | - -HarmonicInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 4, 70, 682, 27600, 5201242310, 1425701518570] | -| RowGcd | [1, 1, -2, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 4, 10, 31, 115, 415, 2051] | -| Middle | [1, 0, -2, 2, 10, -22, -115, 391, 2051] | -| Central | [1, -2, 10, -115, 2051] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, 1, 11, -41, 21, 633, -4253] | -| NegHalf | [1, 1, 5, 17, 71, 355, 2053, 13377, 96503] | -| Bin | [1, 1, -3, -5, 25, 46, -592, 302, 19087] | -| InvBin | [1, 1, 5, 19, 97, 596, 4224, 33608, 294767] | -| TransSqrs | [0, 1, 2, -5, -9, 42, -17, -363, 1396] | -| TransNat0 | [0, 1, 0, -3, 1, 12, -29, -23, 358] | -| TransNat1 | [1, 2, -1, -4, 3, 13, -40, -5, 399] | - -HarmonicInv Diagonals as rows +Harmonic:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -2, 1], [0, 2, -4, 1], [0, -2, 10, -7, 1], [0, 2, -22, 31, -11, 1], [0, -2, 46, -115, 75, -16, 1], [0, 2, -94, 391, -415, 155, -22, 1], [0, -2, 190, -1267, 2051, -1190, 287, -29, 1]] | +| RevTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 2, 0], [1, -7, 10, -2, 0], [1, -11, 31, -22, 2, 0], [1, -16, 75, -115, 46, -2, 0], [1, -22, 155, -415, 391, -94, 2, 0], [1, -29, 287, -1190, 2051, -1267, 190, -2, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -2], [0, 2, 1], [0, -2, -4], [0, 2, 10, 1], [0, -2, -22, -7], [0, 2, 46, 31, 1]] | +| AccTabl | [[1], [0, 1], [0, -2, -1], [0, 2, -2, -1], [0, -2, 8, 1, 2], [0, 2, -20, 11, 0, 1], [0, -2, 44, -71, 4, -12, -11], [0, 2, -92, 299, -116, 39, 17, 18], [0, -2, 188, -1079, 972, -218, 69, 40, 41]] | +| RevAccTabl | [[1], [1, 0], [-1, -2, 0], [-1, -2, 2, 0], [2, 1, 8, -2, 0], [1, 0, 11, -20, 2, 0], [-11, -12, 4, -71, 44, -2, 0], [18, 17, 39, -116, 299, -92, 2, 0], [41, 40, 69, -218, 972, -1079, 188, -2, 0]] | +| AccRevTabl | [[1], [1, 1], [1, -1, -1], [1, -3, -1, -1], [1, -6, 4, 2, 2], [1, -10, 21, -1, 1, 1], [1, -15, 60, -55, -9, -11, -11], [1, -21, 134, -281, 110, 16, 18, 18], [1, -28, 259, -931, 1120, -147, 43, 41, 41]] | + +Harmonic:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -1, 2, 1, -11, 18, 41] | +| EvenSum | [1, 0, 1, -4, 11, -33, 122, -531, 2529] | +| OddSum | [0, 1, -2, 3, -9, 34, -133, 549, -2488] | +| AltSum | [1, -1, 3, -7, 20, -67, 255, -1080, 5017] | +| AccSum | [1, 1, -3, -1, 9, -6, -48, 167, 11] | +| AccRevSum | [1, 2, -1, -4, 3, 13, -40, -5, 399] | +| AntiDiagSum | [1, 0, 1, -2, 3, -6, 13, -31, 80] | + +Harmonic:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 4, 70, 682, 27600, 5201242310, 1425701518570] | +| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 4, 10, 31, 115, 415, 2051] | +| ColMiddle | [1, 0, -2, 2, 10, -22, -115, 391, 2051] | +| ColECenter | [1, -2, 10, -115, 2051] | +| ColOCenter | [0, 2, -22, 391] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -5, -9, 42, -17, -363, 1396] | +| TransNat0 | [0, 1, 0, -3, 1, 12, -29, -23, 358] | +| TransNat1 | [1, 2, -1, -4, 3, 13, -40, -5, 399] | + +Harmonic:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ HarmonicInv Diagonals as rows | DiagRow7 | [0, -2, -382, -12355, -179855, -1626130, -10644964, -55044704, -237540160]| | DiagRow8 | [0, 2, 766, 37831, 757251, 8887901, 72757685, 458070613, 2358391893]| -HarmonicInv Diagonals as columns +Harmonic:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ HarmonicInv Diagonals as columns | DiagCol7 | [1, -29, 490, -6342, 69867, -691383, 6342820, -55044704, 458070613] | | DiagCol8 | [1, -37, 786, -12630, 170907, -2058639, 22811932, -237540160, 2358391893] | -HarmonicInv Polynomial values as rows +Harmonic:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ HarmonicInv Polynomial values as rows | PolyRow7 | [0, 18, -204, -84, 2280, 5910, 2028, -26544, -96624] | | PolyRow8 | [0, 41, 268, -3183, -5608, 18865, 89556, 146153, -34448] | -HarmonicInv Polynomial values as columns +Harmonic:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ HarmonicInv Polynomial values as columns | PolyCol7 | [1, 7, 35, 161, 476, -35, -8393, -26544, 146153] | | PolyCol8 | [1, 8, 48, 272, 1136, 2192, -10896, -96624, -34448] | -# HarmonicRev -True +# Harmonic:Rev +[] -HarmonicRev Triangle view +Harmonic:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ HarmonicRev Triangle view | Row7 | [1, 22, 197, 932, 2556, 4320, 5040, 0] | | Row8 | [1, 29, 351, 2311, 9080, 22212, 35280, 40320, 0] | -HarmonicRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 7, 18, 24, 0, 1, 11, 46, 96, 120, 0, 1, 16, 101, 326, 600, 720, 0, 1, 22, 197, 932, 2556, 4320, 5040, 0, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 0] | -| TriRev | [1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 18, 7, 1, 0, 120, 96, 46, 11, 1, 0, 720, 600, 326, 101, 16, 1, 0, 5040, 4320, 2556, 932, 197, 22, 1, 0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1] | -| TriAcc | [1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 8, 26, 50, 50, 1, 12, 58, 154, 274, 274, 1, 17, 118, 444, 1044, 1764, 1764, 1, 23, 220, 1152, 3708, 8028, 13068, 13068, 1, 30, 381, 2692, 11772, 33984, 69264, 109584, 109584] | -| TriRevAcc | [1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 50, 50, 26, 8, 1, 274, 274, 154, 58, 12, 1, 1764, 1764, 1044, 444, 118, 17, 1, 13068, 13068, 8028, 3708, 1152, 220, 23, 1, 109584, 109584, 69264, 33984, 11772, 2692, 381, 30, 1] | -| TriAccRev | [1, 0, 1, 0, 2, 3, 0, 6, 10, 11, 0, 24, 42, 49, 50, 0, 120, 216, 262, 273, 274, 0, 720, 1320, 1646, 1747, 1763, 1764, 0, 5040, 9360, 11916, 12848, 13045, 13067, 13068, 0, 40320, 75600, 97812, 106892, 109203, 109554, 109583, 109584] | -| TriDiag | [1, 1, 1, 0, 1, 2, 1, 4, 0, 1, 7, 6, 1, 11, 18, 0, 1, 16, 46, 24] | - -HarmonicRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | -| EvenSum | [1, 1, 1, 7, 19, 167, 702, 7794, 44712] | -| OddSum | [0, 0, 2, 4, 31, 107, 1062, 5274, 64872] | -| AltSum | [1, 1, -1, 3, -12, 60, -360, 2520, -20160] | -| AccSum | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | -| AccRevSum | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | -| RevAccSum | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | -| DiagSum | [1, 1, 1, 3, 5, 14, 30, 87] | - -HarmonicRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 504, 121440, 59266800, 1084064113440, 103321840810930560] | -| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 1, 2, 4, 18, 46, 326, 932, 9080] | -| Central | [1, 2, 18, 326, 9080] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 8, 36, 192, 1200, 8640, 70560, 645120] | -| NegHalf | [1, -2, 0, -4, -16, -80, -480, -3360, -26880] | -| Bin | [1, 1, 5, 31, 233, 2076, 21452, 252372, 3329349] | -| InvBin | [1, -1, -3, -7, -15, -46, -420, -5384, -62811] | -| TransSqrs | [0, 0, 2, 28, 295, 2979, 30954, 339534, 3968572] | -| TransNat0 | [0, 0, 2, 16, 115, 871, 7196, 65276, 648964] | -| TransNat1 | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | - -HarmonicRev Diagonals as rows +Harmonic:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 2, 0], [1, 4, 6, 0], [1, 7, 18, 24, 0], [1, 11, 46, 96, 120, 0], [1, 16, 101, 326, 600, 720, 0], [1, 22, 197, 932, 2556, 4320, 5040, 0], [1, 29, 351, 2311, 9080, 22212, 35280, 40320, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 1], [0, 6, 4, 1], [0, 24, 18, 7, 1], [0, 120, 96, 46, 11, 1], [0, 720, 600, 326, 101, 16, 1], [0, 5040, 4320, 2556, 932, 197, 22, 1], [0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 2], [1, 4, 0], [1, 7, 6], [1, 11, 18, 0], [1, 16, 46, 24], [1, 22, 101, 96, 0]] | +| AccTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 11, 11], [1, 8, 26, 50, 50], [1, 12, 58, 154, 274, 274], [1, 17, 118, 444, 1044, 1764, 1764], [1, 23, 220, 1152, 3708, 8028, 13068, 13068], [1, 30, 381, 2692, 11772, 33984, 69264, 109584, 109584]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 1], [11, 11, 5, 1], [50, 50, 26, 8, 1], [274, 274, 154, 58, 12, 1], [1764, 1764, 1044, 444, 118, 17, 1], [13068, 13068, 8028, 3708, 1152, 220, 23, 1], [109584, 109584, 69264, 33984, 11772, 2692, 381, 30, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 10, 11], [0, 24, 42, 49, 50], [0, 120, 216, 262, 273, 274], [0, 720, 1320, 1646, 1747, 1763, 1764], [0, 5040, 9360, 11916, 12848, 13045, 13067, 13068], [0, 40320, 75600, 97812, 106892, 109203, 109554, 109583, 109584]] | + +Harmonic:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | +| EvenSum | [1, 1, 1, 7, 19, 167, 702, 7794, 44712] | +| OddSum | [0, 0, 2, 4, 31, 107, 1062, 5274, 64872] | +| AltSum | [1, 1, -1, 3, -12, 60, -360, 2520, -20160] | +| AccSum | [1, 2, 7, 28, 135, 773, 5152, 39268, 337292] | +| AccRevSum | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | +| AntiDiagSum | [1, 1, 1, 3, 5, 14, 30, 87, 220] | + +Harmonic:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 504, 121440, 59266800, 1084064113440, 103321840810930560] | +| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 1, 2, 4, 18, 46, 326, 932, 9080] | +| ColECenter | [1, 2, 18, 326, 9080] | +| ColOCenter | [1, 4, 46, 932] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 28, 295, 2979, 30954, 339534, 3968572] | +| TransNat0 | [0, 0, 2, 16, 115, 871, 7196, 65276, 648964] | +| TransNat1 | [1, 1, 5, 27, 165, 1145, 8960, 78344, 758548] | + +Harmonic:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ HarmonicRev Diagonals as rows | DiagRow7 | [1, 29, 583, 10366, 177299, 3029643, 52724894, 944218666, 17499398776]| | DiagRow8 | [1, 37, 916, 19526, 392085, 7734663, 153275513, 3090075848, 63850536496]| -HarmonicRev Diagonals as columns +Harmonic:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ HarmonicRev Diagonals as columns | DiagCol7 | [0, 40320, 322560, 2239344, 12905784, 62364908, 258795044, 944218666, 3090075848] | | DiagCol8 | [0, 362880, 3265920, 25659360, 167622984, 916001880, 4280337452, 17499398776, 63850536496] | -HarmonicRev Polynomial values as rows +Harmonic:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ HarmonicRev Polynomial values as rows | PolyRow7 | [1, 13068, 509985, 4957960, 25784745, 93969036, 272259673, 672023640, 1473722865] | | PolyRow8 | [1, 109584, 8294895, 120097600, 830332965, 3776635296, 13118043379, 37751838720, 94571849385] | -HarmonicRev Polynomial values as columns +Harmonic:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ HarmonicRev Polynomial values as columns | PolyCol7 | [1, 1, 15, 323, 9164, 323380, 13658520, 672023640, 37751838720] | | PolyCol8 | [1, 1, 17, 417, 13497, 543705, 26224065, 1473722865, 94571849385] | -# HarmonicInvRev -True +# Harmonic:Inv:Rev +[] -HarmonicInvRev Triangle view +Harmonic:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ HarmonicInvRev Triangle view | Row7 | [1, -22, 155, -415, 391, -94, 2, 0] | | Row8 | [1, -29, 287, -1190, 2051, -1267, 190, -2, 0] | -HarmonicInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -2, 0, 1, -4, 2, 0, 1, -7, 10, -2, 0, 1, -11, 31, -22, 2, 0, 1, -16, 75, -115, 46, -2, 0, 1, -22, 155, -415, 391, -94, 2, 0, 1, -29, 287, -1190, 2051, -1267, 190, -2, 0] | -| TriRev | [1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, -2, 10, -7, 1, 0, 2, -22, 31, -11, 1, 0, -2, 46, -115, 75, -16, 1, 0, 2, -94, 391, -415, 155, -22, 1, 0, -2, 190, -1267, 2051, -1190, 287, -29, 1] | -| TriAcc | [1, 1, 1, 1, -1, -1, 1, -3, -1, -1, 1, -6, 4, 2, 2, 1, -10, 21, -1, 1, 1, 1, -15, 60, -55, -9, -11, -11, 1, -21, 134, -281, 110, 16, 18, 18, 1, -28, 259, -931, 1120, -147, 43, 41, 41] | -| TriRevAcc | [1, 1, 1, -1, -1, 1, -1, -1, -3, 1, 2, 2, 4, -6, 1, 1, 1, -1, 21, -10, 1, -11, -11, -9, -55, 60, -15, 1, 18, 18, 16, 110, -281, 134, -21, 1, 41, 41, 43, -147, 1120, -931, 259, -28, 1] | -| TriAccRev | [1, 0, 1, 0, -2, -1, 0, 2, -2, -1, 0, -2, 8, 1, 2, 0, 2, -20, 11, 0, 1, 0, -2, 44, -71, 4, -12, -11, 0, 2, -92, 299, -116, 39, 17, 18, 0, -2, 188, -1079, 972, -218, 69, 40, 41] | -| TriDiag | [1, 1, 1, 0, 1, -2, 1, -4, 0, 1, -7, 2, 1, -11, 10, 0, 1, -16, 31, -2] | - -HarmonicInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -1, 2, 1, -11, 18, 41] | -| EvenSum | [1, 1, 1, 3, 11, 34, 122, 549, 2529] | -| OddSum | [0, 0, -2, -4, -9, -33, -133, -531, -2488] | -| AltSum | [1, 1, 3, 7, 20, 67, 255, 1080, 5017] | -| AccSum | [1, 2, -1, -4, 3, 13, -40, -5, 399] | -| AccRevSum | [1, 2, -1, -4, 3, 13, -40, -5, 399] | -| RevAccSum | [1, 1, -3, -1, 9, -6, -48, 167, 11] | -| DiagSum | [1, 1, 1, -1, -3, -4, 0, 14] | - -HarmonicInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 4, 70, 682, 27600, 5201242310, 1425701518570] | -| RowGcd | [1, 1, -2, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 4, 10, 31, 115, 415, 2051] | -| Middle | [1, 1, -2, -4, 10, 31, -115, -415, 2051] | -| Central | [1, -2, 10, -115, 2051] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 0, -4, -4, 20, 12, -204, 268] | -| NegHalf | [1, -2, 8, -28, 116, -548, 2884, -16644, 104132] | -| Bin | [1, 1, -3, -5, 25, 46, -592, 302, 19087] | -| InvBin | [1, -1, 5, -19, 97, -596, 4224, -33608, 294767] | -| TransSqrs | [0, 0, -2, 4, 15, -53, -65, 841, -1708] | -| TransNat0 | [0, 0, -2, 0, 7, -7, -37, 149, -30] | -| TransNat1 | [1, 1, -3, -1, 9, -6, -48, 167, 11] | - -HarmonicInvRev Diagonals as rows +Harmonic:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -2, 0], [1, -4, 2, 0], [1, -7, 10, -2, 0], [1, -11, 31, -22, 2, 0], [1, -16, 75, -115, 46, -2, 0], [1, -22, 155, -415, 391, -94, 2, 0], [1, -29, 287, -1190, 2051, -1267, 190, -2, 0]] | +| RevTabl | [[1], [0, 1], [0, -2, 1], [0, 2, -4, 1], [0, -2, 10, -7, 1], [0, 2, -22, 31, -11, 1], [0, -2, 46, -115, 75, -16, 1], [0, 2, -94, 391, -415, 155, -22, 1], [0, -2, 190, -1267, 2051, -1190, 287, -29, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -2], [1, -4, 0], [1, -7, 2], [1, -11, 10, 0], [1, -16, 31, -2], [1, -22, 75, -22, 0]] | +| AccTabl | [[1], [1, 1], [1, -1, -1], [1, -3, -1, -1], [1, -6, 4, 2, 2], [1, -10, 21, -1, 1, 1], [1, -15, 60, -55, -9, -11, -11], [1, -21, 134, -281, 110, 16, 18, 18], [1, -28, 259, -931, 1120, -147, 43, 41, 41]] | +| RevAccTabl | [[1], [1, 1], [-1, -1, 1], [-1, -1, -3, 1], [2, 2, 4, -6, 1], [1, 1, -1, 21, -10, 1], [-11, -11, -9, -55, 60, -15, 1], [18, 18, 16, 110, -281, 134, -21, 1], [41, 41, 43, -147, 1120, -931, 259, -28, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -2, -1], [0, 2, -2, -1], [0, -2, 8, 1, 2], [0, 2, -20, 11, 0, 1], [0, -2, 44, -71, 4, -12, -11], [0, 2, -92, 299, -116, 39, 17, 18], [0, -2, 188, -1079, 972, -218, 69, 40, 41]] | + +Harmonic:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -1, 2, 1, -11, 18, 41] | +| EvenSum | [1, 1, 1, 3, 11, 34, 122, 549, 2529] | +| OddSum | [0, 0, -2, -4, -9, -33, -133, -531, -2488] | +| AltSum | [1, 1, 3, 7, 20, 67, 255, 1080, 5017] | +| AccSum | [1, 2, -1, -4, 3, 13, -40, -5, 399] | +| AccRevSum | [1, 1, -3, -1, 9, -6, -48, 167, 11] | +| AntiDiagSum | [1, 1, 1, -1, -3, -4, 0, 14, 32] | + +Harmonic:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 4, 70, 682, 27600, 5201242310, 1425701518570] | +| RowGcd | [1, 1, 2, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 4, 10, 31, 115, 415, 2051] | +| ColMiddle | [1, 1, -2, -4, 10, 31, -115, -415, 2051] | +| ColECenter | [1, -2, 10, -115, 2051] | +| ColOCenter | [1, -4, 31, -415] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -2, 4, 15, -53, -65, 841, -1708] | +| TransNat0 | [0, 0, -2, 0, 7, -7, -37, 149, -30] | +| TransNat1 | [1, 1, -3, -1, 9, -6, -48, 167, 11] | + +Harmonic:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ HarmonicInvRev Diagonals as rows | DiagRow7 | [1, -29, 490, -6342, 69867, -691383, 6342820, -55044704, 458070613]| | DiagRow8 | [1, -37, 786, -12630, 170907, -2058639, 22811932, -237540160, 2358391893]| -HarmonicInvRev Diagonals as columns +Harmonic:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ HarmonicInvRev Diagonals as columns | DiagCol7 | [0, -2, -382, -12355, -179855, -1626130, -10644964, -55044704, -237540160] | | DiagCol8 | [0, 2, 766, 37831, 757251, 8887901, 72757685, 458070613, 2358391893] | -HarmonicInvRev Polynomial values as rows +Harmonic:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ HarmonicInvRev Polynomial values as rows | PolyRow7 | [1, 18, 633, 412, -12135, -66234, -215087, -540672, -1157103] | | PolyRow8 | [1, 41, -4253, -37247, -98563, -6719, 863791, 3941897, 11905753] | -HarmonicInvRev Polynomial values as columns +Harmonic:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/HermiteE.md b/data/md/HermiteE.md index dc0529a..163d82e 100644 --- a/data/md/HermiteE.md +++ b/data/md/HermiteE.md @@ -1,5 +1,5 @@ # HermiteE -['A066325', 'A073278', 'A099174'] +['A099174', 'A066325', 'A073278'] HermiteE Triangle view | Row | Seq | @@ -14,45 +14,41 @@ HermiteE Triangle view | Row7 | [0, 105, 0, 105, 0, 21, 0, 1] | | Row8 | [105, 0, 420, 0, 210, 0, 28, 0, 1] | -HermiteE Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 3, 9, 9, 10, 0, 15, 15, 25, 25, 26, 15, 15, 60, 60, 75, 75, 76, 0, 105, 105, 210, 210, 231, 231, 232, 105, 105, 525, 525, 735, 735, 763, 763, 764] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 4, 3, 3, 0, 10, 9, 9, 3, 3, 26, 25, 25, 15, 15, 0, 76, 75, 75, 60, 60, 15, 15, 232, 231, 231, 210, 210, 105, 105, 0, 764, 763, 763, 735, 735, 525, 525, 105, 105] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 10, 1, 1, 11, 11, 26, 26, 1, 1, 16, 16, 61, 61, 76, 1, 1, 22, 22, 127, 127, 232, 232, 1, 1, 29, 29, 239, 239, 659, 659, 764] | -| TriDiag | [1, 0, 1, 1, 0, 0, 3, 3, 1, 0, 0, 0, 15, 15, 6, 1, 0, 0, 0, 0] | +HermiteE Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [3, 0, 6, 0, 1], [0, 15, 0, 10, 0, 1], [15, 0, 45, 0, 15, 0, 1], [0, 105, 0, 105, 0, 21, 0, 1], [105, 0, 420, 0, 210, 0, 28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 3], [1, 0, 10, 0, 15, 0], [1, 0, 15, 0, 45, 0, 15], [1, 0, 21, 0, 105, 0, 105, 0], [1, 0, 28, 0, 210, 0, 420, 0, 105]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [3, 3, 1], [0, 0, 0], [15, 15, 6, 1], [0, 0, 0, 0], [105, 105, 45, 10, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [3, 3, 9, 9, 10], [0, 15, 15, 25, 25, 26], [15, 15, 60, 60, 75, 75, 76], [0, 105, 105, 210, 210, 231, 231, 232], [105, 105, 525, 525, 735, 735, 763, 763, 764]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [4, 3, 3, 0], [10, 9, 9, 3, 3], [26, 25, 25, 15, 15, 0], [76, 75, 75, 60, 60, 15, 15], [232, 231, 231, 210, 210, 105, 105, 0], [764, 763, 763, 735, 735, 525, 525, 105, 105]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 10], [1, 1, 11, 11, 26, 26], [1, 1, 16, 16, 61, 61, 76], [1, 1, 22, 22, 127, 127, 232, 232], [1, 1, 29, 29, 239, 239, 659, 659, 764]] | HermiteE Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| EvenSum | [1, 0, 2, 0, 10, 0, 76, 0, 764] | -| OddSum | [0, 1, 0, 4, 0, 26, 0, 232, 0] | -| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | -| AccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| AccRevSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| RevAccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| DiagSum | [1, 0, 2, 0, 7, 0, 37, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| EvenSum | [1, 0, 2, 0, 10, 0, 76, 0, 764] | +| OddSum | [0, 1, 0, 4, 0, 26, 0, 232, 0] | +| AltSum | [1, -1, 2, -4, 10, -26, 76, -232, 764] | +| AccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | +| AccRevSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | +| AntiDiagSum | [1, 0, 2, 0, 7, 0, 37, 0, 266] | HermiteE Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 0, 0, 3, 6, 0, 0, 105, 210] | -| Central | [1, 0, 6, 0, 210] | -| LeftSide | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 13, 73, 281, 1741, 8485, 57233] | -| NegHalf | [1, 1, 5, 13, 73, 281, 1741, 8485, 57233] | -| Bin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| InvBin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| TransSqrs | [0, 1, 4, 12, 40, 130, 456, 1624, 6112] | -| TransNat0 | [0, 1, 2, 6, 16, 50, 156, 532, 1856] | -| TransNat1 | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 0, 0, 3, 6, 0, 0, 105, 210] | +| ColECenter | [1, 0, 6, 0, 210] | +| ColOCenter | [0, 3, 0, 105] | +| ColLeft | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 12, 40, 130, 456, 1624, 6112] | +| TransNat0 | [0, 1, 2, 6, 16, 50, 156, 532, 1856] | +| TransNat1 | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | HermiteE Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ HermiteE Polynomial values as columns | PolyCol7 | [1, 7, 50, 364, 2698, 20342, 155884, 1213240, 9583868] | | PolyCol8 | [1, 8, 65, 536, 4483, 38008, 326479, 2839880, 25004393] | -# HermiteEInv -True +# HermiteE:Inv +[] -HermiteEInv Triangle view +HermiteE:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ HermiteEInv Triangle view | Row7 | [0, -105, 0, 105, 0, -21, 0, 1] | | Row8 | [105, 0, -420, 0, 210, 0, -28, 0, 1] | -HermiteEInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 15, 0, -10, 0, 1, -15, 0, 45, 0, -15, 0, 1, 0, -105, 0, 105, 0, -21, 0, 1, 105, 0, -420, 0, 210, 0, -28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 3, 1, 0, -10, 0, 15, 0, 1, 0, -15, 0, 45, 0, -15, 1, 0, -21, 0, 105, 0, -105, 0, 1, 0, -28, 0, 210, 0, -420, 0, 105] | -| TriAcc | [1, 0, 1, -1, -1, 0, 0, -3, -3, -2, 3, 3, -3, -3, -2, 0, 15, 15, 5, 5, 6, -15, -15, 30, 30, 15, 15, 16, 0, -105, -105, 0, 0, -21, -21, -20, 105, 105, -315, -315, -105, -105, -133, -133, -132] | -| TriRevAcc | [1, 1, 0, 0, -1, -1, -2, -3, -3, 0, -2, -3, -3, 3, 3, 6, 5, 5, 15, 15, 0, 16, 15, 15, 30, 30, -15, -15, -20, -21, -21, 0, 0, -105, -105, 0, -132, -133, -133, -105, -105, -315, -315, 105, 105] | -| TriAccRev | [1, 1, 1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -5, -5, -2, 1, 1, -9, -9, 6, 6, 1, 1, -14, -14, 31, 31, 16, 1, 1, -20, -20, 85, 85, -20, -20, 1, 1, -27, -27, 183, 183, -237, -237, -132] | -| TriDiag | [1, 0, -1, 1, 0, 0, 3, -3, 1, 0, 0, 0, -15, 15, -6, 1, 0, 0, 0, 0] | - -HermiteEInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| EvenSum | [1, 0, 0, 0, -2, 0, 16, 0, -132] | -| OddSum | [0, 1, 0, -2, 0, 6, 0, -20, 0] | -| AltSum | [1, -1, 0, 2, -2, -6, 16, 20, -132] | -| AccSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | -| AccRevSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | -| RevAccSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | -| DiagSum | [1, 0, 0, 0, 1, 0, -5, 0] | - -HermiteEInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, -3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 0, 0, -3, -6, 0, 0, 105, 210] | -| Central | [1, 0, -6, 0, 210] | -| LeftSide | [1, 0, -1, 0, 3, 0, -15, 0, 105] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, -11, 25, 201, -299, -5123, 3249] | -| NegHalf | [1, 1, -3, -11, 25, 201, -299, -5123, 3249] | -| Bin | [1, 1, 0, -8, -32, -24, 436, 2500, 2262] | -| InvBin | [1, 1, 0, -8, -32, -24, 436, 2500, 2262] | -| TransSqrs | [0, 1, 4, 6, -8, -50, -24, 364, 736] | -| TransNat0 | [0, 1, 2, 0, -8, -10, 36, 112, -160] | -| TransNat1 | [1, 2, 2, -2, -10, -4, 52, 92, -292] | - -HermiteEInv Diagonals as rows +HermiteE:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [3, 0, -6, 0, 1], [0, 15, 0, -10, 0, 1], [-15, 0, 45, 0, -15, 0, 1], [0, -105, 0, 105, 0, -21, 0, 1], [105, 0, -420, 0, 210, 0, -28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 3], [1, 0, -10, 0, 15, 0], [1, 0, -15, 0, 45, 0, -15], [1, 0, -21, 0, 105, 0, -105, 0], [1, 0, -28, 0, 210, 0, -420, 0, 105]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [3, -3, 1], [0, 0, 0], [-15, 15, -6, 1], [0, 0, 0, 0], [105, -105, 45, -10, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [3, 3, -3, -3, -2], [0, 15, 15, 5, 5, 6], [-15, -15, 30, 30, 15, 15, 16], [0, -105, -105, 0, 0, -21, -21, -20], [105, 105, -315, -315, -105, -105, -133, -133, -132]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-2, -3, -3, 0], [-2, -3, -3, 3, 3], [6, 5, 5, 15, 15, 0], [16, 15, 15, 30, 30, -15, -15], [-20, -21, -21, 0, 0, -105, -105, 0], [-132, -133, -133, -105, -105, -315, -315, 105, 105]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, -2], [1, 1, -9, -9, 6, 6], [1, 1, -14, -14, 31, 31, 16], [1, 1, -20, -20, 85, 85, -20, -20], [1, 1, -27, -27, 183, 183, -237, -237, -132]] | + +HermiteE:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| EvenSum | [1, 0, 0, 0, -2, 0, 16, 0, -132] | +| OddSum | [0, 1, 0, -2, 0, 6, 0, -20, 0] | +| AltSum | [1, -1, 0, 2, -2, -6, 16, 20, -132] | +| AccSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | +| AccRevSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | +| AntiDiagSum | [1, 0, 0, 0, 1, 0, -5, 0, 36] | + +HermiteE:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 0, 0, -3, -6, 0, 0, 105, 210] | +| ColECenter | [1, 0, -6, 0, 210] | +| ColOCenter | [0, -3, 0, 105] | +| ColLeft | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 6, -8, -50, -24, 364, 736] | +| TransNat0 | [0, 1, 2, 0, -8, -10, 36, 112, -160] | +| TransNat1 | [1, 2, 2, -2, -10, -4, 52, 92, -292] | + +HermiteE:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ HermiteEInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [105, 945, 4725, 17325, 51975, 135135, 315315, 675675, 1351350]| -HermiteEInv Diagonals as columns +HermiteE:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, -1, 0, 3, 0, -15, 0, 105] | @@ -188,7 +180,7 @@ HermiteEInv Diagonals as columns | DiagCol7 | [1, 0, -36, 0, 990, 0, -25740, 0, 675675] | | DiagCol8 | [1, 0, -45, 0, 1485, 0, -45045, 0, 1351350] | -HermiteEInv Polynomial values as rows +HermiteE:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ HermiteEInv Polynomial values as rows | PolyRow7 | [0, -20, 86, -396, 1180, 25100, 138690, 505876, 1461944] | | PolyRow8 | [105, -132, 249, -516, -2007, 73980, 630393, 2954364, 10270569] | -HermiteEInv Polynomial values as columns +HermiteE:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, -1, 0, 3, 0, -15, 0, 105] | @@ -214,10 +206,10 @@ HermiteEInv Polynomial values as columns | PolyCol7 | [1, 7, 48, 322, 2110, 13482, 83824, 505876, 2954364] | | PolyCol8 | [1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569] | -# HermiteERev -True +# HermiteE:Rev +[] -HermiteERev Triangle view +HermiteE:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ HermiteERev Triangle view | Row7 | [1, 0, 21, 0, 105, 0, 105, 0] | | Row8 | [1, 0, 28, 0, 210, 0, 420, 0, 105] | -HermiteERev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 10, 1, 1, 11, 11, 26, 26, 1, 1, 16, 16, 61, 61, 76, 1, 1, 22, 22, 127, 127, 232, 232, 1, 1, 29, 29, 239, 239, 659, 659, 764] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 7, 7, 1, 1, 26, 26, 11, 11, 1, 1, 76, 61, 61, 16, 16, 1, 1, 232, 232, 127, 127, 22, 22, 1, 1, 764, 659, 659, 239, 239, 29, 29, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 3, 9, 9, 10, 0, 15, 15, 25, 25, 26, 15, 15, 60, 60, 75, 75, 76, 0, 105, 105, 210, 210, 231, 231, 232, 105, 105, 525, 525, 735, 735, 763, 763, 764] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 6, 0, 1, 0, 10, 0] | - -HermiteERev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| EvenSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | -| AccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| AccRevSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | -| RevAccSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | -| DiagSum | [1, 1, 1, 1, 2, 4, 7, 11] | - -HermiteERev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 1, 0, 0, 6, 10, 0, 0, 210] | -| Central | [1, 0, 6, 0, 210] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| PosHalf | [1, 2, 5, 14, 43, 142, 499, 1850, 7193] | -| NegHalf | [1, -2, 5, -14, 43, -142, 499, -1850, 7193] | -| Bin | [1, 1, 2, 10, 40, 176, 916, 4852, 27350] | -| InvBin | [1, -1, 2, -10, 40, -176, 916, -4852, 27350] | -| TransSqrs | [0, 0, 4, 12, 72, 280, 1320, 5544, 25312] | -| TransNat0 | [0, 0, 2, 6, 24, 80, 300, 1092, 4256] | -| TransNat1 | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | - -HermiteERev Diagonals as rows +HermiteE:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 3], [1, 0, 10, 0, 15, 0], [1, 0, 15, 0, 45, 0, 15], [1, 0, 21, 0, 105, 0, 105, 0], [1, 0, 28, 0, 210, 0, 420, 0, 105]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [3, 0, 6, 0, 1], [0, 15, 0, 10, 0, 1], [15, 0, 45, 0, 15, 0, 1], [0, 105, 0, 105, 0, 21, 0, 1], [105, 0, 420, 0, 210, 0, 28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 3], [1, 0, 6, 0], [1, 0, 10, 0], [1, 0, 15, 0, 3]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 10], [1, 1, 11, 11, 26, 26], [1, 1, 16, 16, 61, 61, 76], [1, 1, 22, 22, 127, 127, 232, 232], [1, 1, 29, 29, 239, 239, 659, 659, 764]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [4, 4, 1, 1], [10, 7, 7, 1, 1], [26, 26, 11, 11, 1, 1], [76, 61, 61, 16, 16, 1, 1], [232, 232, 127, 127, 22, 22, 1, 1], [764, 659, 659, 239, 239, 29, 29, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [3, 3, 9, 9, 10], [0, 15, 15, 25, 25, 26], [15, 15, 60, 60, 75, 75, 76], [0, 105, 105, 210, 210, 231, 231, 232], [105, 105, 525, 525, 735, 735, 763, 763, 764]] | + +HermiteE:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| EvenSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 4, 10, 26, 76, 232, 764] | +| AccSum | [1, 2, 4, 10, 26, 76, 232, 764, 2620] | +| AccRevSum | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | +| AntiDiagSum | [1, 1, 1, 1, 2, 4, 7, 11, 19] | + +HermiteE:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 1, 0, 0, 6, 10, 0, 0, 210] | +| ColECenter | [1, 0, 6, 0, 210] | +| ColOCenter | [1, 0, 10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| TransSqrs | [0, 0, 4, 12, 72, 280, 1320, 5544, 25312] | +| TransNat0 | [0, 0, 2, 6, 24, 80, 300, 1092, 4256] | +| TransNat1 | [1, 1, 4, 10, 34, 106, 376, 1324, 5020] | + +HermiteE:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 0, 3, 0, 15, 0, 105]| @@ -283,7 +271,7 @@ HermiteERev Diagonals as rows | DiagRow7 | [1, 0, 36, 0, 990, 0, 25740, 0, 675675]| | DiagRow8 | [1, 0, 45, 0, 1485, 0, 45045, 0, 1351350]| -HermiteERev Diagonals as columns +HermiteE:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ HermiteERev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [105, 945, 4725, 17325, 51975, 135135, 315315, 675675, 1351350] | -HermiteERev Polynomial values as rows +HermiteE:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ HermiteERev Polynomial values as rows | PolyRow7 | [1, 232, 8485, 85240, 457297, 1706776, 5035717, 12606280, 27956545] | | PolyRow8 | [1, 764, 57233, 1012348, 8655809, 47710076, 196228369, 655222268, 1872570113] | -HermiteERev Polynomial values as columns +HermiteE:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ HermiteERev Polynomial values as columns | PolyCol7 | [1, 1, 50, 148, 7498, 36506, 1873516, 12606280, 655222268] | | PolyCol8 | [1, 1, 65, 193, 12673, 62081, 4117441, 27956545, 1872570113] | -# HermiteEInvRev -True +# HermiteE:Inv:Rev +[] -HermiteEInvRev Triangle view +HermiteE:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ HermiteEInvRev Triangle view | Row7 | [1, 0, -21, 0, 105, 0, -105, 0] | | Row8 | [1, 0, -28, 0, 210, 0, -420, 0, 105] | -HermiteEInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 3, 1, 0, -10, 0, 15, 0, 1, 0, -15, 0, 45, 0, -15, 1, 0, -21, 0, 105, 0, -105, 0, 1, 0, -28, 0, 210, 0, -420, 0, 105] | -| TriRev | [1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 15, 0, -10, 0, 1, -15, 0, 45, 0, -15, 0, 1, 0, -105, 0, 105, 0, -21, 0, 1, 105, 0, -420, 0, 210, 0, -28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -5, -5, -2, 1, 1, -9, -9, 6, 6, 1, 1, -14, -14, 31, 31, 16, 1, 1, -20, -20, 85, 85, -20, -20, 1, 1, -27, -27, 183, 183, -237, -237, -132] | -| TriRevAcc | [1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -2, -5, -5, 1, 1, 6, 6, -9, -9, 1, 1, 16, 31, 31, -14, -14, 1, 1, -20, -20, 85, 85, -20, -20, 1, 1, -132, -237, -237, 183, 183, -27, -27, 1, 1] | -| TriAccRev | [1, 0, 1, -1, -1, 0, 0, -3, -3, -2, 3, 3, -3, -3, -2, 0, 15, 15, 5, 5, 6, -15, -15, 30, 30, 15, 15, 16, 0, -105, -105, 0, 0, -21, -21, -20, 105, 105, -315, -315, -105, -105, -133, -133, -132] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -3, 1, 0, -6, 0, 1, 0, -10, 0] | - -HermiteEInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| EvenSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | -| AccSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | -| AccRevSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | -| RevAccSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | -| DiagSum | [1, 1, 1, 1, 0, -2, -5, -9] | - -HermiteEInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 45, 105, 420] | -| RowGcd | [1, 1, 1, -3, 3, 5, 15, 21, 7] | -| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | -| Middle | [1, 1, 0, 0, -6, -10, 0, 0, 210] | -| Central | [1, 0, -6, 0, 210] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, -1, 0, 3, 0, -15, 0, 105] | -| PosHalf | [1, 2, 3, 2, -5, -18, -11, 86, 249] | -| NegHalf | [1, -2, 3, -2, -5, 18, -11, -86, 249] | -| Bin | [1, 1, 0, -8, -32, -24, 436, 2500, 2262] | -| InvBin | [1, -1, 0, 8, -32, 24, 436, -2500, 2262] | -| TransSqrs | [0, 0, -4, -12, 24, 200, 120, -2184, -5152] | -| TransNat0 | [0, 0, -2, -6, 0, 40, 60, -252, -896] | -| TransNat1 | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | - -HermiteEInvRev Diagonals as rows +HermiteE:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 3], [1, 0, -10, 0, 15, 0], [1, 0, -15, 0, 45, 0, -15], [1, 0, -21, 0, 105, 0, -105, 0], [1, 0, -28, 0, 210, 0, -420, 0, 105]] | +| RevTabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [3, 0, -6, 0, 1], [0, 15, 0, -10, 0, 1], [-15, 0, 45, 0, -15, 0, 1], [0, -105, 0, 105, 0, -21, 0, 1], [105, 0, -420, 0, 210, 0, -28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, -1], [1, 0, -3], [1, 0, -6, 0], [1, 0, -10, 0], [1, 0, -15, 0, 3]] | +| AccTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, -2], [1, 1, -9, -9, 6, 6], [1, 1, -14, -14, 31, 31, 16], [1, 1, -20, -20, 85, 85, -20, -20], [1, 1, -27, -27, 183, 183, -237, -237, -132]] | +| RevAccTabl | [[1], [1, 1], [0, 1, 1], [-2, -2, 1, 1], [-2, -5, -5, 1, 1], [6, 6, -9, -9, 1, 1], [16, 31, 31, -14, -14, 1, 1], [-20, -20, 85, 85, -20, -20, 1, 1], [-132, -237, -237, 183, 183, -27, -27, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [3, 3, -3, -3, -2], [0, 15, 15, 5, 5, 6], [-15, -15, 30, 30, 15, 15, 16], [0, -105, -105, 0, 0, -21, -21, -20], [105, 105, -315, -315, -105, -105, -133, -133, -132]] | + +HermiteE:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| EvenSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 0, -2, -2, 6, 16, -20, -132] | +| AccSum | [1, 2, 2, -2, -10, -4, 52, 92, -292] | +| AccRevSum | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | +| AntiDiagSum | [1, 1, 1, 1, 0, -2, -5, -9, -11] | + +HermiteE:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 45, 105, 420] | +| RowGcd | [1, 1, 1, 3, 3, 5, 15, 21, 7] | +| RowMax | [1, 1, 1, 3, 6, 15, 45, 105, 420] | +| ColMiddle | [1, 1, 0, 0, -6, -10, 0, 0, 210] | +| ColECenter | [1, 0, -6, 0, 210] | +| ColOCenter | [1, 0, -10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, -1, 0, 3, 0, -15, 0, 105] | +| TransSqrs | [0, 0, -4, -12, 24, 200, 120, -2184, -5152] | +| TransNat0 | [0, 0, -2, -6, 0, 40, 60, -252, -896] | +| TransNat1 | [1, 1, -2, -8, -2, 46, 76, -272, -1028] | + +HermiteE:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, 0, 3, 0, -15, 0, 105]| @@ -391,7 +375,7 @@ HermiteEInvRev Diagonals as rows | DiagRow7 | [1, 0, -36, 0, 990, 0, -25740, 0, 675675]| | DiagRow8 | [1, 0, -45, 0, 1485, 0, -45045, 0, 1351350]| -HermiteEInvRev Diagonals as columns +HermiteE:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ HermiteEInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [105, 945, 4725, 17325, 51975, 135135, 315315, 675675, 1351350] | -HermiteEInvRev Polynomial values as rows +HermiteE:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ HermiteEInvRev Polynomial values as rows | PolyRow7 | [1, -20, -5123, -68228, -403535, -1575524, -4763555, -12102068, -27096383] | | PolyRow8 | [1, -132, 3249, 399484, 5214273, 34583676, 157035313, 556394364, 1652365569] | -HermiteEInvRev Polynomial values as columns +HermiteE:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/HermiteH.md b/data/md/HermiteH.md index eb474f3..ea3726e 100644 --- a/data/md/HermiteH.md +++ b/data/md/HermiteH.md @@ -14,45 +14,41 @@ HermiteH Triangle view | Row7 | [0, 1680, 0, 3360, 0, 1344, 0, 128] | | Row8 | [1680, 0, 13440, 0, 13440, 0, 3584, 0, 256] | -HermiteH Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 2, 2, 0, 4, 0, 12, 0, 8, 12, 0, 48, 0, 16, 0, 120, 0, 160, 0, 32, 120, 0, 720, 0, 480, 0, 64, 0, 1680, 0, 3360, 0, 1344, 0, 128, 1680, 0, 13440, 0, 13440, 0, 3584, 0, 256] | -| TriRev | [1, 2, 0, 4, 0, 2, 8, 0, 12, 0, 16, 0, 48, 0, 12, 32, 0, 160, 0, 120, 0, 64, 0, 480, 0, 720, 0, 120, 128, 0, 1344, 0, 3360, 0, 1680, 0, 256, 0, 3584, 0, 13440, 0, 13440, 0, 1680] | -| TriAcc | [1, 0, 2, 2, 2, 6, 0, 12, 12, 20, 12, 12, 60, 60, 76, 0, 120, 120, 280, 280, 312, 120, 120, 840, 840, 1320, 1320, 1384, 0, 1680, 1680, 5040, 5040, 6384, 6384, 6512, 1680, 1680, 15120, 15120, 28560, 28560, 32144, 32144, 32400] | -| TriRevAcc | [1, 2, 0, 6, 2, 2, 20, 12, 12, 0, 76, 60, 60, 12, 12, 312, 280, 280, 120, 120, 0, 1384, 1320, 1320, 840, 840, 120, 120, 6512, 6384, 6384, 5040, 5040, 1680, 1680, 0, 32400, 32144, 32144, 28560, 28560, 15120, 15120, 1680, 1680] | -| TriAccRev | [1, 2, 2, 4, 4, 6, 8, 8, 20, 20, 16, 16, 64, 64, 76, 32, 32, 192, 192, 312, 312, 64, 64, 544, 544, 1264, 1264, 1384, 128, 128, 1472, 1472, 4832, 4832, 6512, 6512, 256, 256, 3840, 3840, 17280, 17280, 30720, 30720, 32400] | -| TriDiag | [1, 0, 2, 2, 0, 0, 12, 12, 4, 0, 0, 0, 120, 120, 48, 8, 0, 0, 0, 0] | +HermiteH Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 2], [2, 0, 4], [0, 12, 0, 8], [12, 0, 48, 0, 16], [0, 120, 0, 160, 0, 32], [120, 0, 720, 0, 480, 0, 64], [0, 1680, 0, 3360, 0, 1344, 0, 128], [1680, 0, 13440, 0, 13440, 0, 3584, 0, 256]] | +| RevTabl | [[1], [2, 0], [4, 0, 2], [8, 0, 12, 0], [16, 0, 48, 0, 12], [32, 0, 160, 0, 120, 0], [64, 0, 480, 0, 720, 0, 120], [128, 0, 1344, 0, 3360, 0, 1680, 0], [256, 0, 3584, 0, 13440, 0, 13440, 0, 1680]] | +| AntiDiag | [[1], [0], [2, 2], [0, 0], [12, 12, 4], [0, 0, 0], [120, 120, 48, 8], [0, 0, 0, 0], [1680, 1680, 720, 160, 16]] | +| AccTabl | [[1], [0, 2], [2, 2, 6], [0, 12, 12, 20], [12, 12, 60, 60, 76], [0, 120, 120, 280, 280, 312], [120, 120, 840, 840, 1320, 1320, 1384], [0, 1680, 1680, 5040, 5040, 6384, 6384, 6512], [1680, 1680, 15120, 15120, 28560, 28560, 32144, 32144, 32400]] | +| RevAccTabl | [[1], [2, 0], [6, 2, 2], [20, 12, 12, 0], [76, 60, 60, 12, 12], [312, 280, 280, 120, 120, 0], [1384, 1320, 1320, 840, 840, 120, 120], [6512, 6384, 6384, 5040, 5040, 1680, 1680, 0], [32400, 32144, 32144, 28560, 28560, 15120, 15120, 1680, 1680]] | +| AccRevTabl | [[1], [2, 2], [4, 4, 6], [8, 8, 20, 20], [16, 16, 64, 64, 76], [32, 32, 192, 192, 312, 312], [64, 64, 544, 544, 1264, 1264, 1384], [128, 128, 1472, 1472, 4832, 4832, 6512, 6512], [256, 256, 3840, 3840, 17280, 17280, 30720, 30720, 32400]] | HermiteH Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | -| EvenSum | [1, 0, 6, 0, 76, 0, 1384, 0, 32400] | -| OddSum | [0, 2, 0, 20, 0, 312, 0, 6512, 0] | -| AltSum | [1, -2, 6, -20, 76, -312, 1384, -6512, 32400] | -| AccSum | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | -| AccRevSum | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | -| RevAccSum | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | -| DiagSum | [1, 0, 4, 0, 28, 0, 296, 0] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | +| EvenSum | [1, 0, 6, 0, 76, 0, 1384, 0, 32400] | +| OddSum | [0, 2, 0, 20, 0, 312, 0, 6512, 0] | +| AltSum | [1, -2, 6, -20, 76, -312, 1384, -6512, 32400] | +| AccSum | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | +| AccRevSum | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | +| AntiDiagSum | [1, 0, 4, 0, 28, 0, 296, 0, 4256] | HermiteH Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 24, 48, 480, 2880, 13440, 53760] | -| RowGcd | [1, 2, 2, 4, 4, 8, 8, 16, 16] | -| RowMax | [1, 2, 4, 12, 48, 160, 720, 3360, 13440] | -| Middle | [1, 0, 0, 12, 48, 0, 0, 3360, 13440] | -| Central | [1, 0, 48, 0, 13440] | -| LeftSide | [1, 0, 2, 0, 12, 0, 120, 0, 1680] | -| RightSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| PosHalf | [1, 2, 12, 56, 400, 2592, 21184, 166784, 1519872] | -| NegHalf | [1, 2, 12, 56, 400, 2592, 21184, 166784, 1519872] | -| Bin | [1, 2, 6, 44, 316, 2232, 18184, 157712, 1419408] | -| InvBin | [1, 2, 6, 44, 316, 2232, 18184, 157712, 1419408] | -| TransSqrs | [0, 2, 16, 84, 448, 2360, 12864, 71792, 414208] | -| TransNat0 | [0, 2, 8, 36, 160, 760, 3744, 19376, 104192] | -| TransNat1 | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 24, 48, 480, 2880, 13440, 53760] | +| RowGcd | [1, 2, 2, 4, 4, 8, 8, 16, 16] | +| RowMax | [1, 2, 4, 12, 48, 160, 720, 3360, 13440] | +| ColMiddle | [1, 0, 0, 12, 48, 0, 0, 3360, 13440] | +| ColECenter | [1, 0, 48, 0, 13440] | +| ColOCenter | [0, 12, 0, 3360] | +| ColLeft | [1, 0, 2, 0, 12, 0, 120, 0, 1680] | +| ColRight | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| TransSqrs | [0, 2, 16, 84, 448, 2360, 12864, 71792, 414208] | +| TransNat0 | [0, 2, 8, 36, 160, 760, 3744, 19376, 104192] | +| TransNat1 | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | HermiteH Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ HermiteH Polynomial values as columns | PolyCol7 | [1, 14, 198, 2828, 40780, 593544, 8717416, 129166352, 1930372752] | | PolyCol8 | [1, 16, 258, 4192, 68620, 1131456, 18789496, 314209408, 5290403472] | -# HermiteHRev -True +# HermiteH:Rev +[] -HermiteHRev Triangle view +HermiteH:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ HermiteHRev Triangle view | Row7 | [128, 0, 1344, 0, 3360, 0, 1680, 0] | | Row8 | [256, 0, 3584, 0, 13440, 0, 13440, 0, 1680] | -HermiteHRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 0, 4, 0, 2, 8, 0, 12, 0, 16, 0, 48, 0, 12, 32, 0, 160, 0, 120, 0, 64, 0, 480, 0, 720, 0, 120, 128, 0, 1344, 0, 3360, 0, 1680, 0, 256, 0, 3584, 0, 13440, 0, 13440, 0, 1680] | -| TriRev | [1, 0, 2, 2, 0, 4, 0, 12, 0, 8, 12, 0, 48, 0, 16, 0, 120, 0, 160, 0, 32, 120, 0, 720, 0, 480, 0, 64, 0, 1680, 0, 3360, 0, 1344, 0, 128, 1680, 0, 13440, 0, 13440, 0, 3584, 0, 256] | -| TriAcc | [1, 2, 2, 4, 4, 6, 8, 8, 20, 20, 16, 16, 64, 64, 76, 32, 32, 192, 192, 312, 312, 64, 64, 544, 544, 1264, 1264, 1384, 128, 128, 1472, 1472, 4832, 4832, 6512, 6512, 256, 256, 3840, 3840, 17280, 17280, 30720, 30720, 32400] | -| TriRevAcc | [1, 2, 2, 6, 4, 4, 20, 20, 8, 8, 76, 64, 64, 16, 16, 312, 312, 192, 192, 32, 32, 1384, 1264, 1264, 544, 544, 64, 64, 6512, 6512, 4832, 4832, 1472, 1472, 128, 128, 32400, 30720, 30720, 17280, 17280, 3840, 3840, 256, 256] | -| TriAccRev | [1, 0, 2, 2, 2, 6, 0, 12, 12, 20, 12, 12, 60, 60, 76, 0, 120, 120, 280, 280, 312, 120, 120, 840, 840, 1320, 1320, 1384, 0, 1680, 1680, 5040, 5040, 6384, 6384, 6512, 1680, 1680, 15120, 15120, 28560, 28560, 32144, 32144, 32400] | -| TriDiag | [1, 2, 4, 0, 8, 0, 16, 0, 2, 32, 0, 12, 64, 0, 48, 0, 128, 0, 160, 0] | - -HermiteHRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | -| EvenSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | -| AccSum | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | -| AccRevSum | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | -| RevAccSum | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | -| DiagSum | [1, 2, 4, 8, 18, 44, 112, 288] | - -HermiteHRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 24, 48, 480, 2880, 13440, 53760] | -| RowGcd | [1, 2, 2, 4, 4, 8, 8, 16, 16] | -| RowMax | [1, 2, 4, 12, 48, 160, 720, 3360, 13440] | -| Middle | [1, 2, 0, 0, 48, 160, 0, 0, 13440] | -| Central | [1, 0, 48, 0, 13440] | -| LeftSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RightSide | [1, 0, 2, 0, 12, 0, 120, 0, 1680] | -| PosHalf | [1, 4, 18, 88, 460, 2544, 14776, 89632, 565392] | -| NegHalf | [1, -4, 18, -88, 460, -2544, 14776, -89632, 565392] | -| Bin | [1, 2, 6, 44, 316, 2232, 18184, 157712, 1419408] | -| InvBin | [1, -2, 6, -44, 316, -2232, 18184, -157712, 1419408] | -| TransSqrs | [0, 0, 8, 48, 384, 2560, 17760, 119616, 820736] | -| TransNat0 | [0, 0, 4, 24, 144, 800, 4560, 26208, 155008] | -| TransNat1 | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | - -HermiteHRev Diagonals as rows +HermiteH:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 0], [4, 0, 2], [8, 0, 12, 0], [16, 0, 48, 0, 12], [32, 0, 160, 0, 120, 0], [64, 0, 480, 0, 720, 0, 120], [128, 0, 1344, 0, 3360, 0, 1680, 0], [256, 0, 3584, 0, 13440, 0, 13440, 0, 1680]] | +| RevTabl | [[1], [0, 2], [2, 0, 4], [0, 12, 0, 8], [12, 0, 48, 0, 16], [0, 120, 0, 160, 0, 32], [120, 0, 720, 0, 480, 0, 64], [0, 1680, 0, 3360, 0, 1344, 0, 128], [1680, 0, 13440, 0, 13440, 0, 3584, 0, 256]] | +| AntiDiag | [[1], [2], [4, 0], [8, 0], [16, 0, 2], [32, 0, 12], [64, 0, 48, 0], [128, 0, 160, 0], [256, 0, 480, 0, 12]] | +| AccTabl | [[1], [2, 2], [4, 4, 6], [8, 8, 20, 20], [16, 16, 64, 64, 76], [32, 32, 192, 192, 312, 312], [64, 64, 544, 544, 1264, 1264, 1384], [128, 128, 1472, 1472, 4832, 4832, 6512, 6512], [256, 256, 3840, 3840, 17280, 17280, 30720, 30720, 32400]] | +| RevAccTabl | [[1], [2, 2], [6, 4, 4], [20, 20, 8, 8], [76, 64, 64, 16, 16], [312, 312, 192, 192, 32, 32], [1384, 1264, 1264, 544, 544, 64, 64], [6512, 6512, 4832, 4832, 1472, 1472, 128, 128], [32400, 30720, 30720, 17280, 17280, 3840, 3840, 256, 256]] | +| AccRevTabl | [[1], [0, 2], [2, 2, 6], [0, 12, 12, 20], [12, 12, 60, 60, 76], [0, 120, 120, 280, 280, 312], [120, 120, 840, 840, 1320, 1320, 1384], [0, 1680, 1680, 5040, 5040, 6384, 6384, 6512], [1680, 1680, 15120, 15120, 28560, 28560, 32144, 32144, 32400]] | + +HermiteH:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | +| EvenSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 2, 6, 20, 76, 312, 1384, 6512, 32400] | +| AccSum | [1, 4, 14, 56, 236, 1072, 5128, 25888, 136592] | +| AccRevSum | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | +| AntiDiagSum | [1, 2, 4, 8, 18, 44, 112, 288, 748] | + +HermiteH:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 24, 48, 480, 2880, 13440, 53760] | +| RowGcd | [1, 2, 2, 4, 4, 8, 8, 16, 16] | +| RowMax | [1, 2, 4, 12, 48, 160, 720, 3360, 13440] | +| ColMiddle | [1, 2, 0, 0, 48, 160, 0, 0, 13440] | +| ColECenter | [1, 0, 48, 0, 13440] | +| ColOCenter | [2, 0, 160, 0] | +| ColLeft | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| ColRight | [1, 0, 2, 0, 12, 0, 120, 0, 1680] | +| TransSqrs | [0, 0, 8, 48, 384, 2560, 17760, 119616, 820736] | +| TransNat0 | [0, 0, 4, 24, 144, 800, 4560, 26208, 155008] | +| TransNat1 | [1, 2, 10, 44, 220, 1112, 5944, 32720, 187408] | + +HermiteH:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 2, 0, 12, 0, 120, 0, 1680]| @@ -175,7 +167,7 @@ HermiteHRev Diagonals as rows | DiagRow7 | [128, 0, 9216, 0, 506880, 0, 26357760, 0, 1383782400]| | DiagRow8 | [256, 0, 23040, 0, 1520640, 0, 92252160, 0, 5535129600]| -HermiteHRev Diagonals as columns +HermiteH:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | @@ -188,7 +180,7 @@ HermiteHRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1680, 30240, 302400, 2217600, 13305600, 69189120, 322882560, 1383782400, 5535129600] | -HermiteHRev Polynomial values as rows +HermiteH:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ HermiteHRev Polynomial values as rows | PolyRow7 | [128, 6512, 166784, 1509104, 7763072, 28383728, 82785152, 205783664, 454250624] | | PolyRow8 | [256, 32400, 1519872, 21941392, 168648960, 874739856, 3466359040, 11298513552, 31764218112] | -HermiteHRev Polynomial values as columns +HermiteH:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | diff --git a/data/md/LabeledGraphs.md b/data/md/LabeledGraphs.md new file mode 100644 index 0000000..b19bebd --- /dev/null +++ b/data/md/LabeledGraphs.md @@ -0,0 +1,208 @@ +# LabeledGraphs +['A360603'] + +LabeledGraphs Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [0, 1, 1] | +| Row3 | [0, 2, 2, 4] | +| Row4 | [0, 8, 6, 12, 38] | +| Row5 | [0, 64, 32, 48, 152, 728] | +| Row6 | [0, 1024, 320, 320, 760, 3640, 26704] | +| Row7 | [0, 32768, 6144, 3840, 6080, 21840, 160224, 1866256] | +| Row8 | [0, 2097152, 229376, 86016, 85120, 203840, 1121568, 13063792, 251548592] | + +LabeledGraphs Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 2, 2, 4], [0, 8, 6, 12, 38], [0, 64, 32, 48, 152, 728], [0, 1024, 320, 320, 760, 3640, 26704], [0, 32768, 6144, 3840, 6080, 21840, 160224, 1866256], [0, 2097152, 229376, 86016, 85120, 203840, 1121568, 13063792, 251548592]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [4, 2, 2, 0], [38, 12, 6, 8, 0], [728, 152, 48, 32, 64, 0], [26704, 3640, 760, 320, 320, 1024, 0], [1866256, 160224, 21840, 6080, 3840, 6144, 32768, 0], [251548592, 13063792, 1121568, 203840, 85120, 86016, 229376, 2097152, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 2, 1], [0, 8, 2], [0, 64, 6, 4], [0, 1024, 32, 12], [0, 32768, 320, 48, 38]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 2, 4, 8], [0, 8, 14, 26, 64], [0, 64, 96, 144, 296, 1024], [0, 1024, 1344, 1664, 2424, 6064, 32768], [0, 32768, 38912, 42752, 48832, 70672, 230896, 2097152], [0, 2097152, 2326528, 2412544, 2497664, 2701504, 3823072, 16886864, 268435456]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [8, 4, 2, 0], [64, 26, 14, 8, 0], [1024, 296, 144, 96, 64, 0], [32768, 6064, 2424, 1664, 1344, 1024, 0], [2097152, 230896, 70672, 48832, 42752, 38912, 32768, 0], [268435456, 16886864, 3823072, 2701504, 2497664, 2412544, 2326528, 2097152, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [4, 6, 8, 8], [38, 50, 56, 64, 64], [728, 880, 928, 960, 1024, 1024], [26704, 30344, 31104, 31424, 31744, 32768, 32768], [1866256, 2026480, 2048320, 2054400, 2058240, 2064384, 2097152, 2097152], [251548592, 264612384, 265733952, 265937792, 266022912, 266108928, 266338304, 268435456, 268435456]] | + +LabeledGraphs Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456] | +| EvenSum | [1, 0, 1, 2, 44, 184, 27784, 172448, 252984656] | +| OddSum | [0, 1, 1, 6, 20, 840, 4984, 1924704, 15450800] | +| AltSum | [1, -1, 0, -4, 24, -656, 22800, -1752256, 237533856] | +| AccSum | [1, 1, 3, 14, 112, 1624, 45288, 2561984, 301180784] | +| AccRevSum | [1, 2, 5, 26, 272, 5544, 216856, 16312384, 2383173776] | +| AntiDiagSum | [1, 0, 1, 1, 3, 10, 74, 1068, 33174] | + +LabeledGraphs Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 456, 331968, 14774789120, 1243919045591040, 8761370365451477754839040] | +| RowGcd | [1, 1, 1, 2, 2, 8, 8, 16, 16] | +| RowMax | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| ColMiddle | [1, 0, 1, 2, 6, 32, 320, 3840, 85120] | +| ColECenter | [1, 1, 6, 320, 85120] | +| ColOCenter | [0, 2, 32, 3840] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| TransSqrs | [0, 1, 5, 46, 748, 21256, 1069688, 97949792, 16789858864] | +| TransNat0 | [0, 1, 3, 18, 208, 4520, 184088, 14215232, 2114738320] | +| TransNat1 | [1, 2, 5, 26, 272, 5544, 216856, 16312384, 2383173776] | + +LabeledGraphs Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592]| +| DiagRow1 | [0, 1, 2, 12, 152, 3640, 160224, 13063792, 2012388736]| +| DiagRow2 | [0, 2, 6, 48, 760, 21840, 1121568, 104510336, 18111498624]| +| DiagRow3 | [0, 8, 32, 320, 6080, 203840, 11963392, 1254124032, 241486648320]| +| DiagRow4 | [0, 64, 320, 3840, 85120, 3261440, 215341056, 25082480640, 5312706263040]| +| DiagRow5 | [0, 1024, 6144, 86016, 2179072, 93929472, 6890913792, 882903318528, 204007920500736]| +| DiagRow6 | [0, 32768, 229376, 3670016, 104595456, 5009571840, 404266942464, 56505812385792, 14144549154717696]| +| DiagRow7 | [0, 2097152, 16777216, 301989888, 9563013120, 503819796480, 44353858830336, 6716119414996992, 1810502291803865088]| +| DiagRow8 | [0, 268435456, 2415919104, 48318382080, 1683090309120, 96733400924160, 9225602636709888, 1504410748959326208, 434520550032927621120]| + +LabeledGraphs Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456] | +| DiagCol2 | [1, 2, 6, 32, 320, 6144, 229376, 16777216, 2415919104] | +| DiagCol3 | [4, 12, 48, 320, 3840, 86016, 3670016, 301989888, 48318382080] | +| DiagCol4 | [38, 152, 760, 6080, 85120, 2179072, 104595456, 9563013120, 1683090309120] | +| DiagCol5 | [728, 3640, 21840, 203840, 3261440, 93929472, 5009571840, 503819796480, 96733400924160] | +| DiagCol6 | [26704, 160224, 1121568, 11963392, 215341056, 6890913792, 404266942464, 44353858830336, 9225602636709888] | +| DiagCol7 | [1866256, 13063792, 104510336, 1254124032, 25082480640, 882903318528, 56505812385792, 6716119414996992, 1504410748959326208] | +| DiagCol8 | [251548592, 2012388736, 18111498624, 241486648320, 5312706263040, 204007920500736, 14144549154717696, 1810502291803865088, 434520550032927621120] | + +LabeledGraphs Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [0, 2, 6, 12, 20, 30, 42, 56, 72] | +| PolyRow3 | [0, 8, 44, 132, 296, 560, 948, 1484, 2192] | +| PolyRow4 | [0, 64, 744, 3480, 10624, 25440, 52104, 95704, 162240] | +| PolyRow5 | [0, 1024, 26368, 190992, 788224, 2377120, 5869824, 12618928, 24504832] | +| PolyRow6 | [0, 32768, 1843584, 20427888, 113331200, 429153120, 1275278208, 3204833744, 7122874368] | +| PolyRow7 | [0, 2097152, 250052096, 4204362048, 31257411584, 148377597440, 530086605312, 1556175768896, 3956567441408] | +| PolyRow8 | [0, 268435456, 66154070016, 1679865554400, 16704367722496, 99300019172160, 426216128249856, 1461021806253856, 4247982935900160] | + +LabeledGraphs Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456] | +| PolyCol2 | [1, 2, 6, 44, 744, 26368, 1843584, 250052096, 66154070016] | +| PolyCol3 | [1, 3, 12, 132, 3480, 190992, 20427888, 4204362048, 1679865554400] | +| PolyCol4 | [1, 4, 20, 296, 10624, 788224, 113331200, 31257411584, 16704367722496] | +| PolyCol5 | [1, 5, 30, 560, 25440, 2377120, 429153120, 148377597440, 99300019172160] | +| PolyCol6 | [1, 6, 42, 948, 52104, 5869824, 1275278208, 530086605312, 426216128249856] | +| PolyCol7 | [1, 7, 56, 1484, 95704, 12618928, 3204833744, 1556175768896, 1461021806253856] | +| PolyCol8 | [1, 8, 72, 2192, 162240, 24504832, 7122874368, 3956567441408, 4247982935900160] | + +# LabeledGraphs:Rev +[] + +LabeledGraphs:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 0] | +| Row2 | [1, 1, 0] | +| Row3 | [4, 2, 2, 0] | +| Row4 | [38, 12, 6, 8, 0] | +| Row5 | [728, 152, 48, 32, 64, 0] | +| Row6 | [26704, 3640, 760, 320, 320, 1024, 0] | +| Row7 | [1866256, 160224, 21840, 6080, 3840, 6144, 32768, 0] | +| Row8 | [251548592, 13063792, 1121568, 203840, 85120, 86016, 229376, 2097152, 0] | + +LabeledGraphs:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [4, 2, 2, 0], [38, 12, 6, 8, 0], [728, 152, 48, 32, 64, 0], [26704, 3640, 760, 320, 320, 1024, 0], [1866256, 160224, 21840, 6080, 3840, 6144, 32768, 0], [251548592, 13063792, 1121568, 203840, 85120, 86016, 229376, 2097152, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 2, 2, 4], [0, 8, 6, 12, 38], [0, 64, 32, 48, 152, 728], [0, 1024, 320, 320, 760, 3640, 26704], [0, 32768, 6144, 3840, 6080, 21840, 160224, 1866256], [0, 2097152, 229376, 86016, 85120, 203840, 1121568, 13063792, 251548592]] | +| AntiDiag | [[1], [1], [1, 0], [4, 1], [38, 2, 0], [728, 12, 2], [26704, 152, 6, 0], [1866256, 3640, 48, 8], [251548592, 160224, 760, 32, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [4, 6, 8, 8], [38, 50, 56, 64, 64], [728, 880, 928, 960, 1024, 1024], [26704, 30344, 31104, 31424, 31744, 32768, 32768], [1866256, 2026480, 2048320, 2054400, 2058240, 2064384, 2097152, 2097152], [251548592, 264612384, 265733952, 265937792, 266022912, 266108928, 266338304, 268435456, 268435456]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [8, 8, 6, 4], [64, 64, 56, 50, 38], [1024, 1024, 960, 928, 880, 728], [32768, 32768, 31744, 31424, 31104, 30344, 26704], [2097152, 2097152, 2064384, 2058240, 2054400, 2048320, 2026480, 1866256], [268435456, 268435456, 266338304, 266108928, 266022912, 265937792, 265733952, 264612384, 251548592]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 2, 4, 8], [0, 8, 14, 26, 64], [0, 64, 96, 144, 296, 1024], [0, 1024, 1344, 1664, 2424, 6064, 32768], [0, 32768, 38912, 42752, 48832, 70672, 230896, 2097152], [0, 2097152, 2326528, 2412544, 2497664, 2701504, 3823072, 16886864, 268435456]] | + +LabeledGraphs:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456] | +| EvenSum | [1, 1, 1, 6, 44, 840, 27784, 1924704, 252984656] | +| OddSum | [0, 0, 1, 2, 20, 184, 4984, 172448, 15450800] | +| AltSum | [1, 1, 0, 4, 24, 656, 22800, 1752256, 237533856] | +| AccSum | [1, 2, 5, 26, 272, 5544, 216856, 16312384, 2383173776] | +| AccRevSum | [1, 1, 3, 14, 112, 1624, 45288, 2561984, 301180784] | +| AntiDiagSum | [1, 1, 1, 5, 40, 742, 26862, 1869952, 251709608] | + +LabeledGraphs:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 4, 456, 331968, 14774789120, 1243919045591040, 8761370365451477754839040] | +| RowGcd | [1, 1, 1, 2, 2, 8, 8, 16, 16] | +| RowMax | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| ColMiddle | [1, 1, 1, 2, 6, 48, 320, 6080, 85120] | +| ColECenter | [1, 1, 6, 320, 85120] | +| ColOCenter | [1, 2, 48, 6080] | +| ColLeft | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 10, 108, 1656, 40280, 1696992, 133914928] | +| TransNat0 | [0, 0, 1, 6, 48, 600, 12520, 464832, 32745328] | +| TransNat1 | [1, 1, 3, 14, 112, 1624, 45288, 2561984, 301180784] | + +LabeledGraphs:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow1 | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456]| +| DiagRow2 | [1, 2, 6, 32, 320, 6144, 229376, 16777216, 2415919104]| +| DiagRow3 | [4, 12, 48, 320, 3840, 86016, 3670016, 301989888, 48318382080]| +| DiagRow4 | [38, 152, 760, 6080, 85120, 2179072, 104595456, 9563013120, 1683090309120]| +| DiagRow5 | [728, 3640, 21840, 203840, 3261440, 93929472, 5009571840, 503819796480, 96733400924160]| +| DiagRow6 | [26704, 160224, 1121568, 11963392, 215341056, 6890913792, 404266942464, 44353858830336, 9225602636709888]| +| DiagRow7 | [1866256, 13063792, 104510336, 1254124032, 25082480640, 882903318528, 56505812385792, 6716119414996992, 1504410748959326208]| +| DiagRow8 | [251548592, 2012388736, 18111498624, 241486648320, 5312706263040, 204007920500736, 14144549154717696, 1810502291803865088, 434520550032927621120]| + +LabeledGraphs:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| DiagCol1 | [0, 1, 2, 12, 152, 3640, 160224, 13063792, 2012388736] | +| DiagCol2 | [0, 2, 6, 48, 760, 21840, 1121568, 104510336, 18111498624] | +| DiagCol3 | [0, 8, 32, 320, 6080, 203840, 11963392, 1254124032, 241486648320] | +| DiagCol4 | [0, 64, 320, 3840, 85120, 3261440, 215341056, 25082480640, 5312706263040] | +| DiagCol5 | [0, 1024, 6144, 86016, 2179072, 93929472, 6890913792, 882903318528, 204007920500736] | +| DiagCol6 | [0, 32768, 229376, 3670016, 104595456, 5009571840, 404266942464, 56505812385792, 14144549154717696] | +| DiagCol7 | [0, 2097152, 16777216, 301989888, 9563013120, 503819796480, 44353858830336, 6716119414996992, 1810502291803865088] | +| DiagCol8 | [0, 268435456, 2415919104, 48318382080, 1683090309120, 96733400924160, 9225602636709888, 1504410748959326208, 434520550032927621120] | + +LabeledGraphs:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow2 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| PolyRow3 | [4, 8, 16, 28, 44, 64, 88, 116, 148] | +| PolyRow4 | [38, 64, 150, 344, 694, 1248, 2054, 3160, 4614] | +| PolyRow5 | [728, 1024, 2504, 7664, 20536, 46688, 93224, 168784, 283544] | +| PolyRow6 | [26704, 32768, 77472, 327856, 1204400, 3503904, 8522368, 18177872, 35133456] | +| PolyRow7 | [1866256, 2097152, 4677904, 28399552, 144737936, 537573376, 1586503312, 3973747904, 8814648592] | +| PolyRow8 | [251548592, 268435456, 571023120, 5087820896, 35743928176, 168116386752, 598963622096, 1756198619680, 4461875482416] | + +LabeledGraphs:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592] | +| PolyCol1 | [1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456] | +| PolyCol2 | [1, 1, 3, 16, 150, 2504, 77472, 4677904, 571023120] | +| PolyCol3 | [1, 1, 4, 28, 344, 7664, 327856, 28399552, 5087820896] | +| PolyCol4 | [1, 1, 5, 44, 694, 20536, 1204400, 144737936, 35743928176] | +| PolyCol5 | [1, 1, 6, 64, 1248, 46688, 3503904, 537573376, 168116386752] | +| PolyCol6 | [1, 1, 7, 88, 2054, 93224, 8522368, 1586503312, 598963622096] | +| PolyCol7 | [1, 1, 8, 116, 3160, 168784, 18177872, 3973747904, 1756198619680] | +| PolyCol8 | [1, 1, 9, 148, 4614, 283544, 35133456, 8814648592, 4461875482416] | + diff --git a/data/md/Laguerre.md b/data/md/Laguerre.md index d15a535..f8a82a9 100644 --- a/data/md/Laguerre.md +++ b/data/md/Laguerre.md @@ -14,45 +14,41 @@ Laguerre Triangle view | Row7 | [5040, 35280, 52920, 29400, 7350, 882, 49, 1] | | Row8 | [40320, 322560, 564480, 376320, 117600, 18816, 1568, 64, 1] | -Laguerre Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 4, 1, 6, 18, 9, 1, 24, 96, 72, 16, 1, 120, 600, 600, 200, 25, 1, 720, 4320, 5400, 2400, 450, 36, 1, 5040, 35280, 52920, 29400, 7350, 882, 49, 1, 40320, 322560, 564480, 376320, 117600, 18816, 1568, 64, 1] | -| TriRev | [1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320] | -| TriAcc | [1, 1, 2, 2, 6, 7, 6, 24, 33, 34, 24, 120, 192, 208, 209, 120, 720, 1320, 1520, 1545, 1546, 720, 5040, 10440, 12840, 13290, 13326, 13327, 5040, 40320, 93240, 122640, 129990, 130872, 130921, 130922, 40320, 362880, 927360, 1303680, 1421280, 1440096, 1441664, 1441728, 1441729] | -| TriRevAcc | [1, 2, 1, 7, 6, 2, 34, 33, 24, 6, 209, 208, 192, 120, 24, 1546, 1545, 1520, 1320, 720, 120, 13327, 13326, 13290, 12840, 10440, 5040, 720, 130922, 130921, 130872, 129990, 122640, 93240, 40320, 5040, 1441729, 1441728, 1441664, 1440096, 1421280, 1303680, 927360, 362880, 40320] | -| TriAccRev | [1, 1, 2, 1, 5, 7, 1, 10, 28, 34, 1, 17, 89, 185, 209, 1, 26, 226, 826, 1426, 1546, 1, 37, 487, 2887, 8287, 12607, 13327, 1, 50, 932, 8282, 37682, 90602, 125882, 130922, 1, 65, 1633, 20449, 138049, 514369, 1078849, 1401409, 1441729] | -| TriDiag | [1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16] | +Laguerre Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 4, 1], [6, 18, 9, 1], [24, 96, 72, 16, 1], [120, 600, 600, 200, 25, 1], [720, 4320, 5400, 2400, 450, 36, 1], [5040, 35280, 52920, 29400, 7350, 882, 49, 1], [40320, 322560, 564480, 376320, 117600, 18816, 1568, 64, 1]] | +| RevTabl | [[1], [1, 1], [1, 4, 2], [1, 9, 18, 6], [1, 16, 72, 96, 24], [1, 25, 200, 600, 600, 120], [1, 36, 450, 2400, 5400, 4320, 720], [1, 49, 882, 7350, 29400, 52920, 35280, 5040], [1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 4], [24, 18, 1], [120, 96, 9], [720, 600, 72, 1], [5040, 4320, 600, 16], [40320, 35280, 5400, 200, 1]] | +| AccTabl | [[1], [1, 2], [2, 6, 7], [6, 24, 33, 34], [24, 120, 192, 208, 209], [120, 720, 1320, 1520, 1545, 1546], [720, 5040, 10440, 12840, 13290, 13326, 13327], [5040, 40320, 93240, 122640, 129990, 130872, 130921, 130922], [40320, 362880, 927360, 1303680, 1421280, 1440096, 1441664, 1441728, 1441729]] | +| RevAccTabl | [[1], [2, 1], [7, 6, 2], [34, 33, 24, 6], [209, 208, 192, 120, 24], [1546, 1545, 1520, 1320, 720, 120], [13327, 13326, 13290, 12840, 10440, 5040, 720], [130922, 130921, 130872, 129990, 122640, 93240, 40320, 5040], [1441729, 1441728, 1441664, 1440096, 1421280, 1303680, 927360, 362880, 40320]] | +| AccRevTabl | [[1], [1, 2], [1, 5, 7], [1, 10, 28, 34], [1, 17, 89, 185, 209], [1, 26, 226, 826, 1426, 1546], [1, 37, 487, 2887, 8287, 12607, 13327], [1, 50, 932, 8282, 37682, 90602, 125882, 130922], [1, 65, 1633, 20449, 138049, 514369, 1078849, 1401409, 1441729]] | Laguerre Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| EvenSum | [1, 1, 3, 15, 97, 745, 6571, 65359, 723969] | -| OddSum | [0, 1, 4, 19, 112, 801, 6756, 65563, 717760] | -| AltSum | [1, 0, -1, -4, -15, -56, -185, -204, 6209] | -| AccSum | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | -| AccRevSum | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | -| RevAccSum | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | -| DiagSum | [1, 1, 3, 10, 43, 225, 1393, 9976] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| EvenSum | [1, 1, 3, 15, 97, 745, 6571, 65359, 723969] | +| OddSum | [0, 1, 4, 19, 112, 801, 6756, 65563, 717760] | +| AltSum | [1, 0, -1, -4, -15, -56, -185, -204, 6209] | +| AccSum | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | +| AccRevSum | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | +| AntiDiagSum | [1, 1, 3, 10, 43, 225, 1393, 9976, 81201] | Laguerre Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | -| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | -| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | -| Middle | [1, 1, 4, 18, 72, 600, 2400, 29400, 117600] | -| Central | [1, 4, 72, 2400, 117600] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089] | -| NegHalf | [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063] | -| Bin | [1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273] | -| InvBin | [1, 0, -5, 22, 9, -1244, 14335, -79470, -586943] | -| TransSqrs | [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008] | -| TransNat0 | [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824] | -| TransNat1 | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | +| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | +| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | +| ColMiddle | [1, 1, 4, 18, 72, 600, 2400, 29400, 117600] | +| ColECenter | [1, 4, 72, 2400, 117600] | +| ColOCenter | [1, 18, 600, 29400] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008] | +| TransNat0 | [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824] | +| TransNat1 | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | Laguerre Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Laguerre Polynomial values as columns | PolyCol7 | [1, 8, 79, 916, 12113, 179152, 2921911, 51988748, 1000578817] | | PolyCol8 | [1, 9, 98, 1238, 17688, 280888, 4894672, 92676144, 1891712384] | -# LaguerreInv -True +# Laguerre:Inv +[] -LaguerreInv Triangle view +Laguerre:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LaguerreInv Triangle view | Row7 | [-5040, 35280, -52920, 29400, -7350, 882, -49, 1] | | Row8 | [40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1] | -LaguerreInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 2, -4, 1, -6, 18, -9, 1, 24, -96, 72, -16, 1, -120, 600, -600, 200, -25, 1, 720, -4320, 5400, -2400, 450, -36, 1, -5040, 35280, -52920, 29400, -7350, 882, -49, 1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1] | -| TriRev | [1, 1, -1, 1, -4, 2, 1, -9, 18, -6, 1, -16, 72, -96, 24, 1, -25, 200, -600, 600, -120, 1, -36, 450, -2400, 5400, -4320, 720, 1, -49, 882, -7350, 29400, -52920, 35280, -5040, 1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320] | -| TriAcc | [1, -1, 0, 2, -2, -1, -6, 12, 3, 4, 24, -72, 0, -16, -15, -120, 480, -120, 80, 55, 56, 720, -3600, 1800, -600, -150, -186, -185, -5040, 30240, -22680, 6720, -630, 252, 203, 204, 40320, -282240, 282240, -94080, 23520, 4704, 6272, 6208, 6209] | -| TriRevAcc | [1, 0, -1, -1, -2, 2, 4, 3, 12, -6, -15, -16, 0, -72, 24, 56, 55, 80, -120, 480, -120, -185, -186, -150, -600, 1800, -3600, 720, 204, 203, 252, -630, 6720, -22680, 30240, -5040, 6209, 6208, 6272, 4704, 23520, -94080, 282240, -282240, 40320] | -| TriAccRev | [1, 1, 0, 1, -3, -1, 1, -8, 10, 4, 1, -15, 57, -39, -15, 1, -24, 176, -424, 176, 56, 1, -35, 415, -1985, 3415, -905, -185, 1, -48, 834, -6516, 22884, -30036, 5244, 204, 1, -63, 1505, -17311, 100289, -276031, 288449, -34111, 6209] | -| TriDiag | [1, -1, 2, 1, -6, -4, 24, 18, 1, -120, -96, -9, 720, 600, 72, 1, -5040, -4320, -600, -16] | - -LaguerreInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | -| EvenSum | [1, -1, 3, -15, 97, -745, 6571, -65359, 723969] | -| OddSum | [0, 1, -4, 19, -112, 801, -6756, 65563, -717760] | -| AltSum | [1, -2, 7, -34, 209, -1546, 13327, -130922, 1441729] | -| AccSum | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | -| AccRevSum | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | -| RevAccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| DiagSum | [1, -1, 3, -10, 43, -225, 1393, -9976] | - -LaguerreInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | -| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | -| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | -| Middle | [1, -1, -4, 18, 72, -600, -2400, 29400, 117600] | -| Central | [1, -4, 72, -2400, 117600] | -| LeftSide | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063] | -| NegHalf | [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089] | -| Bin | [1, 0, -5, 22, 9, -1244, 14335, -79470, -586943] | -| InvBin | [1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273] | -| TransSqrs | [0, 1, 0, -9, 64, -375, 2016, -9065, 13056] | -| TransNat0 | [0, 1, -2, 3, 4, -95, 906, -7637, 62728] | -| TransNat1 | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | - -LaguerreInv Diagonals as rows +Laguerre:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [2, -4, 1], [-6, 18, -9, 1], [24, -96, 72, -16, 1], [-120, 600, -600, 200, -25, 1], [720, -4320, 5400, -2400, 450, -36, 1], [-5040, 35280, -52920, 29400, -7350, 882, -49, 1], [40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1]] | +| RevTabl | [[1], [1, -1], [1, -4, 2], [1, -9, 18, -6], [1, -16, 72, -96, 24], [1, -25, 200, -600, 600, -120], [1, -36, 450, -2400, 5400, -4320, 720], [1, -49, 882, -7350, 29400, -52920, 35280, -5040], [1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320]] | +| AntiDiag | [[1], [-1], [2, 1], [-6, -4], [24, 18, 1], [-120, -96, -9], [720, 600, 72, 1], [-5040, -4320, -600, -16], [40320, 35280, 5400, 200, 1]] | +| AccTabl | [[1], [-1, 0], [2, -2, -1], [-6, 12, 3, 4], [24, -72, 0, -16, -15], [-120, 480, -120, 80, 55, 56], [720, -3600, 1800, -600, -150, -186, -185], [-5040, 30240, -22680, 6720, -630, 252, 203, 204], [40320, -282240, 282240, -94080, 23520, 4704, 6272, 6208, 6209]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 2], [4, 3, 12, -6], [-15, -16, 0, -72, 24], [56, 55, 80, -120, 480, -120], [-185, -186, -150, -600, 1800, -3600, 720], [204, 203, 252, -630, 6720, -22680, 30240, -5040], [6209, 6208, 6272, 4704, 23520, -94080, 282240, -282240, 40320]] | +| AccRevTabl | [[1], [1, 0], [1, -3, -1], [1, -8, 10, 4], [1, -15, 57, -39, -15], [1, -24, 176, -424, 176, 56], [1, -35, 415, -1985, 3415, -905, -185], [1, -48, 834, -6516, 22884, -30036, 5244, 204], [1, -63, 1505, -17311, 100289, -276031, 288449, -34111, 6209]] | + +Laguerre:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | +| EvenSum | [1, -1, 3, -15, 97, -745, 6571, -65359, 723969] | +| OddSum | [0, 1, -4, 19, -112, 801, -6756, 65563, -717760] | +| AltSum | [1, -2, 7, -34, 209, -1546, 13327, -130922, 1441729] | +| AccSum | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | +| AccRevSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | +| AntiDiagSum | [1, -1, 3, -10, 43, -225, 1393, -9976, 81201] | + +Laguerre:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | +| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | +| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | +| ColMiddle | [1, -1, -4, 18, 72, -600, -2400, 29400, 117600] | +| ColECenter | [1, -4, 72, -2400, 117600] | +| ColOCenter | [-1, 18, -600, 29400] | +| ColLeft | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 0, -9, 64, -375, 2016, -9065, 13056] | +| TransNat0 | [0, 1, -2, 3, 4, -95, 906, -7637, 62728] | +| TransNat1 | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | + +Laguerre:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ LaguerreInv Diagonals as rows | DiagRow7 | [-5040, -322560, -6531840, -72576000, -548856000, -3161410560, -14841066240, -59364264960, -208702494000]| | DiagRow8 | [40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000]| -LaguerreInv Diagonals as columns +Laguerre:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | @@ -188,7 +180,7 @@ LaguerreInv Diagonals as columns | DiagCol7 | [1, -64, 2592, -86400, 2613600, -75271680, 2120152320, -59364264960, 1669619952000] | | DiagCol8 | [1, -81, 4050, -163350, 5880600, -198764280, 6492966480, -208702494000, 6678479808000] | -LaguerreInv Polynomial values as rows +Laguerre:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LaguerreInv Polynomial values as rows | PolyRow7 | [-5040, 204, -3344, 3762, 8208, -1640, -21456, -31794, -9008] | | PolyRow8 | [40320, 6209, 14464, -44703, -25216, 90145, 157824, 12929, -361088] | -LaguerreInv Polynomial values as columns +Laguerre:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | @@ -214,10 +206,10 @@ LaguerreInv Polynomial values as columns | PolyCol7 | [1, 6, 23, 22, -207, 62, 4927, -31794, 12929] | | PolyCol8 | [1, 7, 34, 74, -232, -952, 8656, -9008, -361088] | -# LaguerreRev -True +# Laguerre:Rev +[] -LaguerreRev Triangle view +Laguerre:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ LaguerreRev Triangle view | Row7 | [1, 49, 882, 7350, 29400, 52920, 35280, 5040] | | Row8 | [1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320] | -LaguerreRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040, 1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320] | -| TriRev | [1, 1, 1, 2, 4, 1, 6, 18, 9, 1, 24, 96, 72, 16, 1, 120, 600, 600, 200, 25, 1, 720, 4320, 5400, 2400, 450, 36, 1, 5040, 35280, 52920, 29400, 7350, 882, 49, 1, 40320, 322560, 564480, 376320, 117600, 18816, 1568, 64, 1] | -| TriAcc | [1, 1, 2, 1, 5, 7, 1, 10, 28, 34, 1, 17, 89, 185, 209, 1, 26, 226, 826, 1426, 1546, 1, 37, 487, 2887, 8287, 12607, 13327, 1, 50, 932, 8282, 37682, 90602, 125882, 130922, 1, 65, 1633, 20449, 138049, 514369, 1078849, 1401409, 1441729] | -| TriRevAcc | [1, 2, 1, 7, 5, 1, 34, 28, 10, 1, 209, 185, 89, 17, 1, 1546, 1426, 826, 226, 26, 1, 13327, 12607, 8287, 2887, 487, 37, 1, 130922, 125882, 90602, 37682, 8282, 932, 50, 1, 1441729, 1401409, 1078849, 514369, 138049, 20449, 1633, 65, 1] | -| TriAccRev | [1, 1, 2, 2, 6, 7, 6, 24, 33, 34, 24, 120, 192, 208, 209, 120, 720, 1320, 1520, 1545, 1546, 720, 5040, 10440, 12840, 13290, 13326, 13327, 5040, 40320, 93240, 122640, 129990, 130872, 130921, 130922, 40320, 362880, 927360, 1303680, 1421280, 1440096, 1441664, 1441728, 1441729] | -| TriDiag | [1, 1, 1, 1, 1, 4, 1, 9, 2, 1, 16, 18, 1, 25, 72, 6, 1, 36, 200, 96] | - -LaguerreRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| EvenSum | [1, 1, 3, 19, 97, 801, 6571, 65563, 723969] | -| OddSum | [0, 1, 4, 15, 112, 745, 6756, 65359, 717760] | -| AltSum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | -| AccSum | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | -| AccRevSum | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | -| RevAccSum | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | -| DiagSum | [1, 1, 2, 5, 12, 35, 104, 333] | - -LaguerreRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | -| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | -| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | -| Middle | [1, 1, 4, 9, 72, 200, 2400, 7350, 117600] | -| Central | [1, 4, 72, 2400, 117600] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432] | -| NegHalf | [1, -1, -2, -2, 8, 88, 592, 3344, 14464] | -| Bin | [1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273] | -| InvBin | [1, 0, -5, -22, 9, 1244, 14335, 79470, -586943] | -| TransSqrs | [0, 1, 12, 135, 1552, 18825, 243756, 3380167, 50172480] | -| TransNat0 | [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008] | -| TransNat1 | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | - -LaguerreRev Diagonals as rows +Laguerre:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 4, 2], [1, 9, 18, 6], [1, 16, 72, 96, 24], [1, 25, 200, 600, 600, 120], [1, 36, 450, 2400, 5400, 4320, 720], [1, 49, 882, 7350, 29400, 52920, 35280, 5040], [1, 64, 1568, 18816, 117600, 376320, 564480, 322560, 40320]] | +| RevTabl | [[1], [1, 1], [2, 4, 1], [6, 18, 9, 1], [24, 96, 72, 16, 1], [120, 600, 600, 200, 25, 1], [720, 4320, 5400, 2400, 450, 36, 1], [5040, 35280, 52920, 29400, 7350, 882, 49, 1], [40320, 322560, 564480, 376320, 117600, 18816, 1568, 64, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 4], [1, 9, 2], [1, 16, 18], [1, 25, 72, 6], [1, 36, 200, 96], [1, 49, 450, 600, 24]] | +| AccTabl | [[1], [1, 2], [1, 5, 7], [1, 10, 28, 34], [1, 17, 89, 185, 209], [1, 26, 226, 826, 1426, 1546], [1, 37, 487, 2887, 8287, 12607, 13327], [1, 50, 932, 8282, 37682, 90602, 125882, 130922], [1, 65, 1633, 20449, 138049, 514369, 1078849, 1401409, 1441729]] | +| RevAccTabl | [[1], [2, 1], [7, 5, 1], [34, 28, 10, 1], [209, 185, 89, 17, 1], [1546, 1426, 826, 226, 26, 1], [13327, 12607, 8287, 2887, 487, 37, 1], [130922, 125882, 90602, 37682, 8282, 932, 50, 1], [1441729, 1401409, 1078849, 514369, 138049, 20449, 1633, 65, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 6, 7], [6, 24, 33, 34], [24, 120, 192, 208, 209], [120, 720, 1320, 1520, 1545, 1546], [720, 5040, 10440, 12840, 13290, 13326, 13327], [5040, 40320, 93240, 122640, 129990, 130872, 130921, 130922], [40320, 362880, 927360, 1303680, 1421280, 1440096, 1441664, 1441728, 1441729]] | + +Laguerre:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| EvenSum | [1, 1, 3, 19, 97, 801, 6571, 65563, 723969] | +| OddSum | [0, 1, 4, 15, 112, 745, 6756, 65359, 717760] | +| AltSum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | +| AccSum | [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553] | +| AccRevSum | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | +| AntiDiagSum | [1, 1, 2, 5, 12, 35, 104, 333, 1124] | + +Laguerre:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | +| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | +| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | +| ColMiddle | [1, 1, 4, 9, 72, 200, 2400, 7350, 117600] | +| ColECenter | [1, 4, 72, 2400, 117600] | +| ColOCenter | [1, 9, 200, 7350] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 12, 135, 1552, 18825, 243756, 3380167, 50172480] | +| TransNat0 | [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008] | +| TransNat1 | [1, 3, 15, 97, 753, 6771, 68983, 783945, 9820737] | + +Laguerre:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| @@ -283,7 +271,7 @@ LaguerreRev Diagonals as rows | DiagRow7 | [1, 64, 2592, 86400, 2613600, 75271680, 2120152320, 59364264960, 1669619952000]| | DiagRow8 | [1, 81, 4050, 163350, 5880600, 198764280, 6492966480, 208702494000, 6678479808000]| -LaguerreRev Diagonals as columns +Laguerre:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ LaguerreRev Diagonals as columns | DiagCol7 | [5040, 322560, 6531840, 72576000, 548856000, 3161410560, 14841066240, 59364264960, 208702494000] | | DiagCol8 | [40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000] | -LaguerreRev Polynomial values as rows +Laguerre:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ LaguerreRev Polynomial values as rows | PolyRow7 | [1, 130922, 5129307, 52189096, 289283429, 1129691046, 3508129087, 9263893892, 21676411401] | | PolyRow8 | [1, 1441729, 101817089, 1482977857, 10656031489, 51021891521, 187437447169, 571103012929, 1513712421377] | -LaguerreRev Polynomial values as columns +Laguerre:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ LaguerreRev Polynomial values as columns | PolyCol7 | [1, 8, 127, 3004, 94193, 3673216, 171124423, 9263893892, 571103012929] | | PolyCol8 | [1, 9, 161, 4297, 152193, 6709961, 353677729, 21676411401, 1513712421377] | -# LaguerreInvRev -True +# Laguerre:Inv:Rev +[] -LaguerreInvRev Triangle view +Laguerre:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ LaguerreInvRev Triangle view | Row7 | [1, -49, 882, -7350, 29400, -52920, 35280, -5040] | | Row8 | [1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320] | -LaguerreInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -4, 2, 1, -9, 18, -6, 1, -16, 72, -96, 24, 1, -25, 200, -600, 600, -120, 1, -36, 450, -2400, 5400, -4320, 720, 1, -49, 882, -7350, 29400, -52920, 35280, -5040, 1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320] | -| TriRev | [1, -1, 1, 2, -4, 1, -6, 18, -9, 1, 24, -96, 72, -16, 1, -120, 600, -600, 200, -25, 1, 720, -4320, 5400, -2400, 450, -36, 1, -5040, 35280, -52920, 29400, -7350, 882, -49, 1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1] | -| TriAcc | [1, 1, 0, 1, -3, -1, 1, -8, 10, 4, 1, -15, 57, -39, -15, 1, -24, 176, -424, 176, 56, 1, -35, 415, -1985, 3415, -905, -185, 1, -48, 834, -6516, 22884, -30036, 5244, 204, 1, -63, 1505, -17311, 100289, -276031, 288449, -34111, 6209] | -| TriRevAcc | [1, 0, 1, -1, -3, 1, 4, 10, -8, 1, -15, -39, 57, -15, 1, 56, 176, -424, 176, -24, 1, -185, -905, 3415, -1985, 415, -35, 1, 204, 5244, -30036, 22884, -6516, 834, -48, 1, 6209, -34111, 288449, -276031, 100289, -17311, 1505, -63, 1] | -| TriAccRev | [1, -1, 0, 2, -2, -1, -6, 12, 3, 4, 24, -72, 0, -16, -15, -120, 480, -120, 80, 55, 56, 720, -3600, 1800, -600, -150, -186, -185, -5040, 30240, -22680, 6720, -630, 252, 203, 204, 40320, -282240, 282240, -94080, 23520, 4704, 6272, 6208, 6209] | -| TriDiag | [1, 1, 1, -1, 1, -4, 1, -9, 2, 1, -16, 18, 1, -25, 72, -6, 1, -36, 200, -96] | - -LaguerreInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | -| EvenSum | [1, 1, 3, 19, 97, 801, 6571, 65563, 723969] | -| OddSum | [0, -1, -4, -15, -112, -745, -6756, -65359, -717760] | -| AltSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| AccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| AccRevSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| RevAccSum | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | -| DiagSum | [1, 1, 0, -3, -6, 3, 42, 69] | - -LaguerreInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | -| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | -| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | -| Middle | [1, 1, -4, -9, 72, 200, -2400, -7350, 117600] | -| Central | [1, -4, 72, -2400, 117600] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| PosHalf | [1, 1, -2, 2, 8, -88, 592, -3344, 14464] | -| NegHalf | [1, -3, 14, -86, 648, -5752, 58576, -671568, 8546432] | -| Bin | [1, 0, -5, 22, 9, -1244, 14335, -79470, -586943] | -| InvBin | [1, -2, 11, -88, 905, -11246, 162607, -2668436, 48830273] | -| TransSqrs | [0, -1, 4, 9, -208, 1975, -15516, 107849, -593216] | -| TransNat0 | [0, -1, 0, 9, -64, 375, -2016, 9065, -13056] | -| TransNat1 | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | - -LaguerreInvRev Diagonals as rows +Laguerre:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -4, 2], [1, -9, 18, -6], [1, -16, 72, -96, 24], [1, -25, 200, -600, 600, -120], [1, -36, 450, -2400, 5400, -4320, 720], [1, -49, 882, -7350, 29400, -52920, 35280, -5040], [1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320]] | +| RevTabl | [[1], [-1, 1], [2, -4, 1], [-6, 18, -9, 1], [24, -96, 72, -16, 1], [-120, 600, -600, 200, -25, 1], [720, -4320, 5400, -2400, 450, -36, 1], [-5040, 35280, -52920, 29400, -7350, 882, -49, 1], [40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -4], [1, -9, 2], [1, -16, 18], [1, -25, 72, -6], [1, -36, 200, -96], [1, -49, 450, -600, 24]] | +| AccTabl | [[1], [1, 0], [1, -3, -1], [1, -8, 10, 4], [1, -15, 57, -39, -15], [1, -24, 176, -424, 176, 56], [1, -35, 415, -1985, 3415, -905, -185], [1, -48, 834, -6516, 22884, -30036, 5244, 204], [1, -63, 1505, -17311, 100289, -276031, 288449, -34111, 6209]] | +| RevAccTabl | [[1], [0, 1], [-1, -3, 1], [4, 10, -8, 1], [-15, -39, 57, -15, 1], [56, 176, -424, 176, -24, 1], [-185, -905, 3415, -1985, 415, -35, 1], [204, 5244, -30036, 22884, -6516, 834, -48, 1], [6209, -34111, 288449, -276031, 100289, -17311, 1505, -63, 1]] | +| AccRevTabl | [[1], [-1, 0], [2, -2, -1], [-6, 12, 3, 4], [24, -72, 0, -16, -15], [-120, 480, -120, 80, 55, 56], [720, -3600, 1800, -600, -150, -186, -185], [-5040, 30240, -22680, 6720, -630, 252, 203, 204], [40320, -282240, 282240, -94080, 23520, 4704, 6272, 6208, 6209]] | + +Laguerre:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | +| EvenSum | [1, 1, 3, 19, 97, 801, 6571, 65563, 723969] | +| OddSum | [0, -1, -4, -15, -112, -745, -6756, -65359, -717760] | +| AltSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| AccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | +| AccRevSum | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | +| AntiDiagSum | [1, 1, 0, -3, -6, 3, 42, 69, -174] | + +Laguerre:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 600, 21600, 529200, 11289600] | +| RowGcd | [1, 1, 2, 3, 8, 5, 6, 7, 32] | +| RowMax | [1, 1, 4, 18, 96, 600, 5400, 52920, 564480] | +| ColMiddle | [1, 1, -4, -9, 72, 200, -2400, -7350, 117600] | +| ColECenter | [1, -4, 72, -2400, 117600] | +| ColOCenter | [1, -9, 200, -7350] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | +| TransSqrs | [0, -1, 4, 9, -208, 1975, -15516, 107849, -593216] | +| TransNat0 | [0, -1, 0, 9, -64, 375, -2016, 9065, -13056] | +| TransNat1 | [1, -1, -1, 13, -79, 431, -2201, 9269, -6847] | + +Laguerre:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 2, -6, 24, -120, 720, -5040, 40320]| @@ -391,7 +375,7 @@ LaguerreInvRev Diagonals as rows | DiagRow7 | [1, -64, 2592, -86400, 2613600, -75271680, 2120152320, -59364264960, 1669619952000]| | DiagRow8 | [1, -81, 4050, -163350, 5880600, -198764280, 6492966480, -208702494000, 6678479808000]| -LaguerreInvRev Diagonals as columns +Laguerre:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ LaguerreInvRev Diagonals as columns | DiagCol7 | [-5040, -322560, -6531840, -72576000, -548856000, -3161410560, -14841066240, -59364264960, -208702494000] | | DiagCol8 | [40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000] | -LaguerreInvRev Polynomial values as rows +Laguerre:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ LaguerreInvRev Polynomial values as rows | PolyRow7 | [1, 204, 334391, 4027822, 14811357, 9603056, -139813421, -821315214, -2938573063] | | PolyRow8 | [1, 6209, -5144063, -111807551, -686726911, -1734813119, 984476929, 27156423617, 136115947009] | -LaguerreInvRev Polynomial values as columns +Laguerre:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Lah.md b/data/md/Lah.md index 3ad57ea..c5c6c57 100644 --- a/data/md/Lah.md +++ b/data/md/Lah.md @@ -1,5 +1,5 @@ # Lah -['A008297', 'A066667', 'A089231', 'A105278', 'A111596', 'A271703'] +['A271703', 'A008297', 'A066667', 'A089231', 'A105278', 'A111596'] Lah Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Lah Triangle view | Row7 | [0, 5040, 15120, 12600, 4200, 630, 42, 1] | | Row8 | [0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1] | -Lah Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 12, 1, 0, 120, 240, 120, 20, 1, 0, 720, 1800, 1200, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1] | -| TriRev | [1, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 12, 36, 24, 0, 1, 20, 120, 240, 120, 0, 1, 30, 300, 1200, 1800, 720, 0, 1, 42, 630, 4200, 12600, 15120, 5040, 0, 1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 0] | -| TriAcc | [1, 0, 1, 0, 2, 3, 0, 6, 12, 13, 0, 24, 60, 72, 73, 0, 120, 360, 480, 500, 501, 0, 720, 2520, 3720, 4020, 4050, 4051, 0, 5040, 20160, 32760, 36960, 37590, 37632, 37633, 0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353] | -| TriRevAcc | [1, 1, 0, 3, 2, 0, 13, 12, 6, 0, 73, 72, 60, 24, 0, 501, 500, 480, 360, 120, 0, 4051, 4050, 4020, 3720, 2520, 720, 0, 37633, 37632, 37590, 36960, 32760, 20160, 5040, 0, 394353, 394352, 394296, 393120, 381360, 322560, 181440, 40320, 0] | -| TriAccRev | [1, 1, 1, 1, 3, 3, 1, 7, 13, 13, 1, 13, 49, 73, 73, 1, 21, 141, 381, 501, 501, 1, 31, 331, 1531, 3331, 4051, 4051, 1, 43, 673, 4873, 17473, 32593, 37633, 37633, 1, 57, 1233, 12993, 71793, 212913, 354033, 394353, 394353] | -| TriDiag | [1, 0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 6, 0, 120, 36, 1, 0, 720, 240, 12] | +Lah Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 1], [0, 6, 6, 1], [0, 24, 36, 12, 1], [0, 120, 240, 120, 20, 1], [0, 720, 1800, 1200, 300, 30, 1], [0, 5040, 15120, 12600, 4200, 630, 42, 1], [0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1]] | +| RevTabl | [[1], [1, 0], [1, 2, 0], [1, 6, 6, 0], [1, 12, 36, 24, 0], [1, 20, 120, 240, 120, 0], [1, 30, 300, 1200, 1800, 720, 0], [1, 42, 630, 4200, 12600, 15120, 5040, 0], [1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 6, 1], [0, 24, 6], [0, 120, 36, 1], [0, 720, 240, 12], [0, 5040, 1800, 120, 1]] | +| AccTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 12, 13], [0, 24, 60, 72, 73], [0, 120, 360, 480, 500, 501], [0, 720, 2520, 3720, 4020, 4050, 4051], [0, 5040, 20160, 32760, 36960, 37590, 37632, 37633], [0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 0], [13, 12, 6, 0], [73, 72, 60, 24, 0], [501, 500, 480, 360, 120, 0], [4051, 4050, 4020, 3720, 2520, 720, 0], [37633, 37632, 37590, 36960, 32760, 20160, 5040, 0], [394353, 394352, 394296, 393120, 381360, 322560, 181440, 40320, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 3, 3], [1, 7, 13, 13], [1, 13, 49, 73, 73], [1, 21, 141, 381, 501, 501], [1, 31, 331, 1531, 3331, 4051, 4051], [1, 43, 673, 4873, 17473, 32593, 37633, 37633], [1, 57, 1233, 12993, 71793, 212913, 354033, 394353, 394353]] | Lah Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | -| EvenSum | [1, 0, 1, 6, 37, 260, 2101, 19362, 201097] | -| OddSum | [0, 1, 2, 7, 36, 241, 1950, 18271, 193256] | -| AltSum | [1, -1, -1, -1, 1, 19, 151, 1091, 7841] | -| AccSum | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | -| AccRevSum | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | -| RevAccSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| DiagSum | [1, 0, 1, 2, 7, 30, 157, 972] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | +| EvenSum | [1, 0, 1, 6, 37, 260, 2101, 19362, 201097] | +| OddSum | [0, 1, 2, 7, 36, 241, 1950, 18271, 193256] | +| AltSum | [1, -1, -1, -1, 1, 19, 151, 1091, 7841] | +| AccSum | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | +| AccRevSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| AntiDiagSum | [1, 0, 1, 2, 7, 30, 157, 972, 6961] | Lah Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | -| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | -| Middle | [1, 0, 2, 6, 36, 240, 1200, 12600, 58800] | -| Central | [1, 2, 36, 1200, 58800] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177] | -| NegHalf | [1, 1, -3, 13, -71, 441, -2699, 9157, 206193] | -| Bin | [1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577] | -| InvBin | [1, 1, -3, 1, 73, -699, 3001, 24697, -783999] | -| TransSqrs | [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824] | -| TransNat0 | [0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376] | -| TransNat1 | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | +| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | +| ColMiddle | [1, 0, 2, 6, 36, 240, 1200, 12600, 58800] | +| ColECenter | [1, 2, 36, 1200, 58800] | +| ColOCenter | [0, 6, 240, 12600] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824] | +| TransNat0 | [0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376] | +| TransNat1 | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | Lah Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Lah Polynomial values as columns | PolyCol7 | [1, 7, 63, 679, 8449, 118587, 1846999, 31535371, 584668833] | | PolyCol8 | [1, 8, 80, 944, 12736, 192448, 3209344, 58413440, 1150303232] | -# LahInv -True +# Lah:Inv +[] -LahInv Triangle view +Lah:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LahInv Triangle view | Row7 | [0, 5040, -15120, 12600, -4200, 630, -42, 1] | | Row8 | [0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1] | -LahInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1] | -| TriRev | [1, 1, 0, 1, -2, 0, 1, -6, 6, 0, 1, -12, 36, -24, 0, 1, -20, 120, -240, 120, 0, 1, -30, 300, -1200, 1800, -720, 0, 1, -42, 630, -4200, 12600, -15120, 5040, 0, 1, -56, 1176, -11760, 58800, -141120, 141120, -40320, 0] | -| TriAcc | [1, 0, 1, 0, -2, -1, 0, 6, 0, 1, 0, -24, 12, 0, 1, 0, 120, -120, 0, -20, -19, 0, -720, 1080, -120, 180, 150, 151, 0, 5040, -10080, 2520, -1680, -1050, -1092, -1091, 0, -40320, 100800, -40320, 18480, 6720, 7896, 7840, 7841] | -| TriRevAcc | [1, 1, 0, -1, -2, 0, 1, 0, 6, 0, 1, 0, 12, -24, 0, -19, -20, 0, -120, 120, 0, 151, 150, 180, -120, 1080, -720, 0, -1091, -1092, -1050, -1680, 2520, -10080, 5040, 0, 7841, 7840, 7896, 6720, 18480, -40320, 100800, -40320, 0] | -| TriAccRev | [1, 1, 1, 1, -1, -1, 1, -5, 1, 1, 1, -11, 25, 1, 1, 1, -19, 101, -139, -19, -19, 1, -29, 271, -929, 871, 151, 151, 1, -41, 589, -3611, 8989, -6131, -1091, -1091, 1, -55, 1121, -10639, 48161, -92959, 48161, 7841, 7841] | -| TriDiag | [1, 0, 0, 1, 0, -2, 0, 6, 1, 0, -24, -6, 0, 120, 36, 1, 0, -720, -240, -12] | - -LahInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | -| EvenSum | [1, 0, 1, -6, 37, -260, 2101, -19362, 201097] | -| OddSum | [0, 1, -2, 7, -36, 241, -1950, 18271, -193256] | -| AltSum | [1, -1, 3, -13, 73, -501, 4051, -37633, 394353] | -| AccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| AccRevSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| RevAccSum | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | -| DiagSum | [1, 0, 1, -2, 7, -30, 157, -972] | - -LahInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 72, 240, 3600, 75600, 1411200] | -| RowGcd | [1, 1, -2, 6, 12, 20, 30, 42, 56] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | -| Middle | [1, 0, -2, 6, 36, -240, -1200, 12600, 58800] | -| Central | [1, -2, 36, -1200, 58800] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, 13, -71, 441, -2699, 9157, 206193] | -| NegHalf | [1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177] | -| Bin | [1, 1, -3, 1, 73, -699, 3001, 24697, -783999] | -| InvBin | [1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577] | -| TransSqrs | [0, 1, 2, -9, 28, -55, -234, 5047, -59464] | -| TransNat0 | [0, 1, 0, -3, 16, -75, 336, -1295, 1632] | -| TransNat1 | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | - -LahInv Diagonals as rows +Lah:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -2, 1], [0, 6, -6, 1], [0, -24, 36, -12, 1], [0, 120, -240, 120, -20, 1], [0, -720, 1800, -1200, 300, -30, 1], [0, 5040, -15120, 12600, -4200, 630, -42, 1], [0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1]] | +| RevTabl | [[1], [1, 0], [1, -2, 0], [1, -6, 6, 0], [1, -12, 36, -24, 0], [1, -20, 120, -240, 120, 0], [1, -30, 300, -1200, 1800, -720, 0], [1, -42, 630, -4200, 12600, -15120, 5040, 0], [1, -56, 1176, -11760, 58800, -141120, 141120, -40320, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -2], [0, 6, 1], [0, -24, -6], [0, 120, 36, 1], [0, -720, -240, -12], [0, 5040, 1800, 120, 1]] | +| AccTabl | [[1], [0, 1], [0, -2, -1], [0, 6, 0, 1], [0, -24, 12, 0, 1], [0, 120, -120, 0, -20, -19], [0, -720, 1080, -120, 180, 150, 151], [0, 5040, -10080, 2520, -1680, -1050, -1092, -1091], [0, -40320, 100800, -40320, 18480, 6720, 7896, 7840, 7841]] | +| RevAccTabl | [[1], [1, 0], [-1, -2, 0], [1, 0, 6, 0], [1, 0, 12, -24, 0], [-19, -20, 0, -120, 120, 0], [151, 150, 180, -120, 1080, -720, 0], [-1091, -1092, -1050, -1680, 2520, -10080, 5040, 0], [7841, 7840, 7896, 6720, 18480, -40320, 100800, -40320, 0]] | +| AccRevTabl | [[1], [1, 1], [1, -1, -1], [1, -5, 1, 1], [1, -11, 25, 1, 1], [1, -19, 101, -139, -19, -19], [1, -29, 271, -929, 871, 151, 151], [1, -41, 589, -3611, 8989, -6131, -1091, -1091], [1, -55, 1121, -10639, 48161, -92959, 48161, 7841, 7841]] | + +Lah:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | +| EvenSum | [1, 0, 1, -6, 37, -260, 2101, -19362, 201097] | +| OddSum | [0, 1, -2, 7, -36, 241, -1950, 18271, -193256] | +| AltSum | [1, -1, 3, -13, 73, -501, 4051, -37633, 394353] | +| AccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | +| AccRevSum | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | +| AntiDiagSum | [1, 0, 1, -2, 7, -30, 157, -972, 6961] | + +Lah:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | +| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | +| ColMiddle | [1, 0, -2, 6, 36, -240, -1200, 12600, 58800] | +| ColECenter | [1, -2, 36, -1200, 58800] | +| ColOCenter | [0, 6, -240, 12600] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -9, 28, -55, -234, 5047, -59464] | +| TransNat0 | [0, 1, 0, -3, 16, -75, 336, -1295, 1632] | +| TransNat1 | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | + +Lah:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ LahInv Diagonals as rows | DiagRow7 | [0, -40320, -1451520, -21772800, -199584000, -1317254400, -6849722880, -29682132480, -111307996800]| | DiagRow8 | [0, 362880, 16329600, 299376000, 3293136000, 25686460800, 155831195520, 779155977600, 3339239904000]| -LahInv Diagonals as columns +Lah:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ LahInv Diagonals as columns | DiagCol7 | [1, -56, 2016, -60480, 1663200, -43908480, 1141620480, -29682132480, 779155977600] | | DiagCol8 | [1, -72, 3240, -118800, 3920400, -122316480, 3710266560, -111307996800, 3339239904000] | -LahInv Polynomial values as rows +Lah:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LahInv Polynomial values as rows | PolyRow7 | [0, -1091, 800, 3699, -1088, -12175, -16416, 2695, 51584] | | PolyRow8 | [0, 7841, -12288, -14607, 40448, 77025, -13824, -241423, -433152] | -LahInv Polynomial values as columns +Lah:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ LahInv Polynomial values as columns | PolyCol7 | [1, 7, 35, 91, -119, -973, 5299, 2695, -241423] | | PolyCol8 | [1, 8, 48, 176, 64, -2112, 2944, 51584, -433152] | -# LahRev -True +# Lah:Rev +[] -LahRev Triangle view +Lah:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ LahRev Triangle view | Row7 | [1, 42, 630, 4200, 12600, 15120, 5040, 0] | | Row8 | [1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 0] | -LahRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 12, 36, 24, 0, 1, 20, 120, 240, 120, 0, 1, 30, 300, 1200, 1800, 720, 0, 1, 42, 630, 4200, 12600, 15120, 5040, 0, 1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 0] | -| TriRev | [1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 12, 1, 0, 120, 240, 120, 20, 1, 0, 720, 1800, 1200, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1] | -| TriAcc | [1, 1, 1, 1, 3, 3, 1, 7, 13, 13, 1, 13, 49, 73, 73, 1, 21, 141, 381, 501, 501, 1, 31, 331, 1531, 3331, 4051, 4051, 1, 43, 673, 4873, 17473, 32593, 37633, 37633, 1, 57, 1233, 12993, 71793, 212913, 354033, 394353, 394353] | -| TriRevAcc | [1, 1, 1, 3, 3, 1, 13, 13, 7, 1, 73, 73, 49, 13, 1, 501, 501, 381, 141, 21, 1, 4051, 4051, 3331, 1531, 331, 31, 1, 37633, 37633, 32593, 17473, 4873, 673, 43, 1, 394353, 394353, 354033, 212913, 71793, 12993, 1233, 57, 1] | -| TriAccRev | [1, 0, 1, 0, 2, 3, 0, 6, 12, 13, 0, 24, 60, 72, 73, 0, 120, 360, 480, 500, 501, 0, 720, 2520, 3720, 4020, 4050, 4051, 0, 5040, 20160, 32760, 36960, 37590, 37632, 37633, 0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353] | -| TriDiag | [1, 1, 1, 0, 1, 2, 1, 6, 0, 1, 12, 6, 1, 20, 36, 0, 1, 30, 120, 24] | - -LahRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | -| EvenSum | [1, 1, 1, 7, 37, 241, 2101, 18271, 201097] | -| OddSum | [0, 0, 2, 6, 36, 260, 1950, 19362, 193256] | -| AltSum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | -| AccSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| AccRevSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| RevAccSum | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | -| DiagSum | [1, 1, 1, 3, 7, 19, 57, 175] | - -LahRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | -| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | -| Middle | [1, 1, 2, 6, 36, 120, 1200, 4200, 58800] | -| Central | [1, 2, 36, 1200, 58800] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888] | -| NegHalf | [1, -2, 0, 4, 16, 48, 64, -800, -12288] | -| Bin | [1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577] | -| InvBin | [1, -1, -3, -1, 73, 699, 3001, -24697, -783999] | -| TransSqrs | [0, 0, 2, 30, 372, 4580, 58830, 801402, 11635400] | -| TransNat0 | [0, 0, 2, 18, 156, 1460, 15030, 170142, 2107448] | -| TransNat1 | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | - -LahRev Diagonals as rows +Lah:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 2, 0], [1, 6, 6, 0], [1, 12, 36, 24, 0], [1, 20, 120, 240, 120, 0], [1, 30, 300, 1200, 1800, 720, 0], [1, 42, 630, 4200, 12600, 15120, 5040, 0], [1, 56, 1176, 11760, 58800, 141120, 141120, 40320, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 1], [0, 6, 6, 1], [0, 24, 36, 12, 1], [0, 120, 240, 120, 20, 1], [0, 720, 1800, 1200, 300, 30, 1], [0, 5040, 15120, 12600, 4200, 630, 42, 1], [0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 2], [1, 6, 0], [1, 12, 6], [1, 20, 36, 0], [1, 30, 120, 24], [1, 42, 300, 240, 0]] | +| AccTabl | [[1], [1, 1], [1, 3, 3], [1, 7, 13, 13], [1, 13, 49, 73, 73], [1, 21, 141, 381, 501, 501], [1, 31, 331, 1531, 3331, 4051, 4051], [1, 43, 673, 4873, 17473, 32593, 37633, 37633], [1, 57, 1233, 12993, 71793, 212913, 354033, 394353, 394353]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 1], [13, 13, 7, 1], [73, 73, 49, 13, 1], [501, 501, 381, 141, 21, 1], [4051, 4051, 3331, 1531, 331, 31, 1], [37633, 37633, 32593, 17473, 4873, 673, 43, 1], [394353, 394353, 354033, 212913, 71793, 12993, 1233, 57, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 12, 13], [0, 24, 60, 72, 73], [0, 120, 360, 480, 500, 501], [0, 720, 2520, 3720, 4020, 4050, 4051], [0, 5040, 20160, 32760, 36960, 37590, 37632, 37633], [0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353]] | + +Lah:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | +| EvenSum | [1, 1, 1, 7, 37, 241, 2101, 18271, 201097] | +| OddSum | [0, 0, 2, 6, 36, 260, 1950, 19362, 193256] | +| AltSum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | +| AccSum | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | +| AccRevSum | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | +| AntiDiagSum | [1, 1, 1, 3, 7, 19, 57, 175, 583] | + +Lah:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | +| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | +| ColMiddle | [1, 1, 2, 6, 36, 120, 1200, 4200, 58800] | +| ColECenter | [1, 2, 36, 1200, 58800] | +| ColOCenter | [1, 6, 120, 4200] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 30, 372, 4580, 58830, 801402, 11635400] | +| TransNat0 | [0, 0, 2, 18, 156, 1460, 15030, 170142, 2107448] | +| TransNat1 | [1, 1, 5, 31, 229, 1961, 19081, 207775, 2501801] | + +Lah:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ LahRev Diagonals as rows | DiagRow7 | [1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480, 779155977600]| | DiagRow8 | [1, 72, 3240, 118800, 3920400, 122316480, 3710266560, 111307996800, 3339239904000]| -LahRev Diagonals as columns +Lah:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ LahRev Diagonals as columns | DiagCol7 | [0, 40320, 1451520, 21772800, 199584000, 1317254400, 6849722880, 29682132480, 111307996800] | | DiagCol8 | [0, 362880, 16329600, 299376000, 3293136000, 25686460800, 155831195520, 779155977600, 3339239904000] | -LahRev Polynomial values as rows +Lah:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ LahRev Polynomial values as rows | PolyRow7 | [1, 37633, 1044205, 8488117, 39631369, 134415961, 369979093, 878797165, 1870458577] | | PolyRow8 | [1, 394353, 19748177, 230439553, 1398961761, 5834249681, 19047250993, 52324954977, 126422091713] | -LahRev Polynomial values as columns +Lah:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ LahRev Polynomial values as columns | PolyCol7 | [1, 1, 15, 337, 10081, 376461, 16849351, 878797165, 52324954977] | | PolyCol8 | [1, 1, 17, 433, 14689, 622241, 31599601, 1870458577, 126422091713] | -# LahInvRev -True +# Lah:Inv:Rev +[] -LahInvRev Triangle view +Lah:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ LahInvRev Triangle view | Row7 | [1, -42, 630, -4200, 12600, -15120, 5040, 0] | | Row8 | [1, -56, 1176, -11760, 58800, -141120, 141120, -40320, 0] | -LahInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -2, 0, 1, -6, 6, 0, 1, -12, 36, -24, 0, 1, -20, 120, -240, 120, 0, 1, -30, 300, -1200, 1800, -720, 0, 1, -42, 630, -4200, 12600, -15120, 5040, 0, 1, -56, 1176, -11760, 58800, -141120, 141120, -40320, 0] | -| TriRev | [1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1] | -| TriAcc | [1, 1, 1, 1, -1, -1, 1, -5, 1, 1, 1, -11, 25, 1, 1, 1, -19, 101, -139, -19, -19, 1, -29, 271, -929, 871, 151, 151, 1, -41, 589, -3611, 8989, -6131, -1091, -1091, 1, -55, 1121, -10639, 48161, -92959, 48161, 7841, 7841] | -| TriRevAcc | [1, 1, 1, -1, -1, 1, 1, 1, -5, 1, 1, 1, 25, -11, 1, -19, -19, -139, 101, -19, 1, 151, 151, 871, -929, 271, -29, 1, -1091, -1091, -6131, 8989, -3611, 589, -41, 1, 7841, 7841, 48161, -92959, 48161, -10639, 1121, -55, 1] | -| TriAccRev | [1, 0, 1, 0, -2, -1, 0, 6, 0, 1, 0, -24, 12, 0, 1, 0, 120, -120, 0, -20, -19, 0, -720, 1080, -120, 180, 150, 151, 0, 5040, -10080, 2520, -1680, -1050, -1092, -1091, 0, -40320, 100800, -40320, 18480, 6720, 7896, 7840, 7841] | -| TriDiag | [1, 1, 1, 0, 1, -2, 1, -6, 0, 1, -12, 6, 1, -20, 36, 0, 1, -30, 120, -24] | - -LahInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | -| EvenSum | [1, 1, 1, 7, 37, 241, 2101, 18271, 201097] | -| OddSum | [0, 0, -2, -6, -36, -260, -1950, -19362, -193256] | -| AltSum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | -| AccSum | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | -| AccRevSum | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | -| RevAccSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | -| DiagSum | [1, 1, 1, -1, -5, -5, 17, 67] | - -LahInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 6, 72, 240, 3600, 75600, 1411200] | -| RowGcd | [1, 1, -2, 6, 12, 20, 30, 42, 56] | -| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | -| Middle | [1, 1, -2, -6, 36, 120, -1200, -4200, 58800] | -| Central | [1, -2, 36, -1200, 58800] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 0, -4, 16, -48, 64, 800, -12288] | -| NegHalf | [1, -2, 8, -44, 304, -2512, 24064, -261536, 3173888] | -| Bin | [1, 1, -3, 1, 73, -699, 3001, 24697, -783999] | -| InvBin | [1, -1, 5, -37, 361, -4301, 60001, -954325, 16984577] | -| TransSqrs | [0, 0, -2, 18, -84, 220, 1170, -30282, 416248] | -| TransNat0 | [0, 0, -2, 6, -12, -20, 570, -6342, 61096] | -| TransNat1 | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | - -LahInvRev Diagonals as rows +Lah:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -2, 0], [1, -6, 6, 0], [1, -12, 36, -24, 0], [1, -20, 120, -240, 120, 0], [1, -30, 300, -1200, 1800, -720, 0], [1, -42, 630, -4200, 12600, -15120, 5040, 0], [1, -56, 1176, -11760, 58800, -141120, 141120, -40320, 0]] | +| RevTabl | [[1], [0, 1], [0, -2, 1], [0, 6, -6, 1], [0, -24, 36, -12, 1], [0, 120, -240, 120, -20, 1], [0, -720, 1800, -1200, 300, -30, 1], [0, 5040, -15120, 12600, -4200, 630, -42, 1], [0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -2], [1, -6, 0], [1, -12, 6], [1, -20, 36, 0], [1, -30, 120, -24], [1, -42, 300, -240, 0]] | +| AccTabl | [[1], [1, 1], [1, -1, -1], [1, -5, 1, 1], [1, -11, 25, 1, 1], [1, -19, 101, -139, -19, -19], [1, -29, 271, -929, 871, 151, 151], [1, -41, 589, -3611, 8989, -6131, -1091, -1091], [1, -55, 1121, -10639, 48161, -92959, 48161, 7841, 7841]] | +| RevAccTabl | [[1], [1, 1], [-1, -1, 1], [1, 1, -5, 1], [1, 1, 25, -11, 1], [-19, -19, -139, 101, -19, 1], [151, 151, 871, -929, 271, -29, 1], [-1091, -1091, -6131, 8989, -3611, 589, -41, 1], [7841, 7841, 48161, -92959, 48161, -10639, 1121, -55, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -2, -1], [0, 6, 0, 1], [0, -24, 12, 0, 1], [0, 120, -120, 0, -20, -19], [0, -720, 1080, -120, 180, 150, 151], [0, 5040, -10080, 2520, -1680, -1050, -1092, -1091], [0, -40320, 100800, -40320, 18480, 6720, 7896, 7840, 7841]] | + +Lah:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, 1, 1, -19, 151, -1091, 7841] | +| EvenSum | [1, 1, 1, 7, 37, 241, 2101, 18271, 201097] | +| OddSum | [0, 0, -2, -6, -36, -260, -1950, -19362, -193256] | +| AltSum | [1, 1, 3, 13, 73, 501, 4051, 37633, 394353] | +| AccSum | [1, 2, -1, -2, 17, -94, 487, -2386, 9473] | +| AccRevSum | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | +| AntiDiagSum | [1, 1, 1, -1, -5, -5, 17, 67, 19] | + +Lah:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 72, 240, 3600, 75600, 1411200] | +| RowGcd | [1, 1, 2, 6, 12, 20, 30, 42, 56] | +| RowMax | [1, 1, 2, 6, 36, 240, 1800, 15120, 141120] | +| ColMiddle | [1, 1, -2, -6, 36, 120, -1200, -4200, 58800] | +| ColECenter | [1, -2, 36, -1200, 58800] | +| ColOCenter | [1, -6, 120, -4200] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -2, 18, -84, 220, 1170, -30282, 416248] | +| TransNat0 | [0, 0, -2, 6, -12, -20, 570, -6342, 61096] | +| TransNat1 | [1, 1, -3, 7, -11, -39, 721, -7433, 68937] | + +Lah:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ LahInvRev Diagonals as rows | DiagRow7 | [1, -56, 2016, -60480, 1663200, -43908480, 1141620480, -29682132480, 779155977600]| | DiagRow8 | [1, -72, 3240, -118800, 3920400, -122316480, 3710266560, -111307996800, 3339239904000]| -LahInvRev Diagonals as columns +Lah:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ LahInvRev Diagonals as columns | DiagCol7 | [0, -40320, -1451520, -21772800, -199584000, -1317254400, -6849722880, -29682132480, -111307996800] | | DiagCol8 | [0, 362880, 16329600, 299376000, 3293136000, 25686460800, 155831195520, 779155977600, 3339239904000] | -LahInvRev Polynomial values as rows +Lah:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ LahInvRev Polynomial values as rows | PolyRow7 | [1, -1091, 9157, 912745, 8127673, 38865541, 133017949, 367671697, 875252785] | | PolyRow8 | [1, 7841, 206193, -15139823, -212763487, -1350690879, -5726567279, -18837228367, -51952729023] | -LahInvRev Polynomial values as columns +Lah:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Lehmer.md b/data/md/Lehmer.md index ac00a0f..4c50ce3 100644 --- a/data/md/Lehmer.md +++ b/data/md/Lehmer.md @@ -1,5 +1,5 @@ # Lehmer -['A039621', 'A354794'] +['A354794', 'A039621'] Lehmer Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Lehmer Triangle view | Row7 | [0, 46656, 31031, 9751, 1890, 245, 21, 1] | | Row8 | [0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1] | -Lehmer Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 27, 19, 6, 1, 0, 256, 175, 55, 10, 1, 0, 3125, 2101, 660, 125, 15, 1, 0, 46656, 31031, 9751, 1890, 245, 21, 1, 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 6, 19, 27, 0, 1, 10, 55, 175, 256, 0, 1, 15, 125, 660, 2101, 3125, 0, 1, 21, 245, 1890, 9751, 31031, 46656, 0, 1, 28, 434, 4550, 33621, 170898, 543607, 823543, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 4, 7, 8, 0, 27, 46, 52, 53, 0, 256, 431, 486, 496, 497, 0, 3125, 5226, 5886, 6011, 6026, 6027, 0, 46656, 77687, 87438, 89328, 89573, 89594, 89595, 0, 823543, 1367150, 1538048, 1571669, 1576219, 1576653, 1576681, 1576682] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 8, 7, 4, 0, 53, 52, 46, 27, 0, 497, 496, 486, 431, 256, 0, 6027, 6026, 6011, 5886, 5226, 3125, 0, 89595, 89594, 89573, 89328, 87438, 77687, 46656, 0, 1576682, 1576681, 1576653, 1576219, 1571669, 1538048, 1367150, 823543, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 4, 8, 8, 1, 7, 26, 53, 53, 1, 11, 66, 241, 497, 497, 1, 16, 141, 801, 2902, 6027, 6027, 1, 22, 267, 2157, 11908, 42939, 89595, 89595, 1, 29, 463, 5013, 38634, 209532, 753139, 1576682, 1576682] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 4, 1, 0, 27, 3, 0, 256, 19, 1, 0, 3125, 175, 6] | +Lehmer Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 4, 3, 1], [0, 27, 19, 6, 1], [0, 256, 175, 55, 10, 1], [0, 3125, 2101, 660, 125, 15, 1], [0, 46656, 31031, 9751, 1890, 245, 21, 1], [0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 3, 4, 0], [1, 6, 19, 27, 0], [1, 10, 55, 175, 256, 0], [1, 15, 125, 660, 2101, 3125, 0], [1, 21, 245, 1890, 9751, 31031, 46656, 0], [1, 28, 434, 4550, 33621, 170898, 543607, 823543, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 4, 1], [0, 27, 3], [0, 256, 19, 1], [0, 3125, 175, 6], [0, 46656, 2101, 55, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 4, 7, 8], [0, 27, 46, 52, 53], [0, 256, 431, 486, 496, 497], [0, 3125, 5226, 5886, 6011, 6026, 6027], [0, 46656, 77687, 87438, 89328, 89573, 89594, 89595], [0, 823543, 1367150, 1538048, 1571669, 1576219, 1576653, 1576681, 1576682]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [8, 7, 4, 0], [53, 52, 46, 27, 0], [497, 496, 486, 431, 256, 0], [6027, 6026, 6011, 5886, 5226, 3125, 0], [89595, 89594, 89573, 89328, 87438, 77687, 46656, 0], [1576682, 1576681, 1576653, 1576219, 1571669, 1538048, 1367150, 823543, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 8, 8], [1, 7, 26, 53, 53], [1, 11, 66, 241, 497, 497], [1, 16, 141, 801, 2902, 6027, 6027], [1, 22, 267, 2157, 11908, 42939, 89595, 89595], [1, 29, 463, 5013, 38634, 209532, 753139, 1576682, 1576682]] | Lehmer Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 8, 53, 497, 6027, 89595, 1576682] | -| EvenSum | [1, 0, 1, 3, 20, 185, 2227, 32942, 577663] | -| OddSum | [0, 1, 1, 5, 33, 312, 3800, 56653, 999019] | -| AltSum | [1, -1, 0, -2, -13, -127, -1573, -23711, -421356] | -| AccSum | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | -| AccRevSum | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | -| RevAccSum | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | -| DiagSum | [1, 0, 1, 1, 5, 30, 276, 3306] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 8, 53, 497, 6027, 89595, 1576682] | +| EvenSum | [1, 0, 1, 3, 20, 185, 2227, 32942, 577663] | +| OddSum | [0, 1, 1, 5, 33, 312, 3800, 56653, 999019] | +| AltSum | [1, -1, 0, -2, -13, -127, -1573, -23711, -421356] | +| AccSum | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | +| AccRevSum | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | +| AntiDiagSum | [1, 0, 1, 1, 5, 30, 276, 3306, 48813] | Lehmer Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 12, 1026, 492800, 78787500, 10083803970240, 27122723143943517881700] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 4, 27, 256, 3125, 46656, 823543] | -| Middle | [1, 0, 1, 4, 19, 175, 660, 9751, 33621] | -| Central | [1, 1, 19, 660, 33621] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 23, 305, 5737, 139427, 4151135, 146249217] | -| NegHalf | [1, 1, -1, 11, -151, 2897, -71193, 2134827, -75588175] | -| Bin | [1, 1, 3, 22, 247, 3631, 65431, 1390971, 34000275] | -| InvBin | [1, 1, -1, 4, -17, 31, 1351, -44925, 1172963] | -| TransSqrs | [0, 1, 5, 25, 173, 1636, 19880, 295709, 5204799] | -| TransNat0 | [0, 1, 3, 13, 87, 816, 9888, 146889, 2583493] | -| TransNat1 | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 12, 1026, 492800, 78787500, 10083803970240, 27122723143943517881700] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 4, 27, 256, 3125, 46656, 823543] | +| ColMiddle | [1, 0, 1, 4, 19, 175, 660, 9751, 33621] | +| ColECenter | [1, 1, 19, 660, 33621] | +| ColOCenter | [0, 4, 175, 9751] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 25, 173, 1636, 19880, 295709, 5204799] | +| TransNat0 | [0, 1, 3, 13, 87, 816, 9888, 146889, 2583493] | +| TransNat1 | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | Lehmer Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Lehmer Polynomial values as columns | PolyCol7 | [1, 7, 56, 518, 5579, 70049, 1021083, 17141481, 328099100] | | PolyCol8 | [1, 8, 72, 736, 8600, 115136, 1763048, 30723520, 604952952] | -# LehmerInv -True +# Lehmer:Inv +[] -LehmerInv Triangle view +Lehmer:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LehmerInv Triangle view | Row7 | [0, -120, 28, 49, 35, 70, -21, 1] | | Row8 | [0, -720, 188, 196, 49, 0, 154, -28, 1] | -LehmerInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, -1, -3, 1, 0, -2, -1, -6, 1, 0, -6, 0, 5, -10, 1, 0, -24, 4, 15, 25, -15, 1, 0, -120, 28, 49, 35, 70, -21, 1, 0, -720, 188, 196, 49, 0, 154, -28, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -3, -1, 0, 1, -6, -1, -2, 0, 1, -10, 5, 0, -6, 0, 1, -15, 25, 15, 4, -24, 0, 1, -21, 70, 35, 49, 28, -120, 0, 1, -28, 154, 0, 49, 196, 188, -720, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, -1, -4, -3, 0, -2, -3, -9, -8, 0, -6, -6, -1, -11, -10, 0, -24, -20, -5, 20, 5, 6, 0, -120, -92, -43, -8, 62, 41, 42, 0, -720, -532, -336, -287, -287, -133, -161, -160] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, -3, -4, -1, 0, -8, -9, -3, -2, 0, -10, -11, -1, -6, -6, 0, 6, 5, 20, -5, -20, -24, 0, 42, 41, 62, -8, -43, -92, -120, 0, -160, -161, -133, -287, -287, -336, -532, -720, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -2, -3, -3, 1, -5, -6, -8, -8, 1, -9, -4, -4, -10, -10, 1, -14, 11, 26, 30, 6, 6, 1, -20, 50, 85, 134, 162, 42, 42, 1, -27, 127, 127, 176, 372, 560, -160, -160] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, -3, 0, -6, -1, 1, 0, -24, 0, -6] | - -LehmerInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -3, -8, -10, 6, 42, -160] | -| EvenSum | [1, 0, 1, -3, 0, -10, 30, 42, 392] | -| OddSum | [0, 1, -1, 0, -8, 0, -24, 0, -552] | -| AltSum | [1, -1, 2, -3, 8, -10, 54, 42, 944] | -| AccSum | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | -| AccRevSum | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | -| RevAccSum | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | -| DiagSum | [1, 0, 1, -1, 0, -5, -6, -30] | - -LehmerInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 600, 5880, 18239760] | -| RowGcd | [1, 1, 1, -3, 2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 6, 10, 25, 120, 720] | -| Middle | [1, 0, -1, -1, -1, 0, 15, 49, 49] | -| Central | [1, -1, -1, 15, 49] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, -9, -31, -95, -513, -5481, -72511] | -| NegHalf | [1, 1, 3, 3, 25, -55, 843, -7749, 99377] | -| Bin | [1, 1, -1, -11, -37, -29, 502, 4012, 17999] | -| InvBin | [1, 1, 3, 7, 27, 71, 370, 680, 8015] | -| TransSqrs | [0, 1, 3, -4, -44, -96, 188, 2036, 6816] | -| TransNat0 | [0, 1, 1, -4, -18, -26, 60, 454, 1176] | -| TransNat1 | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | - -LehmerInv Diagonals as rows +Lehmer:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, -1, -3, 1], [0, -2, -1, -6, 1], [0, -6, 0, 5, -10, 1], [0, -24, 4, 15, 25, -15, 1], [0, -120, 28, 49, 35, 70, -21, 1], [0, -720, 188, 196, 49, 0, 154, -28, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -3, -1, 0], [1, -6, -1, -2, 0], [1, -10, 5, 0, -6, 0], [1, -15, 25, 15, 4, -24, 0], [1, -21, 70, 35, 49, 28, -120, 0], [1, -28, 154, 0, 49, 196, 188, -720, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, -1, 1], [0, -2, -3], [0, -6, -1, 1], [0, -24, 0, -6], [0, -120, 4, 5, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, -1, -4, -3], [0, -2, -3, -9, -8], [0, -6, -6, -1, -11, -10], [0, -24, -20, -5, 20, 5, 6], [0, -120, -92, -43, -8, 62, 41, 42], [0, -720, -532, -336, -287, -287, -133, -161, -160]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [-3, -4, -1, 0], [-8, -9, -3, -2, 0], [-10, -11, -1, -6, -6, 0], [6, 5, 20, -5, -20, -24, 0], [42, 41, 62, -8, -43, -92, -120, 0], [-160, -161, -133, -287, -287, -336, -532, -720, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -3, -3], [1, -5, -6, -8, -8], [1, -9, -4, -4, -10, -10], [1, -14, 11, 26, 30, 6, 6], [1, -20, 50, 85, 134, 162, 42, 42], [1, -27, 127, 127, 176, 372, 560, -160, -160]] | + +Lehmer:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -3, -8, -10, 6, 42, -160] | +| EvenSum | [1, 0, 1, -3, 0, -10, 30, 42, 392] | +| OddSum | [0, 1, -1, 0, -8, 0, -24, 0, -552] | +| AltSum | [1, -1, 2, -3, 8, -10, 54, 42, 944] | +| AccSum | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | +| AccRevSum | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | +| AntiDiagSum | [1, 0, 1, -1, 0, -5, -6, -30, -110] | + +Lehmer:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 600, 5880, 18239760] | +| RowGcd | [1, 1, 1, 3, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 6, 10, 25, 120, 720] | +| ColMiddle | [1, 0, -1, -1, -1, 0, 15, 49, 49] | +| ColECenter | [1, -1, -1, 15, 49] | +| ColOCenter | [0, -1, 0, 49] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -4, -44, -96, 188, 2036, 6816] | +| TransNat0 | [0, 1, 1, -4, -18, -26, 60, 454, 1176] | +| TransNat1 | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | + +Lehmer:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ LehmerInv Diagonals as rows | DiagRow7 | [0, -720, 1368, 5340, -9020, -45870, 0, 311311, 667095]| | DiagRow8 | [0, -5040, 11016, 34716, -87164, -300014, 296296, 2475473, 2475473]| -LehmerInv Diagonals as columns +Lehmer:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ LehmerInv Diagonals as columns | DiagCol7 | [1, -28, 294, -1050, -1617, 1386, 35321, 311311, 2475473] | | DiagCol8 | [1, -36, 510, -2970, 1023, 24882, 141713, 667095, 2475473] | -LehmerInv Polynomial values as rows +Lehmer:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LehmerInv Polynomial values as rows | PolyRow7 | [0, 42, 1848, 7938, 14112, -3150, -99288, -369222, -944832] | | PolyRow8 | [0, -160, 8192, 66384, 262784, 665600, 1134720, 1012592, -1265536] | -LehmerInv Polynomial values as columns +Lehmer:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ LehmerInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 189, 280, -5530, -69258, -369222, 1012592] | | PolyCol8 | [1, 8, 56, 312, 944, -5680, -119232, -944832, -1265536] | -# LehmerRev -True +# Lehmer:Rev +[] -LehmerRev Triangle view +Lehmer:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ LehmerRev Triangle view | Row7 | [1, 21, 245, 1890, 9751, 31031, 46656, 0] | | Row8 | [1, 28, 434, 4550, 33621, 170898, 543607, 823543, 0] | -LehmerRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 6, 19, 27, 0, 1, 10, 55, 175, 256, 0, 1, 15, 125, 660, 2101, 3125, 0, 1, 21, 245, 1890, 9751, 31031, 46656, 0, 1, 28, 434, 4550, 33621, 170898, 543607, 823543, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 27, 19, 6, 1, 0, 256, 175, 55, 10, 1, 0, 3125, 2101, 660, 125, 15, 1, 0, 46656, 31031, 9751, 1890, 245, 21, 1, 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 4, 8, 8, 1, 7, 26, 53, 53, 1, 11, 66, 241, 497, 497, 1, 16, 141, 801, 2902, 6027, 6027, 1, 22, 267, 2157, 11908, 42939, 89595, 89595, 1, 29, 463, 5013, 38634, 209532, 753139, 1576682, 1576682] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 8, 8, 4, 1, 53, 53, 26, 7, 1, 497, 497, 241, 66, 11, 1, 6027, 6027, 2902, 801, 141, 16, 1, 89595, 89595, 42939, 11908, 2157, 267, 22, 1, 1576682, 1576682, 753139, 209532, 38634, 5013, 463, 29, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 4, 7, 8, 0, 27, 46, 52, 53, 0, 256, 431, 486, 496, 497, 0, 3125, 5226, 5886, 6011, 6026, 6027, 0, 46656, 77687, 87438, 89328, 89573, 89594, 89595, 0, 823543, 1367150, 1538048, 1571669, 1576219, 1576653, 1576681, 1576682] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 6, 4, 1, 10, 19, 0, 1, 15, 55, 27] | - -LehmerRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 8, 53, 497, 6027, 89595, 1576682] | -| EvenSum | [1, 1, 1, 5, 20, 312, 2227, 56653, 577663] | -| OddSum | [0, 0, 1, 3, 33, 185, 3800, 32942, 999019] | -| AltSum | [1, 1, 0, 2, -13, 127, -1573, 23711, -421356] | -| AccSum | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | -| AccRevSum | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | -| RevAccSum | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | -| DiagSum | [1, 1, 1, 2, 4, 11, 30, 98] | - -LehmerRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 12, 1026, 492800, 78787500, 10083803970240, 27122723143943517881700] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 4, 27, 256, 3125, 46656, 823543] | -| Middle | [1, 1, 1, 3, 19, 55, 660, 1890, 33621] | -| Central | [1, 1, 19, 660, 33621] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 28, 194, 1844, 22478, 334996, 5903850] | -| NegHalf | [1, -2, 2, -4, -10, -124, -1542, -23580, -423058] | -| Bin | [1, 1, 3, 22, 247, 3631, 65431, 1390971, 34000275] | -| InvBin | [1, -1, -1, -4, -17, -31, 1351, 44925, 1172963] | -| TransSqrs | [0, 0, 1, 19, 325, 5901, 118196, 2629418, 64776559] | -| TransNat0 | [0, 0, 1, 11, 125, 1669, 26274, 480276, 10029963] | -| TransNat1 | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | - -LehmerRev Diagonals as rows +Lehmer:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 3, 4, 0], [1, 6, 19, 27, 0], [1, 10, 55, 175, 256, 0], [1, 15, 125, 660, 2101, 3125, 0], [1, 21, 245, 1890, 9751, 31031, 46656, 0], [1, 28, 434, 4550, 33621, 170898, 543607, 823543, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 4, 3, 1], [0, 27, 19, 6, 1], [0, 256, 175, 55, 10, 1], [0, 3125, 2101, 660, 125, 15, 1], [0, 46656, 31031, 9751, 1890, 245, 21, 1], [0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 3, 0], [1, 6, 4], [1, 10, 19, 0], [1, 15, 55, 27], [1, 21, 125, 175, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 8, 8], [1, 7, 26, 53, 53], [1, 11, 66, 241, 497, 497], [1, 16, 141, 801, 2902, 6027, 6027], [1, 22, 267, 2157, 11908, 42939, 89595, 89595], [1, 29, 463, 5013, 38634, 209532, 753139, 1576682, 1576682]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [8, 8, 4, 1], [53, 53, 26, 7, 1], [497, 497, 241, 66, 11, 1], [6027, 6027, 2902, 801, 141, 16, 1], [89595, 89595, 42939, 11908, 2157, 267, 22, 1], [1576682, 1576682, 753139, 209532, 38634, 5013, 463, 29, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 4, 7, 8], [0, 27, 46, 52, 53], [0, 256, 431, 486, 496, 497], [0, 3125, 5226, 5886, 6011, 6026, 6027], [0, 46656, 77687, 87438, 89328, 89573, 89594, 89595], [0, 823543, 1367150, 1538048, 1571669, 1576219, 1576653, 1576681, 1576682]] | + +Lehmer:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 8, 53, 497, 6027, 89595, 1576682] | +| EvenSum | [1, 1, 1, 5, 20, 312, 2227, 56653, 577663] | +| OddSum | [0, 0, 1, 3, 33, 185, 3800, 32942, 999019] | +| AltSum | [1, 1, 0, 2, -13, 127, -1573, 23711, -421356] | +| AccSum | [1, 2, 5, 21, 140, 1313, 15915, 236484, 4160175] | +| AccRevSum | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | +| AntiDiagSum | [1, 1, 1, 2, 4, 11, 30, 98, 322] | + +Lehmer:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 12, 1026, 492800, 78787500, 10083803970240, 27122723143943517881700] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 4, 27, 256, 3125, 46656, 823543] | +| ColMiddle | [1, 1, 1, 3, 19, 55, 660, 1890, 33621] | +| ColECenter | [1, 1, 19, 660, 33621] | +| ColOCenter | [1, 3, 55, 1890] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 19, 325, 5901, 118196, 2629418, 64776559] | +| TransNat0 | [0, 0, 1, 11, 125, 1669, 26274, 480276, 10029963] | +| TransNat1 | [1, 1, 3, 19, 178, 2166, 32301, 569871, 11606645] | + +Lehmer:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ LehmerRev Diagonals as rows | DiagRow7 | [1, 28, 714, 18900, 535227, 16334472, 537306484, 19004025040, 720369582933]| | DiagRow8 | [1, 36, 1110, 34320, 1109427, 38011116, 1385956000, 53791898160, 2219237925333]| -LehmerRev Diagonals as columns +Lehmer:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ LehmerRev Diagonals as columns | DiagCol7 | [0, 823543, 11012415, 79669320, 413066170, 1717317360, 6078494136, 19004025040, 53791898160] | | DiagCol8 | [0, 16777216, 253202761, 2050086511, 11810819141, 54224511341, 210798120533, 720369582933, 2219237925333] | -LehmerRev Polynomial values as rows +Lehmer:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ LehmerRev Polynomial values as rows | PolyRow7 | [1, 89595, 4151135, 42395887, 225499941, 832308731, 2431133875, 6034642335, 13288337897] | | PolyRow8 | [1, 1576682, 146249217, 2241756400, 15903447569, 73388805366, 257275335697, 745132481612, 1875338217825] | -LehmerRev Polynomial values as columns +Lehmer:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ LehmerRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 218, 10235, 677447, 57798987, 6034642335, 745132481612] | | PolyCol8 | [1, 1, 9, 281, 15089, 1141777, 111351737, 13288337897, 1875338217825] | -# LehmerInvRev -True +# Lehmer:Inv:Rev +[] -LehmerInvRev Triangle view +Lehmer:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ LehmerInvRev Triangle view | Row7 | [1, -21, 70, 35, 49, 28, -120, 0] | | Row8 | [1, -28, 154, 0, 49, 196, 188, -720, 0] | -LehmerInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -3, -1, 0, 1, -6, -1, -2, 0, 1, -10, 5, 0, -6, 0, 1, -15, 25, 15, 4, -24, 0, 1, -21, 70, 35, 49, 28, -120, 0, 1, -28, 154, 0, 49, 196, 188, -720, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, -1, -3, 1, 0, -2, -1, -6, 1, 0, -6, 0, 5, -10, 1, 0, -24, 4, 15, 25, -15, 1, 0, -120, 28, 49, 35, 70, -21, 1, 0, -720, 188, 196, 49, 0, 154, -28, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -2, -3, -3, 1, -5, -6, -8, -8, 1, -9, -4, -4, -10, -10, 1, -14, 11, 26, 30, 6, 6, 1, -20, 50, 85, 134, 162, 42, 42, 1, -27, 127, 127, 176, 372, 560, -160, -160] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, -3, -3, -2, 1, -8, -8, -6, -5, 1, -10, -10, -4, -4, -9, 1, 6, 6, 30, 26, 11, -14, 1, 42, 42, 162, 134, 85, 50, -20, 1, -160, -160, 560, 372, 176, 127, 127, -27, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, -1, -4, -3, 0, -2, -3, -9, -8, 0, -6, -6, -1, -11, -10, 0, -24, -20, -5, 20, 5, 6, 0, -120, -92, -43, -8, 62, 41, 42, 0, -720, -532, -336, -287, -287, -133, -161, -160] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -3, 0, 1, -6, -1, 1, -10, -1, 0, 1, -15, 5, -2] | - -LehmerInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -3, -8, -10, 6, 42, -160] | -| EvenSum | [1, 1, 1, 0, 0, 0, 30, 0, 392] | -| OddSum | [0, 0, -1, -3, -8, -10, -24, 42, -552] | -| AltSum | [1, 1, 2, 3, 8, 10, 54, -42, 944] | -| AccSum | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | -| AccRevSum | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | -| RevAccSum | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | -| DiagSum | [1, 1, 1, 0, -2, -6, -10, -11] | - -LehmerInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 6, 30, 600, 5880, 18239760] | -| RowGcd | [1, 1, 1, -3, 2, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 6, 10, 25, 120, 720] | -| Middle | [1, 1, -1, -3, -1, 5, 15, 35, 49] | -| Central | [1, -1, -1, 15, 49] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, -6, -40, -100, 72, 1848, 8192] | -| NegHalf | [1, -2, 6, -18, 64, -220, 888, -3192, 15104] | -| Bin | [1, 1, -1, -11, -37, -29, 502, 4012, 17999] | -| InvBin | [1, -1, 3, -7, 27, -71, 370, -680, 8015] | -| TransSqrs | [0, 0, -1, -7, -28, -86, -316, -2262, -22240] | -| TransNat0 | [0, 0, -1, -5, -14, -24, -24, -160, -2456] | -| TransNat1 | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | - -LehmerInvRev Diagonals as rows +Lehmer:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -3, -1, 0], [1, -6, -1, -2, 0], [1, -10, 5, 0, -6, 0], [1, -15, 25, 15, 4, -24, 0], [1, -21, 70, 35, 49, 28, -120, 0], [1, -28, 154, 0, 49, 196, 188, -720, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, -1, -3, 1], [0, -2, -1, -6, 1], [0, -6, 0, 5, -10, 1], [0, -24, 4, 15, 25, -15, 1], [0, -120, 28, 49, 35, 70, -21, 1], [0, -720, 188, 196, 49, 0, 154, -28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -3, 0], [1, -6, -1], [1, -10, -1, 0], [1, -15, 5, -2], [1, -21, 25, 0, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -3, -3], [1, -5, -6, -8, -8], [1, -9, -4, -4, -10, -10], [1, -14, 11, 26, 30, 6, 6], [1, -20, 50, 85, 134, 162, 42, 42], [1, -27, 127, 127, 176, 372, 560, -160, -160]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [-3, -3, -2, 1], [-8, -8, -6, -5, 1], [-10, -10, -4, -4, -9, 1], [6, 6, 30, 26, 11, -14, 1], [42, 42, 162, 134, 85, 50, -20, 1], [-160, -160, 560, 372, 176, 127, 127, -27, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, -1, -4, -3], [0, -2, -3, -9, -8], [0, -6, -6, -1, -11, -10], [0, -24, -20, -5, 20, 5, 6], [0, -120, -92, -43, -8, 62, 41, 42], [0, -720, -532, -336, -287, -287, -133, -161, -160]] | + +Lehmer:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -3, -8, -10, 6, 42, -160] | +| EvenSum | [1, 1, 1, 0, 0, 0, 30, 0, 392] | +| OddSum | [0, 0, -1, -3, -8, -10, -24, 42, -552] | +| AltSum | [1, 1, 2, 3, 8, 10, 54, -42, 944] | +| AccSum | [1, 2, 1, -7, -26, -36, 66, 496, 1016] | +| AccRevSum | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | +| AntiDiagSum | [1, 1, 1, 0, -2, -6, -10, -11, 5] | + +Lehmer:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 30, 600, 5880, 18239760] | +| RowGcd | [1, 1, 1, 3, 2, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 6, 10, 25, 120, 720] | +| ColMiddle | [1, 1, -1, -3, -1, 5, 15, 35, 49] | +| ColECenter | [1, -1, -1, 15, 49] | +| ColOCenter | [1, -3, 5, 35] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, -7, -28, -86, -316, -2262, -22240] | +| TransNat0 | [0, 0, -1, -5, -14, -24, -24, -160, -2456] | +| TransNat1 | [1, 1, -1, -8, -22, -34, -18, -118, -2616] | + +Lehmer:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ LehmerInvRev Diagonals as rows | DiagRow7 | [1, -28, 294, -1050, -1617, 1386, 35321, 311311, 2475473]| | DiagRow8 | [1, -36, 510, -2970, 1023, 24882, 141713, 667095, 2475473]| -LehmerInvRev Diagonals as columns +Lehmer:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ LehmerInvRev Diagonals as columns | DiagCol7 | [0, -720, 1368, 5340, -9020, -45870, 0, 311311, 667095] | | DiagCol8 | [0, -5040, 11016, 34716, -87164, -300014, 296296, 2475473, 2475473] | -LehmerInvRev Polynomial values as rows +Lehmer:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ LehmerInvRev Polynomial values as rows | PolyRow7 | [1, 42, -5481, -75194, -447027, -1750854, -5307533, -13514346, -30316839] | | PolyRow8 | [1, -160, -72511, -1384688, -10810831, -52665664, -191189615, -567413776, -1454033503] | -LehmerInvRev Polynomial values as columns +Lehmer:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Leibniz.md b/data/md/Leibniz.md index 7fa9dec..9773621 100644 --- a/data/md/Leibniz.md +++ b/data/md/Leibniz.md @@ -14,45 +14,41 @@ Leibniz Triangle view | Row7 | [8, 56, 168, 280, 280, 168, 56, 8] | | Row8 | [9, 72, 252, 504, 630, 504, 252, 72, 9] | -Leibniz Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9] | -| TriRev | [1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9] | -| TriAcc | [1, 2, 4, 3, 9, 12, 4, 16, 28, 32, 5, 25, 55, 75, 80, 6, 36, 96, 156, 186, 192, 7, 49, 154, 294, 399, 441, 448, 8, 64, 232, 512, 792, 960, 1016, 1024, 9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304] | -| TriRevAcc | [1, 4, 2, 12, 9, 3, 32, 28, 16, 4, 80, 75, 55, 25, 5, 192, 186, 156, 96, 36, 6, 448, 441, 399, 294, 154, 49, 7, 1024, 1016, 960, 792, 512, 232, 64, 8, 2304, 2295, 2223, 1971, 1467, 837, 333, 81, 9] | -| TriAccRev | [1, 2, 4, 3, 9, 12, 4, 16, 28, 32, 5, 25, 55, 75, 80, 6, 36, 96, 156, 186, 192, 7, 49, 154, 294, 399, 441, 448, 8, 64, 232, 512, 792, 960, 1016, 1024, 9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304] | -| TriDiag | [1, 2, 3, 2, 4, 6, 5, 12, 3, 6, 20, 12, 7, 30, 30, 4, 8, 42, 60, 20] | +Leibniz Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 2], [3, 6, 3], [4, 12, 12, 4], [5, 20, 30, 20, 5], [6, 30, 60, 60, 30, 6], [7, 42, 105, 140, 105, 42, 7], [8, 56, 168, 280, 280, 168, 56, 8], [9, 72, 252, 504, 630, 504, 252, 72, 9]] | +| RevTabl | [[1], [2, 2], [3, 6, 3], [4, 12, 12, 4], [5, 20, 30, 20, 5], [6, 30, 60, 60, 30, 6], [7, 42, 105, 140, 105, 42, 7], [8, 56, 168, 280, 280, 168, 56, 8], [9, 72, 252, 504, 630, 504, 252, 72, 9]] | +| AntiDiag | [[1], [2], [3, 2], [4, 6], [5, 12, 3], [6, 20, 12], [7, 30, 30, 4], [8, 42, 60, 20], [9, 56, 105, 60, 5]] | +| AccTabl | [[1], [2, 4], [3, 9, 12], [4, 16, 28, 32], [5, 25, 55, 75, 80], [6, 36, 96, 156, 186, 192], [7, 49, 154, 294, 399, 441, 448], [8, 64, 232, 512, 792, 960, 1016, 1024], [9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304]] | +| RevAccTabl | [[1], [4, 2], [12, 9, 3], [32, 28, 16, 4], [80, 75, 55, 25, 5], [192, 186, 156, 96, 36, 6], [448, 441, 399, 294, 154, 49, 7], [1024, 1016, 960, 792, 512, 232, 64, 8], [2304, 2295, 2223, 1971, 1467, 837, 333, 81, 9]] | +| AccRevTabl | [[1], [2, 4], [3, 9, 12], [4, 16, 28, 32], [5, 25, 55, 75, 80], [6, 36, 96, 156, 186, 192], [7, 49, 154, 294, 399, 441, 448], [8, 64, 232, 512, 792, 960, 1016, 1024], [9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304]] | Leibniz Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 4, 12, 32, 80, 192, 448, 1024, 2304] | -| EvenSum | [1, 2, 6, 16, 40, 96, 224, 512, 1152] | -| OddSum | [0, 2, 6, 16, 40, 96, 224, 512, 1152] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| AccRevSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| RevAccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| DiagSum | [1, 2, 5, 10, 20, 38, 71, 130] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 4, 12, 32, 80, 192, 448, 1024, 2304] | +| EvenSum | [1, 2, 6, 16, 40, 96, 224, 512, 1152] | +| OddSum | [0, 2, 6, 16, 40, 96, 224, 512, 1152] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | +| AccRevSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | +| AntiDiagSum | [1, 2, 5, 10, 20, 38, 71, 130, 235] | Leibniz Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 6, 12, 60, 60, 420, 840, 2520] | -| RowGcd | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| RowMax | [1, 2, 6, 12, 30, 60, 140, 280, 630] | -| Middle | [1, 2, 6, 12, 30, 60, 140, 280, 630] | -| Central | [1, 6, 30, 140, 630] | -| LeftSide | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| RightSide | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| PosHalf | [1, 6, 27, 108, 405, 1458, 5103, 17496, 59049] | -| NegHalf | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| Bin | [1, 4, 18, 80, 350, 1512, 6468, 27456, 115830] | -| InvBin | [1, 0, -6, 0, 30, 0, -140, 0, 630] | -| TransSqrs | [0, 2, 18, 96, 400, 1440, 4704, 14336, 41472] | -| TransNat0 | [0, 2, 12, 48, 160, 480, 1344, 3584, 9216] | -| TransNat1 | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 12, 60, 60, 420, 840, 2520] | +| RowGcd | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| RowMax | [1, 2, 6, 12, 30, 60, 140, 280, 630] | +| ColMiddle | [1, 2, 6, 12, 30, 60, 140, 280, 630] | +| ColECenter | [1, 6, 30, 140, 630] | +| ColOCenter | [2, 12, 60, 280] | +| ColLeft | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| ColRight | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| TransSqrs | [0, 2, 18, 96, 400, 1440, 4704, 14336, 41472] | +| TransNat0 | [0, 2, 12, 48, 160, 480, 1344, 3584, 9216] | +| TransNat1 | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | Leibniz Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Leibniz Polynomial values as columns | PolyCol7 | [1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944] | | PolyCol8 | [1, 18, 243, 2916, 32805, 354294, 3720087, 38263752, 387420489] | -# LeibnizRev -True +# Leibniz:Rev +[] -LeibnizRev Triangle view +Leibniz:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LeibnizRev Triangle view | Row7 | [8, 56, 168, 280, 280, 168, 56, 8] | | Row8 | [9, 72, 252, 504, 630, 504, 252, 72, 9] | -LeibnizRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9] | -| TriRev | [1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9] | -| TriAcc | [1, 2, 4, 3, 9, 12, 4, 16, 28, 32, 5, 25, 55, 75, 80, 6, 36, 96, 156, 186, 192, 7, 49, 154, 294, 399, 441, 448, 8, 64, 232, 512, 792, 960, 1016, 1024, 9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304] | -| TriRevAcc | [1, 4, 2, 12, 9, 3, 32, 28, 16, 4, 80, 75, 55, 25, 5, 192, 186, 156, 96, 36, 6, 448, 441, 399, 294, 154, 49, 7, 1024, 1016, 960, 792, 512, 232, 64, 8, 2304, 2295, 2223, 1971, 1467, 837, 333, 81, 9] | -| TriAccRev | [1, 2, 4, 3, 9, 12, 4, 16, 28, 32, 5, 25, 55, 75, 80, 6, 36, 96, 156, 186, 192, 7, 49, 154, 294, 399, 441, 448, 8, 64, 232, 512, 792, 960, 1016, 1024, 9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304] | -| TriDiag | [1, 2, 3, 2, 4, 6, 5, 12, 3, 6, 20, 12, 7, 30, 30, 4, 8, 42, 60, 20] | - -LeibnizRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 4, 12, 32, 80, 192, 448, 1024, 2304] | -| EvenSum | [1, 2, 6, 16, 40, 96, 224, 512, 1152] | -| OddSum | [0, 2, 6, 16, 40, 96, 224, 512, 1152] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| AccRevSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| RevAccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | -| DiagSum | [1, 2, 5, 10, 20, 38, 71, 130] | - -LeibnizRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 6, 12, 60, 60, 420, 840, 2520] | -| RowGcd | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| RowMax | [1, 2, 6, 12, 30, 60, 140, 280, 630] | -| Middle | [1, 2, 6, 12, 30, 60, 140, 280, 630] | -| Central | [1, 6, 30, 140, 630] | -| LeftSide | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| RightSide | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| PosHalf | [1, 6, 27, 108, 405, 1458, 5103, 17496, 59049] | -| NegHalf | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| Bin | [1, 4, 18, 80, 350, 1512, 6468, 27456, 115830] | -| InvBin | [1, 0, -6, 0, 30, 0, -140, 0, 630] | -| TransSqrs | [0, 2, 18, 96, 400, 1440, 4704, 14336, 41472] | -| TransNat0 | [0, 2, 12, 48, 160, 480, 1344, 3584, 9216] | -| TransNat1 | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | - -LeibnizRev Diagonals as rows +Leibniz:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 2], [3, 6, 3], [4, 12, 12, 4], [5, 20, 30, 20, 5], [6, 30, 60, 60, 30, 6], [7, 42, 105, 140, 105, 42, 7], [8, 56, 168, 280, 280, 168, 56, 8], [9, 72, 252, 504, 630, 504, 252, 72, 9]] | +| RevTabl | [[1], [2, 2], [3, 6, 3], [4, 12, 12, 4], [5, 20, 30, 20, 5], [6, 30, 60, 60, 30, 6], [7, 42, 105, 140, 105, 42, 7], [8, 56, 168, 280, 280, 168, 56, 8], [9, 72, 252, 504, 630, 504, 252, 72, 9]] | +| AntiDiag | [[1], [2], [3, 2], [4, 6], [5, 12, 3], [6, 20, 12], [7, 30, 30, 4], [8, 42, 60, 20], [9, 56, 105, 60, 5]] | +| AccTabl | [[1], [2, 4], [3, 9, 12], [4, 16, 28, 32], [5, 25, 55, 75, 80], [6, 36, 96, 156, 186, 192], [7, 49, 154, 294, 399, 441, 448], [8, 64, 232, 512, 792, 960, 1016, 1024], [9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304]] | +| RevAccTabl | [[1], [4, 2], [12, 9, 3], [32, 28, 16, 4], [80, 75, 55, 25, 5], [192, 186, 156, 96, 36, 6], [448, 441, 399, 294, 154, 49, 7], [1024, 1016, 960, 792, 512, 232, 64, 8], [2304, 2295, 2223, 1971, 1467, 837, 333, 81, 9]] | +| AccRevTabl | [[1], [2, 4], [3, 9, 12], [4, 16, 28, 32], [5, 25, 55, 75, 80], [6, 36, 96, 156, 186, 192], [7, 49, 154, 294, 399, 441, 448], [8, 64, 232, 512, 792, 960, 1016, 1024], [9, 81, 333, 837, 1467, 1971, 2223, 2295, 2304]] | + +Leibniz:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 4, 12, 32, 80, 192, 448, 1024, 2304] | +| EvenSum | [1, 2, 6, 16, 40, 96, 224, 512, 1152] | +| OddSum | [0, 2, 6, 16, 40, 96, 224, 512, 1152] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | +| AccRevSum | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | +| AntiDiagSum | [1, 2, 5, 10, 20, 38, 71, 130, 235] | + +Leibniz:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 12, 60, 60, 420, 840, 2520] | +| RowGcd | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| RowMax | [1, 2, 6, 12, 30, 60, 140, 280, 630] | +| ColMiddle | [1, 2, 6, 12, 30, 60, 140, 280, 630] | +| ColECenter | [1, 6, 30, 140, 630] | +| ColOCenter | [2, 12, 60, 280] | +| ColLeft | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| ColRight | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| TransSqrs | [0, 2, 18, 96, 400, 1440, 4704, 14336, 41472] | +| TransNat0 | [0, 2, 12, 48, 160, 480, 1344, 3584, 9216] | +| TransNat1 | [1, 6, 24, 80, 240, 672, 1792, 4608, 11520] | + +Leibniz:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 2, 3, 4, 5, 6, 7, 8, 9]| @@ -175,7 +167,7 @@ LeibnizRev Diagonals as rows | DiagRow7 | [8, 72, 360, 1320, 3960, 10296, 24024, 51480, 102960]| | DiagRow8 | [9, 90, 495, 1980, 6435, 18018, 45045, 102960, 218790]| -LeibnizRev Diagonals as columns +Leibniz:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | @@ -188,7 +180,7 @@ LeibnizRev Diagonals as columns | DiagCol7 | [8, 72, 360, 1320, 3960, 10296, 24024, 51480, 102960] | | DiagCol8 | [9, 90, 495, 1980, 6435, 18018, 45045, 102960, 218790] | -LeibnizRev Polynomial values as rows +Leibniz:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LeibnizRev Polynomial values as rows | PolyRow7 | [8, 1024, 17496, 131072, 625000, 2239488, 6588344, 16777216, 38263752] | | PolyRow8 | [9, 2304, 59049, 589824, 3515625, 15116544, 51883209, 150994944, 387420489] | -LeibnizRev Polynomial values as columns +Leibniz:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | diff --git a/data/md/Levin.md b/data/md/Levin.md index a2a88aa..2b2c760 100644 --- a/data/md/Levin.md +++ b/data/md/Levin.md @@ -14,45 +14,41 @@ Levin Triangle view | Row7 | [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432] | | Row8 | [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -Levin Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -| TriRev | [1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -| TriAcc | [1, 2, 4, 6, 18, 24, 20, 80, 140, 160, 70, 350, 770, 1050, 1120, 252, 1512, 4032, 6552, 7812, 8064, 924, 6468, 20328, 38808, 52668, 58212, 59136, 3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296, 12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720] | -| TriRevAcc | [1, 4, 2, 24, 18, 6, 160, 140, 80, 20, 1120, 1050, 770, 350, 70, 8064, 7812, 6552, 4032, 1512, 252, 59136, 58212, 52668, 38808, 20328, 6468, 924, 439296, 435864, 411840, 339768, 219648, 99528, 27456, 3432, 3294720, 3281850, 3178890, 2818530, 2097810, 1196910, 476190, 115830, 12870] | -| TriAccRev | [1, 2, 4, 6, 18, 24, 20, 80, 140, 160, 70, 350, 770, 1050, 1120, 252, 1512, 4032, 6552, 7812, 8064, 924, 6468, 20328, 38808, 52668, 58212, 59136, 3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296, 12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720] | -| TriDiag | [1, 2, 6, 2, 20, 12, 70, 60, 6, 252, 280, 60, 924, 1260, 420, 20, 3432, 5544, 2520, 280] | +Levin Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 2], [6, 12, 6], [20, 60, 60, 20], [70, 280, 420, 280, 70], [252, 1260, 2520, 2520, 1260, 252], [924, 5544, 13860, 18480, 13860, 5544, 924], [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432], [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870]] | +| RevTabl | [[1], [2, 2], [6, 12, 6], [20, 60, 60, 20], [70, 280, 420, 280, 70], [252, 1260, 2520, 2520, 1260, 252], [924, 5544, 13860, 18480, 13860, 5544, 924], [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432], [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870]] | +| AntiDiag | [[1], [2], [6, 2], [20, 12], [70, 60, 6], [252, 280, 60], [924, 1260, 420, 20], [3432, 5544, 2520, 280], [12870, 24024, 13860, 2520, 70]] | +| AccTabl | [[1], [2, 4], [6, 18, 24], [20, 80, 140, 160], [70, 350, 770, 1050, 1120], [252, 1512, 4032, 6552, 7812, 8064], [924, 6468, 20328, 38808, 52668, 58212, 59136], [3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296], [12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720]] | +| RevAccTabl | [[1], [4, 2], [24, 18, 6], [160, 140, 80, 20], [1120, 1050, 770, 350, 70], [8064, 7812, 6552, 4032, 1512, 252], [59136, 58212, 52668, 38808, 20328, 6468, 924], [439296, 435864, 411840, 339768, 219648, 99528, 27456, 3432], [3294720, 3281850, 3178890, 2818530, 2097810, 1196910, 476190, 115830, 12870]] | +| AccRevTabl | [[1], [2, 4], [6, 18, 24], [20, 80, 140, 160], [70, 350, 770, 1050, 1120], [252, 1512, 4032, 6552, 7812, 8064], [924, 6468, 20328, 38808, 52668, 58212, 59136], [3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296], [12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720]] | Levin Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720] | -| EvenSum | [1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | -| OddSum | [0, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| AccRevSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| RevAccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| DiagSum | [1, 2, 8, 32, 136, 592, 2624, 11776] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720] | +| EvenSum | [1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | +| OddSum | [0, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | +| AccRevSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | +| AntiDiagSum | [1, 2, 8, 32, 136, 592, 2624, 11776, 53344] | Levin Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 12, 60, 840, 2520, 55440, 360360, 3603600] | -| RowGcd | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| RowMax | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | -| Middle | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | -| Central | [1, 12, 420, 18480, 900900] | -| LeftSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| RightSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| PosHalf | [1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070] | -| NegHalf | [1, -2, 6, -20, 70, -252, 924, -3432, 12870] | -| Bin | [1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900] | -| InvBin | [1, 0, -12, 0, 420, 0, -18480, 0, 900900] | -| TransSqrs | [0, 2, 36, 480, 5600, 60480, 620928, 6150144, 59304960] | -| TransNat0 | [0, 2, 24, 240, 2240, 20160, 177408, 1537536, 13178880] | -| TransNat1 | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 12, 60, 840, 2520, 55440, 360360, 3603600] | +| RowGcd | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| RowMax | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | +| ColMiddle | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | +| ColECenter | [1, 12, 420, 18480, 900900] | +| ColOCenter | [2, 60, 2520, 120120] | +| ColLeft | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| ColRight | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| TransSqrs | [0, 2, 36, 480, 5600, 60480, 620928, 6150144, 59304960] | +| TransNat0 | [0, 2, 24, 240, 2240, 20160, 177408, 1537536, 13178880] | +| TransNat1 | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | Levin Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Levin Polynomial values as columns | PolyCol7 | [1, 16, 384, 10240, 286720, 8257536, 242221056, 7197425664, 215922769920] | | PolyCol8 | [1, 18, 486, 14580, 459270, 14880348, 491051484, 16415149608, 554011299270] | -# LevinRev -True +# Levin:Rev +[] -LevinRev Triangle view +Levin:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LevinRev Triangle view | Row7 | [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432] | | Row8 | [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -LevinRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -| TriRev | [1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870] | -| TriAcc | [1, 2, 4, 6, 18, 24, 20, 80, 140, 160, 70, 350, 770, 1050, 1120, 252, 1512, 4032, 6552, 7812, 8064, 924, 6468, 20328, 38808, 52668, 58212, 59136, 3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296, 12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720] | -| TriRevAcc | [1, 4, 2, 24, 18, 6, 160, 140, 80, 20, 1120, 1050, 770, 350, 70, 8064, 7812, 6552, 4032, 1512, 252, 59136, 58212, 52668, 38808, 20328, 6468, 924, 439296, 435864, 411840, 339768, 219648, 99528, 27456, 3432, 3294720, 3281850, 3178890, 2818530, 2097810, 1196910, 476190, 115830, 12870] | -| TriAccRev | [1, 2, 4, 6, 18, 24, 20, 80, 140, 160, 70, 350, 770, 1050, 1120, 252, 1512, 4032, 6552, 7812, 8064, 924, 6468, 20328, 38808, 52668, 58212, 59136, 3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296, 12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720] | -| TriDiag | [1, 2, 6, 2, 20, 12, 70, 60, 6, 252, 280, 60, 924, 1260, 420, 20, 3432, 5544, 2520, 280] | - -LevinRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720] | -| EvenSum | [1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | -| OddSum | [0, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| AccRevSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| RevAccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | -| DiagSum | [1, 2, 8, 32, 136, 592, 2624, 11776] | - -LevinRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 12, 60, 840, 2520, 55440, 360360, 3603600] | -| RowGcd | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| RowMax | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | -| Middle | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | -| Central | [1, 12, 420, 18480, 900900] | -| LeftSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| RightSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| PosHalf | [1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070] | -| NegHalf | [1, -2, 6, -20, 70, -252, 924, -3432, 12870] | -| Bin | [1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900] | -| InvBin | [1, 0, -12, 0, 420, 0, -18480, 0, 900900] | -| TransSqrs | [0, 2, 36, 480, 5600, 60480, 620928, 6150144, 59304960] | -| TransNat0 | [0, 2, 24, 240, 2240, 20160, 177408, 1537536, 13178880] | -| TransNat1 | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | - -LevinRev Diagonals as rows +Levin:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 2], [6, 12, 6], [20, 60, 60, 20], [70, 280, 420, 280, 70], [252, 1260, 2520, 2520, 1260, 252], [924, 5544, 13860, 18480, 13860, 5544, 924], [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432], [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870]] | +| RevTabl | [[1], [2, 2], [6, 12, 6], [20, 60, 60, 20], [70, 280, 420, 280, 70], [252, 1260, 2520, 2520, 1260, 252], [924, 5544, 13860, 18480, 13860, 5544, 924], [3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432], [12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870]] | +| AntiDiag | [[1], [2], [6, 2], [20, 12], [70, 60, 6], [252, 280, 60], [924, 1260, 420, 20], [3432, 5544, 2520, 280], [12870, 24024, 13860, 2520, 70]] | +| AccTabl | [[1], [2, 4], [6, 18, 24], [20, 80, 140, 160], [70, 350, 770, 1050, 1120], [252, 1512, 4032, 6552, 7812, 8064], [924, 6468, 20328, 38808, 52668, 58212, 59136], [3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296], [12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720]] | +| RevAccTabl | [[1], [4, 2], [24, 18, 6], [160, 140, 80, 20], [1120, 1050, 770, 350, 70], [8064, 7812, 6552, 4032, 1512, 252], [59136, 58212, 52668, 38808, 20328, 6468, 924], [439296, 435864, 411840, 339768, 219648, 99528, 27456, 3432], [3294720, 3281850, 3178890, 2818530, 2097810, 1196910, 476190, 115830, 12870]] | +| AccRevTabl | [[1], [2, 4], [6, 18, 24], [20, 80, 140, 160], [70, 350, 770, 1050, 1120], [252, 1512, 4032, 6552, 7812, 8064], [924, 6468, 20328, 38808, 52668, 58212, 59136], [3432, 27456, 99528, 219648, 339768, 411840, 435864, 439296], [12870, 115830, 476190, 1196910, 2097810, 2818530, 3178890, 3281850, 3294720]] | + +Levin:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720] | +| EvenSum | [1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | +| OddSum | [0, 2, 12, 80, 560, 4032, 29568, 219648, 1647360] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | +| AccRevSum | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | +| AntiDiagSum | [1, 2, 8, 32, 136, 592, 2624, 11776, 53344] | + +Levin:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 12, 60, 840, 2520, 55440, 360360, 3603600] | +| RowGcd | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| RowMax | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | +| ColMiddle | [1, 2, 12, 60, 420, 2520, 18480, 120120, 900900] | +| ColECenter | [1, 12, 420, 18480, 900900] | +| ColOCenter | [2, 60, 2520, 120120] | +| ColLeft | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| ColRight | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| TransSqrs | [0, 2, 36, 480, 5600, 60480, 620928, 6150144, 59304960] | +| TransNat0 | [0, 2, 24, 240, 2240, 20160, 177408, 1537536, 13178880] | +| TransNat1 | [1, 6, 48, 400, 3360, 28224, 236544, 1976832, 16473600] | + +Levin:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870]| @@ -175,7 +167,7 @@ LevinRev Diagonals as rows | DiagRow7 | [3432, 102960, 1750320, 22170720, 232792560, 2141691552, 17847429600, 137680171200, 998181241200]| | DiagRow8 | [12870, 437580, 8314020, 116396280, 1338557220, 13385572200, 120470149800, 998181241200, 7735904619300]| -LevinRev Diagonals as columns +Levin:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | @@ -188,7 +180,7 @@ LevinRev Diagonals as columns | DiagCol7 | [3432, 102960, 1750320, 22170720, 232792560, 2141691552, 17847429600, 137680171200, 998181241200] | | DiagCol8 | [12870, 437580, 8314020, 116396280, 1338557220, 13385572200, 120470149800, 998181241200, 7735904619300] | -LevinRev Polynomial values as rows +Levin:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LevinRev Polynomial values as rows | PolyRow7 | [3432, 439296, 7505784, 56229888, 268125000, 960740352, 2826399576, 7197425664, 16415149608] | | PolyRow8 | [12870, 3294720, 84440070, 843448320, 5027343750, 21616657920, 74192988870, 215922769920, 554011299270] | -LevinRev Polynomial values as columns +Levin:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | diff --git a/data/md/Lozanic.md b/data/md/Lozanic.md index f19b150..3ce41d6 100644 --- a/data/md/Lozanic.md +++ b/data/md/Lozanic.md @@ -14,45 +14,41 @@ Lozanic Triangle view | Row7 | [1, 4, 12, 19, 19, 12, 4, 1] | | Row8 | [1, 4, 16, 28, 38, 28, 16, 4, 1] | -Lozanic Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1] | -| TriRev | [1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1] | -| TriAcc | [1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 10, 16, 19, 20, 1, 4, 13, 23, 32, 35, 36, 1, 5, 17, 36, 55, 67, 71, 72, 1, 5, 21, 49, 87, 115, 131, 135, 136] | -| TriRevAcc | [1, 2, 1, 3, 2, 1, 6, 5, 3, 1, 10, 9, 7, 3, 1, 20, 19, 16, 10, 4, 1, 36, 35, 32, 23, 13, 4, 1, 72, 71, 67, 55, 36, 17, 5, 1, 136, 135, 131, 115, 87, 49, 21, 5, 1] | -| TriAccRev | [1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 10, 16, 19, 20, 1, 4, 13, 23, 32, 35, 36, 1, 5, 17, 36, 55, 67, 71, 72, 1, 5, 21, 49, 87, 115, 131, 135, 136] | -| TriDiag | [1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 3, 6, 2] | +Lozanic Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 1, 1], [1, 2, 2, 1], [1, 2, 4, 2, 1], [1, 3, 6, 6, 3, 1], [1, 3, 9, 10, 9, 3, 1], [1, 4, 12, 19, 19, 12, 4, 1], [1, 4, 16, 28, 38, 28, 16, 4, 1]] | +| RevTabl | [[1], [1, 1], [1, 1, 1], [1, 2, 2, 1], [1, 2, 4, 2, 1], [1, 3, 6, 6, 3, 1], [1, 3, 9, 10, 9, 3, 1], [1, 4, 12, 19, 19, 12, 4, 1], [1, 4, 16, 28, 38, 28, 16, 4, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 1], [1, 2, 1], [1, 2, 2], [1, 3, 4, 1], [1, 3, 6, 2], [1, 4, 9, 6, 1]] | +| AccTabl | [[1], [1, 2], [1, 2, 3], [1, 3, 5, 6], [1, 3, 7, 9, 10], [1, 4, 10, 16, 19, 20], [1, 4, 13, 23, 32, 35, 36], [1, 5, 17, 36, 55, 67, 71, 72], [1, 5, 21, 49, 87, 115, 131, 135, 136]] | +| RevAccTabl | [[1], [2, 1], [3, 2, 1], [6, 5, 3, 1], [10, 9, 7, 3, 1], [20, 19, 16, 10, 4, 1], [36, 35, 32, 23, 13, 4, 1], [72, 71, 67, 55, 36, 17, 5, 1], [136, 135, 131, 115, 87, 49, 21, 5, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 2, 3], [1, 3, 5, 6], [1, 3, 7, 9, 10], [1, 4, 10, 16, 19, 20], [1, 4, 13, 23, 32, 35, 36], [1, 5, 17, 36, 55, 67, 71, 72], [1, 5, 21, 49, 87, 115, 131, 135, 136]] | Lozanic Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 6, 10, 20, 36, 72, 136] | -| EvenSum | [1, 1, 2, 3, 6, 10, 20, 36, 72] | -| OddSum | [0, 1, 1, 3, 4, 10, 16, 36, 64] | -| AltSum | [1, 0, 1, 0, 2, 0, 4, 0, 8] | -| AccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| AccRevSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| RevAccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| DiagSum | [1, 1, 2, 2, 4, 5, 9, 12] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 6, 10, 20, 36, 72, 136] | +| EvenSum | [1, 1, 2, 3, 6, 10, 20, 36, 72] | +| OddSum | [0, 1, 1, 3, 4, 10, 16, 36, 64] | +| AltSum | [1, 0, 1, 0, 2, 0, 4, 0, 8] | +| AccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | +| AccRevSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | +| AntiDiagSum | [1, 1, 2, 2, 4, 5, 9, 12, 21] | Lozanic Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 4, 6, 90, 228, 2128] | -| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 2] | -| RowMax | [1, 1, 1, 2, 4, 6, 10, 19, 38] | -| Middle | [1, 1, 1, 2, 4, 6, 10, 19, 38] | -| Central | [1, 1, 4, 10, 38] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 7, 21, 53, 159, 427, 1281, 3593] | -| NegHalf | [1, -1, 3, -3, 13, -13, 63, -63, 313] | -| Bin | [1, 2, 4, 14, 42, 152, 508, 1892, 6758] | -| InvBin | [1, 0, 0, 0, 10, 0, 36, 0, 358] | -| TransSqrs | [0, 1, 5, 19, 52, 154, 384, 1020, 2464] | -| TransNat0 | [0, 1, 3, 9, 20, 50, 108, 252, 544] | -| TransNat1 | [1, 3, 6, 15, 30, 70, 144, 324, 680] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 4, 6, 90, 228, 2128] | +| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 2] | +| RowMax | [1, 1, 1, 2, 4, 6, 10, 19, 38] | +| ColMiddle | [1, 1, 1, 2, 4, 6, 10, 19, 38] | +| ColECenter | [1, 1, 4, 10, 38] | +| ColOCenter | [1, 2, 6, 19] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 19, 52, 154, 384, 1020, 2464] | +| TransNat0 | [0, 1, 3, 9, 20, 50, 108, 252, 544] | +| TransNat1 | [1, 3, 6, 15, 30, 70, 144, 324, 680] | Lozanic Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Lozanic Polynomial values as columns | PolyCol7 | [1, 8, 57, 456, 3298, 26384, 193572, 1548576, 11513608] | | PolyCol8 | [1, 9, 73, 657, 5393, 48537, 403033, 3627297, 30448673] | -# LozanicInv -True +# Lozanic:Inv +[] -LozanicInv Triangle view +Lozanic:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ LozanicInv Triangle view | Row7 | [-1, 18, -18, -13, 17, 0, -4, 1] | | Row8 | [-19, -4, 56, -28, -22, 20, 0, -4, 1] | -LozanicInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 0, -1, 1, 1, 0, -2, 1, -1, 2, 0, -2, 1, -1, -3, 6, 0, -3, 1, 4, -3, -7, 8, 0, -3, 1, -1, 18, -18, -13, 17, 0, -4, 1, -19, -4, 56, -28, -22, 20, 0, -4, 1] | -| TriRev | [1, 1, -1, 1, -1, 0, 1, -2, 0, 1, 1, -2, 0, 2, -1, 1, -3, 0, 6, -3, -1, 1, -3, 0, 8, -7, -3, 4, 1, -4, 0, 17, -13, -18, 18, -1, 1, -4, 0, 20, -22, -28, 56, -4, -19] | -| TriAcc | [1, -1, 0, 0, -1, 0, 1, 1, -1, 0, -1, 1, 1, -1, 0, -1, -4, 2, 2, -1, 0, 4, 1, -6, 2, 2, -1, 0, -1, 17, -1, -14, 3, 3, -1, 0, -19, -23, 33, 5, -17, 3, 3, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, 0, 0, -1, 1, 1, 0, -1, 1, 1, -1, 0, -1, 2, 2, -4, -1, 0, -1, 2, 2, -6, 1, 4, 0, -1, 3, 3, -14, -1, 17, -1, 0, -1, 3, 3, -17, 5, 33, -23, -19] | -| TriAccRev | [1, 1, 0, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 1, 0, 1, -2, -2, 4, 1, 0, 1, -2, -2, 6, -1, -4, 0, 1, -3, -3, 14, 1, -17, 1, 0, 1, -3, -3, 17, -5, -33, 23, 19, 0] | -| TriDiag | [1, -1, 0, 1, 1, -1, -1, 0, 1, -1, 2, -2, 4, -3, 0, 1, -1, -3, 6, -2] | - -LozanicInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 1, -1, 0, 2, -2, -6, 16] | -| OddSum | [0, 1, -1, 1, 0, -2, 2, 6, -16] | -| AltSum | [1, -2, 2, -2, 0, 4, -4, -12, 32] | -| AccSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | -| AccRevSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | -| RevAccSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | -| DiagSum | [1, -1, 1, 0, 0, -1, 2, 0] | - -LozanicInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, 2, 6, 168, 7956, 58520] | -| RowGcd | [1, 1, 1, -2, 2, 3, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 2, 6, 8, 18, 56] | -| Middle | [1, -1, -1, 0, 0, 6, 8, -13, -22] | -| Central | [1, -1, 0, 8, -22] | -| LeftSide | [1, -1, 0, 1, -1, -1, 4, -1, -19] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -1, 5, -3, -37, 107, 369, -2887] | -| NegHalf | [1, 3, 3, -3, -27, -57, 183, 1521, -375] | -| Bin | [1, 0, -1, -4, 0, 30, 24, -140, -502] | -| InvBin | [1, 2, 3, 6, 0, -58, -224, -516, 522] | -| TransSqrs | [0, 1, 3, 1, 0, -2, 2, 6, -16] | -| TransNat0 | [0, 1, 1, -1, 0, 2, -2, -6, 16] | -| TransNat1 | [1, 1, 1, -1, 0, 2, -2, -6, 16] | - -LozanicInv Diagonals as rows +Lozanic:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -1, 1], [1, 0, -2, 1], [-1, 2, 0, -2, 1], [-1, -3, 6, 0, -3, 1], [4, -3, -7, 8, 0, -3, 1], [-1, 18, -18, -13, 17, 0, -4, 1], [-19, -4, 56, -28, -22, 20, 0, -4, 1]] | +| RevTabl | [[1], [1, -1], [1, -1, 0], [1, -2, 0, 1], [1, -2, 0, 2, -1], [1, -3, 0, 6, -3, -1], [1, -3, 0, 8, -7, -3, 4], [1, -4, 0, 17, -13, -18, 18, -1], [1, -4, 0, 20, -22, -28, 56, -4, -19]] | +| AntiDiag | [[1], [-1], [0, 1], [1, -1], [-1, 0, 1], [-1, 2, -2], [4, -3, 0, 1], [-1, -3, 6, -2], [-19, 18, -7, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -1, 0], [1, 1, -1, 0], [-1, 1, 1, -1, 0], [-1, -4, 2, 2, -1, 0], [4, 1, -6, 2, 2, -1, 0], [-1, 17, -1, -14, 3, 3, -1, 0], [-19, -23, 33, 5, -17, 3, 3, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 0], [0, -1, 1, 1], [0, -1, 1, 1, -1], [0, -1, 2, 2, -4, -1], [0, -1, 2, 2, -6, 1, 4], [0, -1, 3, 3, -14, -1, 17, -1], [0, -1, 3, 3, -17, 5, 33, -23, -19]] | +| AccRevTabl | [[1], [1, 0], [1, 0, 0], [1, -1, -1, 0], [1, -1, -1, 1, 0], [1, -2, -2, 4, 1, 0], [1, -2, -2, 6, -1, -4, 0], [1, -3, -3, 14, 1, -17, 1, 0], [1, -3, -3, 17, -5, -33, 23, 19, 0]] | + +Lozanic:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 1, -1, 0, 2, -2, -6, 16] | +| OddSum | [0, 1, -1, 1, 0, -2, 2, 6, -16] | +| AltSum | [1, -2, 2, -2, 0, 4, -4, -12, 32] | +| AccSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | +| AccRevSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | +| AntiDiagSum | [1, -1, 1, 0, 0, -1, 2, 0, -7] | + +Lozanic:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 2, 6, 168, 7956, 58520] | +| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 2, 6, 8, 18, 56] | +| ColMiddle | [1, -1, -1, 0, 0, 6, 8, -13, -22] | +| ColECenter | [1, -1, 0, 8, -22] | +| ColOCenter | [-1, 0, 6, -13] | +| ColLeft | [1, -1, 0, 1, -1, -1, 4, -1, -19] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 1, 0, -2, 2, 6, -16] | +| TransNat0 | [0, 1, 1, -1, 0, 2, -2, -6, 16] | +| TransNat1 | [1, 1, 1, -1, 0, 2, -2, -6, 16] | + +Lozanic:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ LozanicInv Diagonals as rows | DiagRow7 | [-1, -4, 60, 80, 730, 908, 4100, 4808, 15777]| | DiagRow8 | [-19, -127, -511, -1603, -4157, -9633, -20121, -39301, -71858]| -LozanicInv Diagonals as columns +Lozanic:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 0, 1, -1, -1, 4, -1, -19] | @@ -188,7 +180,7 @@ LozanicInv Diagonals as columns | DiagCol7 | [1, -4, 0, 40, -70, -396, 1608, 4808, -39301] | | DiagCol8 | [1, -5, 0, 65, -95, -891, 2548, 15777, -71858] | -LozanicInv Polynomial values as rows +Lozanic:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ LozanicInv Polynomial values as rows | PolyRow7 | [-1, 0, 3, 188, 3303, 24264, 111995, 388548, 1110543] | | PolyRow8 | [-19, 0, 5, 608, 13917, 124736, 682805, 2747040, 8943053] | -LozanicInv Polynomial values as columns +Lozanic:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 0, 1, -1, -1, 4, -1, -19] | @@ -214,10 +206,10 @@ LozanicInv Polynomial values as columns | PolyCol7 | [1, 6, 42, 246, 1728, 9876, 69612, 388548, 2747040] | | PolyCol8 | [1, 7, 56, 385, 3087, 20839, 167468, 1110543, 8943053] | -# LozanicRev -True +# Lozanic:Rev +[] -LozanicRev Triangle view +Lozanic:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ LozanicRev Triangle view | Row7 | [1, 4, 12, 19, 19, 12, 4, 1] | | Row8 | [1, 4, 16, 28, 38, 28, 16, 4, 1] | -LozanicRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1] | -| TriRev | [1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1] | -| TriAcc | [1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 10, 16, 19, 20, 1, 4, 13, 23, 32, 35, 36, 1, 5, 17, 36, 55, 67, 71, 72, 1, 5, 21, 49, 87, 115, 131, 135, 136] | -| TriRevAcc | [1, 2, 1, 3, 2, 1, 6, 5, 3, 1, 10, 9, 7, 3, 1, 20, 19, 16, 10, 4, 1, 36, 35, 32, 23, 13, 4, 1, 72, 71, 67, 55, 36, 17, 5, 1, 136, 135, 131, 115, 87, 49, 21, 5, 1] | -| TriAccRev | [1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 10, 16, 19, 20, 1, 4, 13, 23, 32, 35, 36, 1, 5, 17, 36, 55, 67, 71, 72, 1, 5, 21, 49, 87, 115, 131, 135, 136] | -| TriDiag | [1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 3, 6, 2] | - -LozanicRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 6, 10, 20, 36, 72, 136] | -| EvenSum | [1, 1, 2, 3, 6, 10, 20, 36, 72] | -| OddSum | [0, 1, 1, 3, 4, 10, 16, 36, 64] | -| AltSum | [1, 0, 1, 0, 2, 0, 4, 0, 8] | -| AccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| AccRevSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| RevAccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | -| DiagSum | [1, 1, 2, 2, 4, 5, 9, 12] | - -LozanicRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 4, 6, 90, 228, 2128] | -| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 2] | -| RowMax | [1, 1, 1, 2, 4, 6, 10, 19, 38] | -| Middle | [1, 1, 1, 2, 4, 6, 10, 19, 38] | -| Central | [1, 1, 4, 10, 38] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 7, 21, 53, 159, 427, 1281, 3593] | -| NegHalf | [1, -1, 3, -3, 13, -13, 63, -63, 313] | -| Bin | [1, 2, 4, 14, 42, 152, 508, 1892, 6758] | -| InvBin | [1, 0, 0, 0, 10, 0, 36, 0, 358] | -| TransSqrs | [0, 1, 5, 19, 52, 154, 384, 1020, 2464] | -| TransNat0 | [0, 1, 3, 9, 20, 50, 108, 252, 544] | -| TransNat1 | [1, 3, 6, 15, 30, 70, 144, 324, 680] | - -LozanicRev Diagonals as rows +Lozanic:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 1, 1], [1, 2, 2, 1], [1, 2, 4, 2, 1], [1, 3, 6, 6, 3, 1], [1, 3, 9, 10, 9, 3, 1], [1, 4, 12, 19, 19, 12, 4, 1], [1, 4, 16, 28, 38, 28, 16, 4, 1]] | +| RevTabl | [[1], [1, 1], [1, 1, 1], [1, 2, 2, 1], [1, 2, 4, 2, 1], [1, 3, 6, 6, 3, 1], [1, 3, 9, 10, 9, 3, 1], [1, 4, 12, 19, 19, 12, 4, 1], [1, 4, 16, 28, 38, 28, 16, 4, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 1], [1, 2, 1], [1, 2, 2], [1, 3, 4, 1], [1, 3, 6, 2], [1, 4, 9, 6, 1]] | +| AccTabl | [[1], [1, 2], [1, 2, 3], [1, 3, 5, 6], [1, 3, 7, 9, 10], [1, 4, 10, 16, 19, 20], [1, 4, 13, 23, 32, 35, 36], [1, 5, 17, 36, 55, 67, 71, 72], [1, 5, 21, 49, 87, 115, 131, 135, 136]] | +| RevAccTabl | [[1], [2, 1], [3, 2, 1], [6, 5, 3, 1], [10, 9, 7, 3, 1], [20, 19, 16, 10, 4, 1], [36, 35, 32, 23, 13, 4, 1], [72, 71, 67, 55, 36, 17, 5, 1], [136, 135, 131, 115, 87, 49, 21, 5, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 2, 3], [1, 3, 5, 6], [1, 3, 7, 9, 10], [1, 4, 10, 16, 19, 20], [1, 4, 13, 23, 32, 35, 36], [1, 5, 17, 36, 55, 67, 71, 72], [1, 5, 21, 49, 87, 115, 131, 135, 136]] | + +Lozanic:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 6, 10, 20, 36, 72, 136] | +| EvenSum | [1, 1, 2, 3, 6, 10, 20, 36, 72] | +| OddSum | [0, 1, 1, 3, 4, 10, 16, 36, 64] | +| AltSum | [1, 0, 1, 0, 2, 0, 4, 0, 8] | +| AccSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | +| AccRevSum | [1, 3, 6, 15, 30, 70, 144, 324, 680] | +| AntiDiagSum | [1, 1, 2, 2, 4, 5, 9, 12, 21] | + +Lozanic:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 4, 6, 90, 228, 2128] | +| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 2] | +| RowMax | [1, 1, 1, 2, 4, 6, 10, 19, 38] | +| ColMiddle | [1, 1, 1, 2, 4, 6, 10, 19, 38] | +| ColECenter | [1, 1, 4, 10, 38] | +| ColOCenter | [1, 2, 6, 19] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 19, 52, 154, 384, 1020, 2464] | +| TransNat0 | [0, 1, 3, 9, 20, 50, 108, 252, 544] | +| TransNat1 | [1, 3, 6, 15, 30, 70, 144, 324, 680] | + +Lozanic:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ LozanicRev Diagonals as rows | DiagRow7 | [1, 4, 20, 60, 170, 396, 868, 1716, 3235]| | DiagRow8 | [1, 5, 25, 85, 255, 651, 1519, 3235, 6470]| -LozanicRev Diagonals as columns +Lozanic:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ LozanicRev Diagonals as columns | DiagCol7 | [1, 4, 20, 60, 170, 396, 868, 1716, 3235] | | DiagCol8 | [1, 5, 25, 85, 255, 651, 1519, 3235, 6470] | -LozanicRev Polynomial values as rows +Lozanic:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ LozanicRev Polynomial values as rows | PolyRow7 | [1, 72, 1281, 10192, 51345, 192696, 589057, 1548576, 3627297] | | PolyRow8 | [1, 136, 3593, 37768, 237073, 1068296, 3819481, 11513608, 30448673] | -LozanicRev Polynomial values as columns +Lozanic:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ LozanicRev Polynomial values as columns | PolyCol7 | [1, 8, 57, 456, 3298, 26384, 193572, 1548576, 11513608] | | PolyCol8 | [1, 9, 73, 657, 5393, 48537, 403033, 3627297, 30448673] | -# LozanicInvRev -True +# Lozanic:Inv:Rev +[] -LozanicInvRev Triangle view +Lozanic:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ LozanicInvRev Triangle view | Row7 | [1, -4, 0, 17, -13, -18, 18, -1] | | Row8 | [1, -4, 0, 20, -22, -28, 56, -4, -19] | -LozanicInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -1, 0, 1, -2, 0, 1, 1, -2, 0, 2, -1, 1, -3, 0, 6, -3, -1, 1, -3, 0, 8, -7, -3, 4, 1, -4, 0, 17, -13, -18, 18, -1, 1, -4, 0, 20, -22, -28, 56, -4, -19] | -| TriRev | [1, -1, 1, 0, -1, 1, 1, 0, -2, 1, -1, 2, 0, -2, 1, -1, -3, 6, 0, -3, 1, 4, -3, -7, 8, 0, -3, 1, -1, 18, -18, -13, 17, 0, -4, 1, -19, -4, 56, -28, -22, 20, 0, -4, 1] | -| TriAcc | [1, 1, 0, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 1, 0, 1, -2, -2, 4, 1, 0, 1, -2, -2, 6, -1, -4, 0, 1, -3, -3, 14, 1, -17, 1, 0, 1, -3, -3, 17, -5, -33, 23, 19, 0] | -| TriRevAcc | [1, 0, 1, 0, 0, 1, 0, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, 4, -2, -2, 1, 0, -4, -1, 6, -2, -2, 1, 0, 1, -17, 1, 14, -3, -3, 1, 0, 19, 23, -33, -5, 17, -3, -3, 1] | -| TriAccRev | [1, -1, 0, 0, -1, 0, 1, 1, -1, 0, -1, 1, 1, -1, 0, -1, -4, 2, 2, -1, 0, 4, 1, -6, 2, 2, -1, 0, -1, 17, -1, -14, 3, 3, -1, 0, -19, -23, 33, 5, -17, 3, 3, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, -1, 1, -2, 0, 1, -2, 0, 1, -3, 0, 1, 1, -3, 0, 2] | - -LozanicInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 1, 0, -2, -2, 6, 16] | -| OddSum | [0, -1, -1, -1, 0, 2, 2, -6, -16] | -| AltSum | [1, 2, 2, 2, 0, -4, -4, 12, 32] | -| AccSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | -| AccRevSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | -| RevAccSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | -| DiagSum | [1, 1, 0, 0, -1, -1, -1, 0] | - -LozanicInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -2, 2, 6, 168, 7956, 58520] | -| RowGcd | [1, 1, 1, -2, 2, 3, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 2, 6, 8, 18, 56] | -| Middle | [1, 1, -1, -2, 0, 0, 8, 17, -22] | -| Central | [1, -1, 0, 8, -22] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 0, 1, -1, -1, 4, -1, -19] | -| PosHalf | [1, 1, 2, 1, 3, 1, 2, 3, 5] | -| NegHalf | [1, -3, 6, -15, 27, -51, 78, -117, 213] | -| Bin | [1, 0, -1, -4, 0, 30, 24, -140, -502] | -| InvBin | [1, -2, 3, -6, 0, 58, -224, 516, 522] | -| TransSqrs | [0, -1, -1, 7, 0, -22, 26, 90, -272] | -| TransNat0 | [0, -1, -1, 1, 0, -2, 2, 6, -16] | -| TransNat1 | [1, -1, -1, 1, 0, -2, 2, 6, -16] | - -LozanicInvRev Diagonals as rows +Lozanic:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -1, 0], [1, -2, 0, 1], [1, -2, 0, 2, -1], [1, -3, 0, 6, -3, -1], [1, -3, 0, 8, -7, -3, 4], [1, -4, 0, 17, -13, -18, 18, -1], [1, -4, 0, 20, -22, -28, 56, -4, -19]] | +| RevTabl | [[1], [-1, 1], [0, -1, 1], [1, 0, -2, 1], [-1, 2, 0, -2, 1], [-1, -3, 6, 0, -3, 1], [4, -3, -7, 8, 0, -3, 1], [-1, 18, -18, -13, 17, 0, -4, 1], [-19, -4, 56, -28, -22, 20, 0, -4, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -1], [1, -2, 0], [1, -2, 0], [1, -3, 0, 1], [1, -3, 0, 2], [1, -4, 0, 6, -1]] | +| AccTabl | [[1], [1, 0], [1, 0, 0], [1, -1, -1, 0], [1, -1, -1, 1, 0], [1, -2, -2, 4, 1, 0], [1, -2, -2, 6, -1, -4, 0], [1, -3, -3, 14, 1, -17, 1, 0], [1, -3, -3, 17, -5, -33, 23, 19, 0]] | +| RevAccTabl | [[1], [0, 1], [0, 0, 1], [0, -1, -1, 1], [0, 1, -1, -1, 1], [0, 1, 4, -2, -2, 1], [0, -4, -1, 6, -2, -2, 1], [0, 1, -17, 1, 14, -3, -3, 1], [0, 19, 23, -33, -5, 17, -3, -3, 1]] | +| AccRevTabl | [[1], [-1, 0], [0, -1, 0], [1, 1, -1, 0], [-1, 1, 1, -1, 0], [-1, -4, 2, 2, -1, 0], [4, 1, -6, 2, 2, -1, 0], [-1, 17, -1, -14, 3, 3, -1, 0], [-19, -23, 33, 5, -17, 3, 3, -1, 0]] | + +Lozanic:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 1, 0, -2, -2, 6, 16] | +| OddSum | [0, -1, -1, -1, 0, 2, 2, -6, -16] | +| AltSum | [1, 2, 2, 2, 0, -4, -4, 12, 32] | +| AccSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | +| AccRevSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | +| AntiDiagSum | [1, 1, 0, 0, -1, -1, -1, 0, 2] | + +Lozanic:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 2, 6, 168, 7956, 58520] | +| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 2, 6, 8, 18, 56] | +| ColMiddle | [1, 1, -1, -2, 0, 0, 8, 17, -22] | +| ColECenter | [1, -1, 0, 8, -22] | +| ColOCenter | [1, -2, 0, 17] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 0, 1, -1, -1, 4, -1, -19] | +| TransSqrs | [0, -1, -1, 7, 0, -22, 26, 90, -272] | +| TransNat0 | [0, -1, -1, 1, 0, -2, 2, 6, -16] | +| TransNat1 | [1, -1, -1, 1, 0, -2, 2, 6, -16] | + +Lozanic:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 0, 1, -1, -1, 4, -1, -19]| @@ -391,7 +375,7 @@ LozanicInvRev Diagonals as rows | DiagRow7 | [1, -4, 0, 40, -70, -396, 1608, 4808, -39301]| | DiagRow8 | [1, -5, 0, 65, -95, -891, 2548, 15777, -71858]| -LozanicInvRev Diagonals as columns +Lozanic:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ LozanicInvRev Diagonals as columns | DiagCol7 | [-1, -4, 60, 80, 730, 908, 4100, 4808, 15777] | | DiagCol8 | [-19, -127, -511, -1603, -4157, -9633, -20121, -39301, -71858] | -LozanicInvRev Polynomial values as rows +Lozanic:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ LozanicInvRev Polynomial values as rows | PolyRow7 | [1, 0, 369, 5956, 36657, 140856, 406705, 966204, 1987041] | | PolyRow8 | [1, 0, -2887, -100640, -1114383, -6958144, -30661655, -106753632, -313473055] | -LozanicInvRev Polynomial values as columns +Lozanic:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ LozanicInvRev Polynomial values as columns | PolyCol7 | [1, -6, -6, 330, -1728, -21972, 406092, 966204, -106753632] | | PolyCol8 | [1, -7, -7, 497, -3087, -42007, 925673, 1987041, -313473055] | +# Lozanic:Rev:Inv +[] + +Lozanic:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [0, -1, 1] | +| Row3 | [1, 0, -2, 1] | +| Row4 | [-1, 2, 0, -2, 1] | +| Row5 | [-1, -3, 6, 0, -3, 1] | +| Row6 | [4, -3, -7, 8, 0, -3, 1] | +| Row7 | [-1, 18, -18, -13, 17, 0, -4, 1] | +| Row8 | [-19, -4, 56, -28, -22, 20, 0, -4, 1] | + +Lozanic:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -1, 1], [1, 0, -2, 1], [-1, 2, 0, -2, 1], [-1, -3, 6, 0, -3, 1], [4, -3, -7, 8, 0, -3, 1], [-1, 18, -18, -13, 17, 0, -4, 1], [-19, -4, 56, -28, -22, 20, 0, -4, 1]] | +| RevTabl | [[1], [1, -1], [1, -1, 0], [1, -2, 0, 1], [1, -2, 0, 2, -1], [1, -3, 0, 6, -3, -1], [1, -3, 0, 8, -7, -3, 4], [1, -4, 0, 17, -13, -18, 18, -1], [1, -4, 0, 20, -22, -28, 56, -4, -19]] | +| AntiDiag | [[1], [-1], [0, 1], [1, -1], [-1, 0, 1], [-1, 2, -2], [4, -3, 0, 1], [-1, -3, 6, -2], [-19, 18, -7, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -1, 0], [1, 1, -1, 0], [-1, 1, 1, -1, 0], [-1, -4, 2, 2, -1, 0], [4, 1, -6, 2, 2, -1, 0], [-1, 17, -1, -14, 3, 3, -1, 0], [-19, -23, 33, 5, -17, 3, 3, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 0], [0, -1, 1, 1], [0, -1, 1, 1, -1], [0, -1, 2, 2, -4, -1], [0, -1, 2, 2, -6, 1, 4], [0, -1, 3, 3, -14, -1, 17, -1], [0, -1, 3, 3, -17, 5, 33, -23, -19]] | +| AccRevTabl | [[1], [1, 0], [1, 0, 0], [1, -1, -1, 0], [1, -1, -1, 1, 0], [1, -2, -2, 4, 1, 0], [1, -2, -2, 6, -1, -4, 0], [1, -3, -3, 14, 1, -17, 1, 0], [1, -3, -3, 17, -5, -33, 23, 19, 0]] | + +Lozanic:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 1, -1, 0, 2, -2, -6, 16] | +| OddSum | [0, 1, -1, 1, 0, -2, 2, 6, -16] | +| AltSum | [1, -2, 2, -2, 0, 4, -4, -12, 32] | +| AccSum | [1, -1, -1, 1, 0, -2, 2, 6, -16] | +| AccRevSum | [1, 1, 1, -1, 0, 2, -2, -6, 16] | +| AntiDiagSum | [1, -1, 1, 0, 0, -1, 2, 0, -7] | + +Lozanic:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 2, 6, 168, 7956, 58520] | +| RowGcd | [1, 1, 1, 2, 2, 3, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 2, 6, 8, 18, 56] | +| ColMiddle | [1, -1, -1, 0, 0, 6, 8, -13, -22] | +| ColECenter | [1, -1, 0, 8, -22] | +| ColOCenter | [-1, 0, 6, -13] | +| ColLeft | [1, -1, 0, 1, -1, -1, 4, -1, -19] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 1, 0, -2, 2, 6, -16] | +| TransNat0 | [0, 1, 1, -1, 0, 2, -2, -6, 16] | +| TransNat1 | [1, 1, 1, -1, 0, 2, -2, -6, 16] | + +Lozanic:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -1, -2, -2, -3, -3, -4, -4, -5]| +| DiagRow2 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow3 | [1, 2, 6, 8, 17, 20, 36, 40, 65]| +| DiagRow4 | [-1, -3, -7, -13, -22, -34, -50, -70, -95]| +| DiagRow5 | [-1, -3, -18, -28, -98, -126, -336, -396, -891]| +| DiagRow6 | [4, 18, 56, 136, 288, 548, 968, 1608, 2548]| +| DiagRow7 | [-1, -4, 60, 80, 730, 908, 4100, 4808, 15777]| +| DiagRow8 | [-19, -127, -511, -1603, -4157, -9633, -20121, -39301, -71858]| + +Lozanic:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 0, 1, -1, -1, 4, -1, -19] | +| DiagCol1 | [1, -1, 0, 2, -3, -3, 18, -4, -127] | +| DiagCol2 | [1, -2, 0, 6, -7, -18, 56, 60, -511] | +| DiagCol3 | [1, -2, 0, 8, -13, -28, 136, 80, -1603] | +| DiagCol4 | [1, -3, 0, 17, -22, -98, 288, 730, -4157] | +| DiagCol5 | [1, -3, 0, 20, -34, -126, 548, 908, -9633] | +| DiagCol6 | [1, -4, 0, 36, -50, -336, 968, 4100, -20121] | +| DiagCol7 | [1, -4, 0, 40, -70, -396, 1608, 4808, -39301] | +| DiagCol8 | [1, -5, 0, 65, -95, -891, 2548, 15777, -71858] | + +Lozanic:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [0, 0, 2, 6, 12, 20, 30, 42, 56] | +| PolyRow3 | [1, 0, 1, 10, 33, 76, 145, 246, 385] | +| PolyRow4 | [-1, 0, 3, 32, 135, 384, 875, 1728, 3087] | +| PolyRow5 | [-1, 0, 1, 44, 339, 1384, 4085, 9876, 20839] | +| PolyRow6 | [4, 0, 2, 148, 1416, 7064, 24790, 69612, 167468] | +| PolyRow7 | [-1, 0, 3, 188, 3303, 24264, 111995, 388548, 1110543] | +| PolyRow8 | [-19, 0, 5, 608, 13917, 124736, 682805, 2747040, 8943053] | + +Lozanic:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 0, 1, -1, -1, 4, -1, -19] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 2, 1, 3, 1, 2, 3, 5] | +| PolyCol3 | [1, 2, 6, 10, 32, 44, 148, 188, 608] | +| PolyCol4 | [1, 3, 12, 33, 135, 339, 1416, 3303, 13917] | +| PolyCol5 | [1, 4, 20, 76, 384, 1384, 7064, 24264, 124736] | +| PolyCol6 | [1, 5, 30, 145, 875, 4085, 24790, 111995, 682805] | +| PolyCol7 | [1, 6, 42, 246, 1728, 9876, 69612, 388548, 2747040] | +| PolyCol8 | [1, 7, 56, 385, 3087, 20839, 167468, 1110543, 8943053] | + diff --git a/data/md/Motzkin.md b/data/md/Motzkin.md index dffce81..f0e0422 100644 --- a/data/md/Motzkin.md +++ b/data/md/Motzkin.md @@ -14,45 +14,41 @@ Motzkin Triangle view | Row7 | [1, 0, 21, 0, 70, 0, 35, 0] | | Row8 | [1, 0, 28, 0, 140, 0, 140, 0, 14] | -Motzkin Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14] | -| TriRev | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 5, 0, 30, 0, 15, 0, 1, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 9, 1, 1, 11, 11, 21, 21, 1, 1, 16, 16, 46, 46, 51, 1, 1, 22, 22, 92, 92, 127, 127, 1, 1, 29, 29, 169, 169, 309, 309, 323] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 9, 7, 7, 1, 1, 21, 21, 11, 11, 1, 1, 51, 46, 46, 16, 16, 1, 1, 127, 127, 92, 92, 22, 22, 1, 1, 323, 309, 309, 169, 169, 29, 29, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 2, 2, 8, 8, 9, 0, 10, 10, 20, 20, 21, 5, 5, 35, 35, 50, 50, 51, 0, 35, 35, 105, 105, 126, 126, 127, 14, 14, 154, 154, 294, 294, 322, 322, 323] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 6, 0, 1, 0, 10, 0] | +Motzkin Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 2], [1, 0, 10, 0, 10, 0], [1, 0, 15, 0, 30, 0, 5], [1, 0, 21, 0, 70, 0, 35, 0], [1, 0, 28, 0, 140, 0, 140, 0, 14]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [2, 0, 6, 0, 1], [0, 10, 0, 10, 0, 1], [5, 0, 30, 0, 15, 0, 1], [0, 35, 0, 70, 0, 21, 0, 1], [14, 0, 140, 0, 140, 0, 28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 3], [1, 0, 6, 0], [1, 0, 10, 0], [1, 0, 15, 0, 2]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 9], [1, 1, 11, 11, 21, 21], [1, 1, 16, 16, 46, 46, 51], [1, 1, 22, 22, 92, 92, 127, 127], [1, 1, 29, 29, 169, 169, 309, 309, 323]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [4, 4, 1, 1], [9, 7, 7, 1, 1], [21, 21, 11, 11, 1, 1], [51, 46, 46, 16, 16, 1, 1], [127, 127, 92, 92, 22, 22, 1, 1], [323, 309, 309, 169, 169, 29, 29, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [2, 2, 8, 8, 9], [0, 10, 10, 20, 20, 21], [5, 5, 35, 35, 50, 50, 51], [0, 35, 35, 105, 105, 126, 126, 127], [14, 14, 154, 154, 294, 294, 322, 322, 323]] | Motzkin Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| EvenSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| AccSum | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | -| AccRevSum | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | -| RevAccSum | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | -| DiagSum | [1, 1, 1, 1, 2, 4, 7, 11] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| EvenSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| OddSum | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| AccSum | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | +| AccRevSum | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | +| AntiDiagSum | [1, 1, 1, 1, 2, 4, 7, 11, 18] | Motzkin Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 10, 30, 210, 140] | -| RowGcd | [1, 1, 1, 3, 2, 10, 5, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 10, 30, 70, 140] | -| Middle | [1, 1, 0, 0, 6, 10, 0, 0, 140] | -| Central | [1, 0, 6, 0, 140] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| PosHalf | [1, 2, 5, 14, 42, 132, 429, 1430, 4862] | -| NegHalf | [1, -2, 5, -14, 42, -132, 429, -1430, 4862] | -| Bin | [1, 1, 2, 10, 39, 151, 681, 3137, 14519] | -| InvBin | [1, -1, 2, -10, 39, -151, 681, -3137, 14519] | -| TransSqrs | [0, 0, 4, 12, 56, 200, 720, 2464, 8288] | -| TransNat0 | [0, 0, 2, 6, 20, 60, 180, 532, 1568] | -| TransNat1 | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 10, 30, 210, 140] | +| RowGcd | [1, 1, 1, 3, 2, 10, 5, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 10, 30, 70, 140] | +| ColMiddle | [1, 1, 0, 0, 6, 10, 0, 0, 140] | +| ColECenter | [1, 0, 6, 0, 140] | +| ColOCenter | [1, 0, 10, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| TransSqrs | [0, 0, 4, 12, 56, 200, 720, 2464, 8288] | +| TransNat0 | [0, 0, 2, 6, 20, 60, 180, 532, 1568] | +| TransNat1 | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | Motzkin Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Motzkin Polynomial values as columns | PolyCol7 | [1, 1, 50, 148, 5097, 24501, 661011, 4286815, 97515587] | | PolyCol8 | [1, 1, 65, 193, 8577, 41601, 1434561, 9463105, 272156417] | -# MotzkinRev -True +# Motzkin:Rev +[] -MotzkinRev Triangle view +Motzkin:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ MotzkinRev Triangle view | Row7 | [0, 35, 0, 70, 0, 21, 0, 1] | | Row8 | [14, 0, 140, 0, 140, 0, 28, 0, 1] | -MotzkinRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 5, 0, 30, 0, 15, 0, 1, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14] | -| TriAcc | [1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 2, 2, 8, 8, 9, 0, 10, 10, 20, 20, 21, 5, 5, 35, 35, 50, 50, 51, 0, 35, 35, 105, 105, 126, 126, 127, 14, 14, 154, 154, 294, 294, 322, 322, 323] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 4, 3, 3, 0, 9, 8, 8, 2, 2, 21, 20, 20, 10, 10, 0, 51, 50, 50, 35, 35, 5, 5, 127, 126, 126, 105, 105, 35, 35, 0, 323, 322, 322, 294, 294, 154, 154, 14, 14] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 7, 9, 1, 1, 11, 11, 21, 21, 1, 1, 16, 16, 46, 46, 51, 1, 1, 22, 22, 92, 92, 127, 127, 1, 1, 29, 29, 169, 169, 309, 309, 323] | -| TriDiag | [1, 0, 1, 1, 0, 0, 2, 3, 1, 0, 0, 0, 5, 10, 6, 1, 0, 0, 0, 0] | - -MotzkinRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| EvenSum | [1, 0, 2, 0, 9, 0, 51, 0, 323] | -| OddSum | [0, 1, 0, 4, 0, 21, 0, 127, 0] | -| AltSum | [1, -1, 2, -4, 9, -21, 51, -127, 323] | -| AccSum | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | -| AccRevSum | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | -| RevAccSum | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | -| DiagSum | [1, 0, 2, 0, 6, 0, 22, 0] | - -MotzkinRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 10, 30, 210, 140] | -| RowGcd | [1, 1, 1, 3, 2, 10, 5, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 10, 30, 70, 140] | -| Middle | [1, 0, 0, 3, 6, 0, 0, 70, 140] | -| Central | [1, 0, 6, 0, 140] | -| LeftSide | [1, 0, 1, 0, 2, 0, 5, 0, 14] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 13, 57, 201, 861, 3445, 14897] | -| NegHalf | [1, 1, 5, 13, 57, 201, 861, 3445, 14897] | -| Bin | [1, 1, 2, 10, 39, 151, 681, 3137, 14519] | -| InvBin | [1, 1, 2, 10, 39, 151, 681, 3137, 14519] | -| TransSqrs | [0, 1, 4, 12, 40, 125, 396, 1239, 3872] | -| TransNat0 | [0, 1, 2, 6, 16, 45, 126, 357, 1016] | -| TransNat1 | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | - -MotzkinRev Diagonals as rows +Motzkin:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [0, 3, 0, 1], [2, 0, 6, 0, 1], [0, 10, 0, 10, 0, 1], [5, 0, 30, 0, 15, 0, 1], [0, 35, 0, 70, 0, 21, 0, 1], [14, 0, 140, 0, 140, 0, 28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 0], [1, 0, 6, 0, 2], [1, 0, 10, 0, 10, 0], [1, 0, 15, 0, 30, 0, 5], [1, 0, 21, 0, 70, 0, 35, 0], [1, 0, 28, 0, 140, 0, 140, 0, 14]] | +| AntiDiag | [[1], [0], [1, 1], [0, 0], [2, 3, 1], [0, 0, 0], [5, 10, 6, 1], [0, 0, 0, 0], [14, 35, 30, 10, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [0, 3, 3, 4], [2, 2, 8, 8, 9], [0, 10, 10, 20, 20, 21], [5, 5, 35, 35, 50, 50, 51], [0, 35, 35, 105, 105, 126, 126, 127], [14, 14, 154, 154, 294, 294, 322, 322, 323]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [4, 3, 3, 0], [9, 8, 8, 2, 2], [21, 20, 20, 10, 10, 0], [51, 50, 50, 35, 35, 5, 5], [127, 126, 126, 105, 105, 35, 35, 0], [323, 322, 322, 294, 294, 154, 154, 14, 14]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 4], [1, 1, 7, 7, 9], [1, 1, 11, 11, 21, 21], [1, 1, 16, 16, 46, 46, 51], [1, 1, 22, 22, 92, 92, 127, 127], [1, 1, 29, 29, 169, 169, 309, 309, 323]] | + +Motzkin:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| EvenSum | [1, 0, 2, 0, 9, 0, 51, 0, 323] | +| OddSum | [0, 1, 0, 4, 0, 21, 0, 127, 0] | +| AltSum | [1, -1, 2, -4, 9, -21, 51, -127, 323] | +| AccSum | [1, 1, 4, 10, 29, 81, 231, 659, 1891] | +| AccRevSum | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | +| AntiDiagSum | [1, 0, 2, 0, 6, 0, 22, 0, 90] | + +Motzkin:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 10, 30, 210, 140] | +| RowGcd | [1, 1, 1, 3, 2, 10, 5, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 10, 30, 70, 140] | +| ColMiddle | [1, 0, 0, 3, 6, 0, 0, 70, 140] | +| ColECenter | [1, 0, 6, 0, 140] | +| ColOCenter | [0, 3, 0, 70] | +| ColLeft | [1, 0, 1, 0, 2, 0, 5, 0, 14] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 12, 40, 125, 396, 1239, 3872] | +| TransNat0 | [0, 1, 2, 6, 16, 45, 126, 357, 1016] | +| TransNat1 | [1, 2, 4, 10, 25, 66, 177, 484, 1339] | + +Motzkin:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ MotzkinRev Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [14, 126, 630, 2310, 6930, 18018, 42042, 90090, 180180]| -MotzkinRev Diagonals as columns +Motzkin:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 1, 0, 2, 0, 5, 0, 14] | @@ -188,7 +180,7 @@ MotzkinRev Diagonals as columns | DiagCol7 | [1, 0, 36, 0, 660, 0, 8580, 0, 90090] | | DiagCol8 | [1, 0, 45, 0, 990, 0, 15015, 0, 180180] | -MotzkinRev Polynomial values as rows +Motzkin:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ MotzkinRev Polynomial values as rows | PolyRow7 | [0, 127, 1430, 9285, 42508, 152675, 458562, 1200745, 2821400] | | PolyRow8 | [14, 323, 4862, 39587, 218318, 919139, 3172478, 9401987, 24699662] | -MotzkinRev Polynomial values as columns +Motzkin:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 1, 0, 2, 0, 5, 0, 14] | @@ -214,3 +206,107 @@ MotzkinRev Polynomial values as columns | PolyCol7 | [1, 7, 50, 364, 2697, 20307, 155139, 1200745, 9401987] | | PolyCol8 | [1, 8, 65, 536, 4482, 37968, 325509, 2821400, 24699662] | +# Motzkin:Rev:Inv +[] + +Motzkin:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [-1, 0, 1] | +| Row3 | [0, -3, 0, 1] | +| Row4 | [4, 0, -6, 0, 1] | +| Row5 | [0, 20, 0, -10, 0, 1] | +| Row6 | [-35, 0, 60, 0, -15, 0, 1] | +| Row7 | [0, -245, 0, 140, 0, -21, 0, 1] | +| Row8 | [546, 0, -980, 0, 280, 0, -28, 0, 1] | + +Motzkin:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [0, -3, 0, 1], [4, 0, -6, 0, 1], [0, 20, 0, -10, 0, 1], [-35, 0, 60, 0, -15, 0, 1], [0, -245, 0, 140, 0, -21, 0, 1], [546, 0, -980, 0, 280, 0, -28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, 0], [1, 0, -6, 0, 4], [1, 0, -10, 0, 20, 0], [1, 0, -15, 0, 60, 0, -35], [1, 0, -21, 0, 140, 0, -245, 0], [1, 0, -28, 0, 280, 0, -980, 0, 546]] | +| AntiDiag | [[1], [0], [-1, 1], [0, 0], [4, -3, 1], [0, 0, 0], [-35, 20, -6, 1], [0, 0, 0, 0], [546, -245, 60, -10, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [0, -3, -3, -2], [4, 4, -2, -2, -1], [0, 20, 20, 10, 10, 11], [-35, -35, 25, 25, 10, 10, 11], [0, -245, -245, -105, -105, -126, -126, -125], [546, 546, -434, -434, -154, -154, -182, -182, -181]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-2, -3, -3, 0], [-1, -2, -2, 4, 4], [11, 10, 10, 20, 20, 0], [11, 10, 10, 25, 25, -35, -35], [-125, -126, -126, -105, -105, -245, -245, 0], [-181, -182, -182, -154, -154, -434, -434, 546, 546]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -2], [1, 1, -5, -5, -1], [1, 1, -9, -9, 11, 11], [1, 1, -14, -14, 46, 46, 11], [1, 1, -20, -20, 120, 120, -125, -125], [1, 1, -27, -27, 253, 253, -727, -727, -181]] | + +Motzkin:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -2, -1, 11, 11, -125, -181] | +| EvenSum | [1, 0, 0, 0, -1, 0, 11, 0, -181] | +| OddSum | [0, 1, 0, -2, 0, 11, 0, -125, 0] | +| AltSum | [1, -1, 0, 2, -1, -11, 11, 125, -181] | +| AccSum | [1, 1, -2, -8, 3, 71, 11, -1077, -629] | +| AccRevSum | [1, 2, 2, -2, -9, 6, 77, -48, -1181] | +| AntiDiagSum | [1, 0, 0, 0, 2, 0, -20, 0, 352] | + +Motzkin:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 12, 20, 420, 2940, 76440] | +| RowGcd | [1, 1, 1, 3, 2, 10, 5, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 20, 60, 245, 980] | +| ColMiddle | [1, 0, 0, -3, -6, 0, 0, 140, 280] | +| ColECenter | [1, 0, -6, 0, 280] | +| ColOCenter | [0, -3, 0, 140] | +| ColLeft | [1, 0, -1, 0, 4, 0, -35, 0, 546] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 6, -8, -45, 36, 539, -384] | +| TransNat0 | [0, 1, 2, 0, -8, -5, 66, 77, -1000] | +| TransNat1 | [1, 2, 2, -2, -9, 6, 77, -48, -1181] | + +Motzkin:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow2 | [-1, -3, -6, -10, -15, -21, -28, -36, -45]| +| DiagRow3 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow4 | [4, 20, 60, 140, 280, 504, 840, 1320, 1980]| +| DiagRow5 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow6 | [-35, -245, -980, -2940, -7350, -16170, -32340, -60060, -105105]| +| DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow8 | [546, 4914, 24570, 90090, 270270, 702702, 1639638, 3513510, 7027020]| + +Motzkin:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, -1, 0, 4, 0, -35, 0, 546] | +| DiagCol1 | [1, 0, -3, 0, 20, 0, -245, 0, 4914] | +| DiagCol2 | [1, 0, -6, 0, 60, 0, -980, 0, 24570] | +| DiagCol3 | [1, 0, -10, 0, 140, 0, -2940, 0, 90090] | +| DiagCol4 | [1, 0, -15, 0, 280, 0, -7350, 0, 270270] | +| DiagCol5 | [1, 0, -21, 0, 504, 0, -16170, 0, 702702] | +| DiagCol6 | [1, 0, -28, 0, 840, 0, -32340, 0, 1639638] | +| DiagCol7 | [1, 0, -36, 0, 1320, 0, -60060, 0, 3513510] | +| DiagCol8 | [1, 0, -45, 0, 1980, 0, -105105, 0, 7027020] | + +Motzkin:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [-1, 0, 3, 8, 15, 24, 35, 48, 63] | +| PolyRow3 | [0, -2, 2, 18, 52, 110, 198, 322, 488] | +| PolyRow4 | [4, -1, -4, 31, 164, 479, 1084, 2111, 3716] | +| PolyRow5 | [0, 11, -8, 33, 464, 1975, 5736, 13517, 27808] | +| PolyRow6 | [-35, 11, 29, 19, 1181, 7715, 29341, 84539, 204509] | +| PolyRow7 | [0, -125, 86, 129, 2860, 28775, 145410, 516901, 1478744] | +| PolyRow8 | [546, -181, -430, 555, 7394, 104171, 701394, 3095435, 10521890] | + +Motzkin:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, -1, 0, 4, 0, -35, 0, 546] | +| PolyCol1 | [1, 1, 0, -2, -1, 11, 11, -125, -181] | +| PolyCol2 | [1, 2, 3, 2, -4, -8, 29, 86, -430] | +| PolyCol3 | [1, 3, 8, 18, 31, 33, 19, 129, 555] | +| PolyCol4 | [1, 4, 15, 52, 164, 464, 1181, 2860, 7394] | +| PolyCol5 | [1, 5, 24, 110, 479, 1975, 7715, 28775, 104171] | +| PolyCol6 | [1, 6, 35, 198, 1084, 5736, 29341, 145410, 701394] | +| PolyCol7 | [1, 7, 48, 322, 2111, 13517, 84539, 516901, 3095435] | +| PolyCol8 | [1, 8, 63, 488, 3716, 27808, 204509, 1478744, 10521890] | + diff --git a/data/md/MotzkinGF.md b/data/md/MotzkinGF.md index d91011c..5666c64 100644 --- a/data/md/MotzkinGF.md +++ b/data/md/MotzkinGF.md @@ -1,5 +1,5 @@ # MotzkinGF -['A026300', 'A064189', 'A009766'] +['A064189', 'A026300', 'A009766'] MotzkinGF Triangle view | Row | Seq | @@ -14,45 +14,41 @@ MotzkinGF Triangle view | Row7 | [127, 196, 189, 133, 70, 27, 7, 1] | | Row8 | [323, 512, 518, 392, 230, 104, 35, 8, 1] | -MotzkinGF Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 51, 76, 69, 44, 20, 6, 1, 127, 196, 189, 133, 70, 27, 7, 1, 323, 512, 518, 392, 230, 104, 35, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 9, 1, 5, 14, 25, 30, 21, 1, 6, 20, 44, 69, 76, 51, 1, 7, 27, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323] | -| TriAcc | [1, 1, 2, 2, 4, 5, 4, 9, 12, 13, 9, 21, 30, 34, 35, 21, 51, 76, 90, 95, 96, 51, 127, 196, 240, 260, 266, 267, 127, 323, 512, 645, 715, 742, 749, 750, 323, 835, 1353, 1745, 1975, 2079, 2114, 2122, 2123] | -| TriRevAcc | [1, 2, 1, 5, 4, 2, 13, 12, 9, 4, 35, 34, 30, 21, 9, 96, 95, 90, 76, 51, 21, 267, 266, 260, 240, 196, 127, 51, 750, 749, 742, 715, 645, 512, 323, 127, 2123, 2122, 2114, 2079, 1975, 1745, 1353, 835, 323] | -| TriAccRev | [1, 1, 2, 1, 3, 5, 1, 4, 9, 13, 1, 5, 14, 26, 35, 1, 6, 20, 45, 75, 96, 1, 7, 27, 71, 140, 216, 267, 1, 8, 35, 105, 238, 427, 623, 750, 1, 9, 44, 148, 378, 770, 1288, 1800, 2123] | -| TriDiag | [1, 1, 2, 1, 4, 2, 9, 5, 1, 21, 12, 3, 51, 30, 9, 1, 127, 76, 25, 4] | +MotzkinGF Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 2, 1], [4, 5, 3, 1], [9, 12, 9, 4, 1], [21, 30, 25, 14, 5, 1], [51, 76, 69, 44, 20, 6, 1], [127, 196, 189, 133, 70, 27, 7, 1], [323, 512, 518, 392, 230, 104, 35, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 2], [1, 3, 5, 4], [1, 4, 9, 12, 9], [1, 5, 14, 25, 30, 21], [1, 6, 20, 44, 69, 76, 51], [1, 7, 27, 70, 133, 189, 196, 127], [1, 8, 35, 104, 230, 392, 518, 512, 323]] | +| AntiDiag | [[1], [1], [2, 1], [4, 2], [9, 5, 1], [21, 12, 3], [51, 30, 9, 1], [127, 76, 25, 4], [323, 196, 69, 14, 1]] | +| AccTabl | [[1], [1, 2], [2, 4, 5], [4, 9, 12, 13], [9, 21, 30, 34, 35], [21, 51, 76, 90, 95, 96], [51, 127, 196, 240, 260, 266, 267], [127, 323, 512, 645, 715, 742, 749, 750], [323, 835, 1353, 1745, 1975, 2079, 2114, 2122, 2123]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 2], [13, 12, 9, 4], [35, 34, 30, 21, 9], [96, 95, 90, 76, 51, 21], [267, 266, 260, 240, 196, 127, 51], [750, 749, 742, 715, 645, 512, 323, 127], [2123, 2122, 2114, 2079, 1975, 1745, 1353, 835, 323]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 9, 13], [1, 5, 14, 26, 35], [1, 6, 20, 45, 75, 96], [1, 7, 27, 71, 140, 216, 267], [1, 8, 35, 105, 238, 427, 623, 750], [1, 9, 44, 148, 378, 770, 1288, 1800, 2123]] | MotzkinGF Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 13, 35, 96, 267, 750, 2123] | -| EvenSum | [1, 1, 3, 7, 19, 51, 141, 393, 1107] | -| OddSum | [0, 1, 2, 6, 16, 45, 126, 357, 1016] | -| AltSum | [1, 0, 1, 1, 3, 6, 15, 36, 91] | -| AccSum | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | -| AccRevSum | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | -| RevAccSum | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| DiagSum | [1, 1, 3, 6, 15, 36, 91, 232] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 13, 35, 96, 267, 750, 2123] | +| EvenSum | [1, 1, 3, 7, 19, 51, 141, 393, 1107] | +| OddSum | [0, 1, 2, 6, 16, 45, 126, 357, 1016] | +| AltSum | [1, 0, 1, 1, 3, 6, 15, 36, 91] | +| AccSum | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | +| AccRevSum | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| AntiDiagSum | [1, 1, 3, 6, 15, 36, 91, 232, 603] | MotzkinGF Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 60, 36, 1050, 4903140, 63847980, 448240898560] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 5, 12, 30, 76, 196, 518] | -| Middle | [1, 1, 2, 5, 9, 25, 44, 133, 230] | -| Central | [1, 2, 9, 44, 230] | -| LeftSide | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 13, 59, 285, 1419, 7245, 37659, 198589] | -| NegHalf | [1, -1, 5, -17, 77, -345, 1653, -8097, 40733] | -| Bin | [1, 2, 7, 29, 128, 587, 2759, 13190, 63844] | -| InvBin | [1, 0, -1, 3, 0, -5, 15, 0, -28] | -| TransSqrs | [0, 1, 6, 26, 100, 361, 1254, 4245, 14108] | -| TransNat0 | [0, 1, 4, 14, 46, 147, 462, 1437, 4438] | -| TransNat1 | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 60, 36, 1050, 4903140, 63847980, 448240898560] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 5, 12, 30, 76, 196, 518] | +| ColMiddle | [1, 1, 2, 5, 9, 25, 44, 133, 230] | +| ColECenter | [1, 2, 9, 44, 230] | +| ColOCenter | [1, 5, 25, 133] | +| ColLeft | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 26, 100, 361, 1254, 4245, 14108] | +| TransNat0 | [0, 1, 4, 14, 46, 147, 462, 1437, 4438] | +| TransNat1 | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | MotzkinGF Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ MotzkinGF Polynomial values as columns | PolyCol7 | [1, 8, 65, 529, 4307, 35070, 285567, 2325324, 18934763] | | PolyCol8 | [1, 9, 82, 748, 6825, 62277, 568275, 5185503, 47317699] | -# MotzkinGFInv -True +# MotzkinGF:Inv +[] -MotzkinGFInv Triangle view +MotzkinGF:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ MotzkinGFInv Triangle view | Row7 | [-1, 3, 9, -15, -5, 15, -7, 1] | | Row8 | [0, -6, 3, 24, -20, -14, 21, -8, 1] | -MotzkinGFInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 0, -2, 1, 1, 1, -3, 1, -1, 2, 3, -4, 1, 0, -4, 2, 6, -5, 1, 1, 2, -9, 0, 10, -6, 1, -1, 3, 9, -15, -5, 15, -7, 1, 0, -6, 3, 24, -20, -14, 21, -8, 1] | -| TriRev | [1, 1, -1, 1, -2, 0, 1, -3, 1, 1, 1, -4, 3, 2, -1, 1, -5, 6, 2, -4, 0, 1, -6, 10, 0, -9, 2, 1, 1, -7, 15, -5, -15, 9, 3, -1, 1, -8, 21, -14, -20, 24, 3, -6, 0] | -| TriAcc | [1, -1, 0, 0, -2, -1, 1, 2, -1, 0, -1, 1, 4, 0, 1, 0, -4, -2, 4, -1, 0, 1, 3, -6, -6, 4, -2, -1, -1, 2, 11, -4, -9, 6, -1, 0, 0, -6, -3, 21, 1, -13, 8, 0, 1] | -| TriRevAcc | [1, 0, -1, -1, -2, 0, 0, -1, 2, 1, 1, 0, 4, 1, -1, 0, -1, 4, -2, -4, 0, -1, -2, 4, -6, -6, 3, 1, 0, -1, 6, -9, -4, 11, 2, -1, 1, 0, 8, -13, 1, 21, -3, -6, 0] | -| TriAccRev | [1, 1, 0, 1, -1, -1, 1, -2, -1, 0, 1, -3, 0, 2, 1, 1, -4, 2, 4, 0, 0, 1, -5, 5, 5, -4, -2, -1, 1, -6, 9, 4, -11, -2, 1, 0, 1, -7, 14, 0, -20, 4, 7, 1, 1] | -| TriDiag | [1, -1, 0, 1, 1, -2, -1, 1, 1, 0, 2, -3, 1, -4, 3, 1, -1, 2, 2, -4] | - -MotzkinGFInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| EvenSum | [1, -1, 1, -2, 3, -3, 3, -4, 5] | -| OddSum | [0, 1, -2, 2, -2, 3, -4, 4, -4] | -| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| AccSum | [1, -1, -3, 2, 5, -3, -7, 4, 9] | -| AccRevSum | [1, -1, -3, 2, 5, -3, -7, 4, 9] | -| RevAccSum | [1, 1, -1, -2, 1, 3, -1, -4, 1] | -| DiagSum | [1, -1, 1, -1, 1, -1, 1, -1] | - -MotzkinGFInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, -3, 12, 60, 90, 315, 840] | -| RowGcd | [1, 1, -2, -3, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 4, 6, 10, 15, 24] | -| Middle | [1, -1, -2, 1, 3, 2, 0, -15, -20] | -| Central | [1, -2, 3, 0, -20] | -| LeftSide | [1, -1, 0, 1, -1, 0, 1, -1, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, 7, 5, -33, 13, 119, -171] | -| NegHalf | [1, 3, 5, 3, -11, -45, -91, -93, 85] | -| Bin | [1, 0, -3, -4, 10, 36, -7, -224, -279] | -| InvBin | [1, 2, 5, 12, 26, 46, 41, -152, -1175] | -| TransSqrs | [0, 1, 2, -2, -6, 3, 12, -4, -20] | -| TransNat0 | [0, 1, 0, -2, 0, 3, 0, -4, 0] | -| TransNat1 | [1, 1, -1, -2, 1, 3, -1, -4, 1] | - -MotzkinGFInv Diagonals as rows +MotzkinGF:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -2, 1], [1, 1, -3, 1], [-1, 2, 3, -4, 1], [0, -4, 2, 6, -5, 1], [1, 2, -9, 0, 10, -6, 1], [-1, 3, 9, -15, -5, 15, -7, 1], [0, -6, 3, 24, -20, -14, 21, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 0], [1, -3, 1, 1], [1, -4, 3, 2, -1], [1, -5, 6, 2, -4, 0], [1, -6, 10, 0, -9, 2, 1], [1, -7, 15, -5, -15, 9, 3, -1], [1, -8, 21, -14, -20, 24, 3, -6, 0]] | +| AntiDiag | [[1], [-1], [0, 1], [1, -2], [-1, 1, 1], [0, 2, -3], [1, -4, 3, 1], [-1, 2, 2, -4], [0, 3, -9, 6, 1]] | +| AccTabl | [[1], [-1, 0], [0, -2, -1], [1, 2, -1, 0], [-1, 1, 4, 0, 1], [0, -4, -2, 4, -1, 0], [1, 3, -6, -6, 4, -2, -1], [-1, 2, 11, -4, -9, 6, -1, 0], [0, -6, -3, 21, 1, -13, 8, 0, 1]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 0], [0, -1, 2, 1], [1, 0, 4, 1, -1], [0, -1, 4, -2, -4, 0], [-1, -2, 4, -6, -6, 3, 1], [0, -1, 6, -9, -4, 11, 2, -1], [1, 0, 8, -13, 1, 21, -3, -6, 0]] | +| AccRevTabl | [[1], [1, 0], [1, -1, -1], [1, -2, -1, 0], [1, -3, 0, 2, 1], [1, -4, 2, 4, 0, 0], [1, -5, 5, 5, -4, -2, -1], [1, -6, 9, 4, -11, -2, 1, 0], [1, -7, 14, 0, -20, 4, 7, 1, 1]] | + +MotzkinGF:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| EvenSum | [1, -1, 1, -2, 3, -3, 3, -4, 5] | +| OddSum | [0, 1, -2, 2, -2, 3, -4, 4, -4] | +| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| AccSum | [1, -1, -3, 2, 5, -3, -7, 4, 9] | +| AccRevSum | [1, 1, -1, -2, 1, 3, -1, -4, 1] | +| AntiDiagSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | + +MotzkinGF:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 60, 90, 315, 840] | +| RowGcd | [1, 1, 2, 3, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 4, 6, 10, 15, 24] | +| ColMiddle | [1, -1, -2, 1, 3, 2, 0, -15, -20] | +| ColECenter | [1, -2, 3, 0, -20] | +| ColOCenter | [-1, 1, 2, -15] | +| ColLeft | [1, -1, 0, 1, -1, 0, 1, -1, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -2, -6, 3, 12, -4, -20] | +| TransNat0 | [0, 1, 0, -2, 0, 3, 0, -4, 0] | +| TransNat1 | [1, 1, -1, -2, 1, 3, -1, -4, 1] | + +MotzkinGF:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ MotzkinGFInv Diagonals as rows | DiagRow7 | [-1, -6, -18, -36, -50, -36, 48, 264, 693]| | DiagRow8 | [0, 3, 18, 60, 145, 279, 441, 561, 495]| -MotzkinGFInv Diagonals as columns +MotzkinGF:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 0, 1, -1, 0, 1, -1, 0] | @@ -188,7 +180,7 @@ MotzkinGFInv Diagonals as columns | DiagCol7 | [1, -8, 28, -48, 6, 168, -384, 264, 561] | | DiagCol8 | [1, -9, 36, -75, 45, 198, -627, 693, 495] | -MotzkinGFInv Polynomial values as rows +MotzkinGF:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ MotzkinGFInv Polynomial values as rows | PolyRow7 | [-1, 0, 1, 8, 987, 10864, 60605, 235416, 726103] | | PolyRow8 | [0, 1, 0, 9, 2584, 40545, 290376, 1372105, 4976784] | -MotzkinGFInv Polynomial values as columns +MotzkinGF:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 0, 1, -1, 0, 1, -1, 0] | @@ -214,10 +206,10 @@ MotzkinGFInv Polynomial values as columns | PolyCol7 | [1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105] | | PolyCol8 | [1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784] | -# MotzkinGFRev -True +# MotzkinGF:Rev +[] -MotzkinGFRev Triangle view +MotzkinGF:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ MotzkinGFRev Triangle view | Row7 | [1, 7, 27, 70, 133, 189, 196, 127] | | Row8 | [1, 8, 35, 104, 230, 392, 518, 512, 323] | -MotzkinGFRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 9, 1, 5, 14, 25, 30, 21, 1, 6, 20, 44, 69, 76, 51, 1, 7, 27, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323] | -| TriRev | [1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 51, 76, 69, 44, 20, 6, 1, 127, 196, 189, 133, 70, 27, 7, 1, 323, 512, 518, 392, 230, 104, 35, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 5, 1, 4, 9, 13, 1, 5, 14, 26, 35, 1, 6, 20, 45, 75, 96, 1, 7, 27, 71, 140, 216, 267, 1, 8, 35, 105, 238, 427, 623, 750, 1, 9, 44, 148, 378, 770, 1288, 1800, 2123] | -| TriRevAcc | [1, 2, 1, 5, 3, 1, 13, 9, 4, 1, 35, 26, 14, 5, 1, 96, 75, 45, 20, 6, 1, 267, 216, 140, 71, 27, 7, 1, 750, 623, 427, 238, 105, 35, 8, 1, 2123, 1800, 1288, 770, 378, 148, 44, 9, 1] | -| TriAccRev | [1, 1, 2, 2, 4, 5, 4, 9, 12, 13, 9, 21, 30, 34, 35, 21, 51, 76, 90, 95, 96, 51, 127, 196, 240, 260, 266, 267, 127, 323, 512, 645, 715, 742, 749, 750, 323, 835, 1353, 1745, 1975, 2079, 2114, 2122, 2123] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 4, 1, 6, 14, 12] | - -MotzkinGFRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 13, 35, 96, 267, 750, 2123] | -| EvenSum | [1, 1, 3, 6, 19, 45, 141, 357, 1107] | -| OddSum | [0, 1, 2, 7, 16, 51, 126, 393, 1016] | -| AltSum | [1, 0, 1, -1, 3, -6, 15, -36, 91] | -| AccSum | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| AccRevSum | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| RevAccSum | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | -| DiagSum | [1, 1, 2, 3, 6, 10, 19, 33] | - -MotzkinGFRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 60, 36, 1050, 4903140, 63847980, 448240898560] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 5, 12, 30, 76, 196, 518] | -| Middle | [1, 1, 2, 3, 9, 14, 44, 70, 230] | -| Central | [1, 2, 9, 44, 230] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 4, 9, 21, 51, 127, 323] | -| PosHalf | [1, 3, 10, 34, 117, 405, 1407, 4899, 17083] | -| NegHalf | [1, -1, 2, -2, 5, -3, 15, 3, 59] | -| Bin | [1, 2, 7, 29, 128, 587, 2759, 13190, 63844] | -| InvBin | [1, 0, -1, -3, 0, 5, 15, 0, -28] | -| TransSqrs | [0, 1, 10, 59, 292, 1291, 5322, 20877, 78972] | -| TransNat0 | [0, 1, 6, 25, 94, 333, 1140, 3813, 12546] | -| TransNat1 | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | - -MotzkinGFRev Diagonals as rows +MotzkinGF:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 2], [1, 3, 5, 4], [1, 4, 9, 12, 9], [1, 5, 14, 25, 30, 21], [1, 6, 20, 44, 69, 76, 51], [1, 7, 27, 70, 133, 189, 196, 127], [1, 8, 35, 104, 230, 392, 518, 512, 323]] | +| RevTabl | [[1], [1, 1], [2, 2, 1], [4, 5, 3, 1], [9, 12, 9, 4, 1], [21, 30, 25, 14, 5, 1], [51, 76, 69, 44, 20, 6, 1], [127, 196, 189, 133, 70, 27, 7, 1], [323, 512, 518, 392, 230, 104, 35, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 2], [1, 4, 5], [1, 5, 9, 4], [1, 6, 14, 12], [1, 7, 20, 25, 9]] | +| AccTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 9, 13], [1, 5, 14, 26, 35], [1, 6, 20, 45, 75, 96], [1, 7, 27, 71, 140, 216, 267], [1, 8, 35, 105, 238, 427, 623, 750], [1, 9, 44, 148, 378, 770, 1288, 1800, 2123]] | +| RevAccTabl | [[1], [2, 1], [5, 3, 1], [13, 9, 4, 1], [35, 26, 14, 5, 1], [96, 75, 45, 20, 6, 1], [267, 216, 140, 71, 27, 7, 1], [750, 623, 427, 238, 105, 35, 8, 1], [2123, 1800, 1288, 770, 378, 148, 44, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 4, 5], [4, 9, 12, 13], [9, 21, 30, 34, 35], [21, 51, 76, 90, 95, 96], [51, 127, 196, 240, 260, 266, 267], [127, 323, 512, 645, 715, 742, 749, 750], [323, 835, 1353, 1745, 1975, 2079, 2114, 2122, 2123]] | + +MotzkinGF:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 13, 35, 96, 267, 750, 2123] | +| EvenSum | [1, 1, 3, 6, 19, 45, 141, 357, 1107] | +| OddSum | [0, 1, 2, 7, 16, 51, 126, 393, 1016] | +| AltSum | [1, 0, 1, -1, 3, -6, 15, -36, 91] | +| AccSum | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| AccRevSum | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | +| AntiDiagSum | [1, 1, 2, 3, 6, 10, 19, 33, 62] | + +MotzkinGF:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 60, 36, 1050, 4903140, 63847980, 448240898560] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 5, 12, 30, 76, 196, 518] | +| ColMiddle | [1, 1, 2, 3, 9, 14, 44, 70, 230] | +| ColECenter | [1, 2, 9, 44, 230] | +| ColOCenter | [1, 3, 14, 70] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 4, 9, 21, 51, 127, 323] | +| TransSqrs | [0, 1, 10, 59, 292, 1291, 5322, 20877, 78972] | +| TransNat0 | [0, 1, 6, 25, 94, 333, 1140, 3813, 12546] | +| TransNat1 | [1, 3, 11, 38, 129, 429, 1407, 4563, 14669] | + +MotzkinGF:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 4, 9, 21, 51, 127, 323]| @@ -283,7 +271,7 @@ MotzkinGFRev Diagonals as rows | DiagRow7 | [1, 8, 44, 200, 814, 3080, 11076, 38376, 129285]| | DiagRow8 | [1, 9, 54, 264, 1143, 4563, 17199, 62127, 217242]| -MotzkinGFRev Diagonals as columns +MotzkinGF:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ MotzkinGFRev Diagonals as columns | DiagCol7 | [127, 512, 1422, 3288, 6765, 12804, 22737, 38376, 62127] | | DiagCol8 | [323, 1353, 3915, 9438, 20240, 39897, 73710, 129285, 217242] | -MotzkinGFRev Polynomial values as rows +MotzkinGF:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ MotzkinGFRev Polynomial values as rows | PolyRow7 | [1, 750, 37659, 479488, 3116109, 13667586, 46354615, 131170404, 324494073] | | PolyRow8 | [1, 2123, 198589, 3733603, 32146001, 175648291, 713381053, 2351804939, 6642415873] | -MotzkinGFRev Polynomial values as columns +MotzkinGF:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ MotzkinGFRev Polynomial values as columns | PolyCol7 | [1, 8, 113, 1639, 26195, 434274, 7459215, 131170404, 2351804939] | | PolyCol8 | [1, 9, 145, 2393, 43617, 824745, 16166193, 324494073, 6642415873] | -# MotzkinGFInvRev -True +# MotzkinGF:Inv:Rev +[] -MotzkinGFInvRev Triangle view +MotzkinGF:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ MotzkinGFInvRev Triangle view | Row7 | [1, -7, 15, -5, -15, 9, 3, -1] | | Row8 | [1, -8, 21, -14, -20, 24, 3, -6, 0] | -MotzkinGFInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -2, 0, 1, -3, 1, 1, 1, -4, 3, 2, -1, 1, -5, 6, 2, -4, 0, 1, -6, 10, 0, -9, 2, 1, 1, -7, 15, -5, -15, 9, 3, -1, 1, -8, 21, -14, -20, 24, 3, -6, 0] | -| TriRev | [1, -1, 1, 0, -2, 1, 1, 1, -3, 1, -1, 2, 3, -4, 1, 0, -4, 2, 6, -5, 1, 1, 2, -9, 0, 10, -6, 1, -1, 3, 9, -15, -5, 15, -7, 1, 0, -6, 3, 24, -20, -14, 21, -8, 1] | -| TriAcc | [1, 1, 0, 1, -1, -1, 1, -2, -1, 0, 1, -3, 0, 2, 1, 1, -4, 2, 4, 0, 0, 1, -5, 5, 5, -4, -2, -1, 1, -6, 9, 4, -11, -2, 1, 0, 1, -7, 14, 0, -20, 4, 7, 1, 1] | -| TriRevAcc | [1, 0, 1, -1, -1, 1, 0, -1, -2, 1, 1, 2, 0, -3, 1, 0, 0, 4, 2, -4, 1, -1, -2, -4, 5, 5, -5, 1, 0, 1, -2, -11, 4, 9, -6, 1, 1, 1, 7, 4, -20, 0, 14, -7, 1] | -| TriAccRev | [1, -1, 0, 0, -2, -1, 1, 2, -1, 0, -1, 1, 4, 0, 1, 0, -4, -2, 4, -1, 0, 1, 3, -6, -6, 4, -2, -1, -1, 2, 11, -4, -9, 6, -1, 0, 0, -6, -3, 21, 1, -13, 8, 0, 1] | -| TriDiag | [1, 1, 1, -1, 1, -2, 1, -3, 0, 1, -4, 1, 1, -5, 3, 1, 1, -6, 6, 2] | - -MotzkinGFInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | -| EvenSum | [1, 1, 1, 2, 3, 3, 3, 4, 5] | -| OddSum | [0, -1, -2, -2, -2, -3, -4, -4, -4] | -| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| AccSum | [1, 1, -1, -2, 1, 3, -1, -4, 1] | -| AccRevSum | [1, 1, -1, -2, 1, 3, -1, -4, 1] | -| RevAccSum | [1, -1, -3, 2, 5, -3, -7, 4, 9] | -| DiagSum | [1, 1, 0, -1, -2, -2, 0, 3] | - -MotzkinGFInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, -3, 12, 60, 90, 315, 840] | -| RowGcd | [1, 1, -2, -3, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 4, 6, 10, 15, 24] | -| Middle | [1, 1, -2, -3, 3, 6, 0, -5, -20] | -| Central | [1, -2, 3, 0, -20] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 0, 1, -1, 0, 1, -1, 0] | -| PosHalf | [1, 1, 0, -1, -1, 0, 1, 1, 0] | -| NegHalf | [1, -3, 8, -21, 55, -144, 377, -987, 2584] | -| Bin | [1, 0, -3, -4, 10, 36, -7, -224, -279] | -| InvBin | [1, -2, 5, -12, 26, -46, 41, 152, -1175] | -| TransSqrs | [0, -1, -2, 10, 10, -27, -24, 52, 44] | -| TransNat0 | [0, -1, -2, 2, 4, -3, -6, 4, 8] | -| TransNat1 | [1, -1, -3, 2, 5, -3, -7, 4, 9] | - -MotzkinGFInvRev Diagonals as rows +MotzkinGF:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -2, 0], [1, -3, 1, 1], [1, -4, 3, 2, -1], [1, -5, 6, 2, -4, 0], [1, -6, 10, 0, -9, 2, 1], [1, -7, 15, -5, -15, 9, 3, -1], [1, -8, 21, -14, -20, 24, 3, -6, 0]] | +| RevTabl | [[1], [-1, 1], [0, -2, 1], [1, 1, -3, 1], [-1, 2, 3, -4, 1], [0, -4, 2, 6, -5, 1], [1, 2, -9, 0, 10, -6, 1], [-1, 3, 9, -15, -5, 15, -7, 1], [0, -6, 3, 24, -20, -14, 21, -8, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -2], [1, -3, 0], [1, -4, 1], [1, -5, 3, 1], [1, -6, 6, 2], [1, -7, 10, 2, -1]] | +| AccTabl | [[1], [1, 0], [1, -1, -1], [1, -2, -1, 0], [1, -3, 0, 2, 1], [1, -4, 2, 4, 0, 0], [1, -5, 5, 5, -4, -2, -1], [1, -6, 9, 4, -11, -2, 1, 0], [1, -7, 14, 0, -20, 4, 7, 1, 1]] | +| RevAccTabl | [[1], [0, 1], [-1, -1, 1], [0, -1, -2, 1], [1, 2, 0, -3, 1], [0, 0, 4, 2, -4, 1], [-1, -2, -4, 5, 5, -5, 1], [0, 1, -2, -11, 4, 9, -6, 1], [1, 1, 7, 4, -20, 0, 14, -7, 1]] | +| AccRevTabl | [[1], [-1, 0], [0, -2, -1], [1, 2, -1, 0], [-1, 1, 4, 0, 1], [0, -4, -2, 4, -1, 0], [1, 3, -6, -6, 4, -2, -1], [-1, 2, 11, -4, -9, 6, -1, 0], [0, -6, -3, 21, 1, -13, 8, 0, 1]] | + +MotzkinGF:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 0, 1, 0, -1, 0, 1] | +| EvenSum | [1, 1, 1, 2, 3, 3, 3, 4, 5] | +| OddSum | [0, -1, -2, -2, -2, -3, -4, -4, -4] | +| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| AccSum | [1, 1, -1, -2, 1, 3, -1, -4, 1] | +| AccRevSum | [1, -1, -3, 2, 5, -3, -7, 4, 9] | +| AntiDiagSum | [1, 1, 0, -1, -2, -2, 0, 3, 5] | + +MotzkinGF:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 60, 90, 315, 840] | +| RowGcd | [1, 1, 2, 3, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 4, 6, 10, 15, 24] | +| ColMiddle | [1, 1, -2, -3, 3, 6, 0, -5, -20] | +| ColECenter | [1, -2, 3, 0, -20] | +| ColOCenter | [1, -3, 6, -5] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 0, 1, -1, 0, 1, -1, 0] | +| TransSqrs | [0, -1, -2, 10, 10, -27, -24, 52, 44] | +| TransNat0 | [0, -1, -2, 2, 4, -3, -6, 4, 8] | +| TransNat1 | [1, -1, -3, 2, 5, -3, -7, 4, 9] | + +MotzkinGF:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 0, 1, -1, 0, 1, -1, 0]| @@ -391,7 +375,7 @@ MotzkinGFInvRev Diagonals as rows | DiagRow7 | [1, -8, 28, -48, 6, 168, -384, 264, 561]| | DiagRow8 | [1, -9, 36, -75, 45, 198, -627, 693, 495]| -MotzkinGFInvRev Diagonals as columns +MotzkinGF:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ MotzkinGFInvRev Diagonals as columns | DiagCol7 | [-1, -6, -18, -36, -50, -36, 48, 264, 693] | | DiagCol8 | [0, 3, 18, 60, 145, 279, 441, 561, 495] | -MotzkinGFInvRev Polynomial values as rows +MotzkinGF:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ MotzkinGFInvRev Polynomial values as rows | PolyRow7 | [1, 0, 119, 952, 1173, -12784, -90005, -356376, -1078903] | | PolyRow8 | [1, 1, -171, -6935, -67151, -360639, -1381259, -4236791, -11097855] | -MotzkinGFInvRev Polynomial values as columns +MotzkinGF:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Narayana.md b/data/md/Narayana.md index 5ce53e4..78925ed 100644 --- a/data/md/Narayana.md +++ b/data/md/Narayana.md @@ -1,5 +1,5 @@ # Narayana -['A001263', 'A090181', 'A131198'] +['A090181', 'A001263', 'A131198'] Narayana Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Narayana Triangle view | Row7 | [0, 1, 21, 105, 175, 105, 21, 1] | | Row8 | [0, 1, 28, 196, 490, 490, 196, 28, 1] | -Narayana Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 13, 14, 0, 1, 11, 31, 41, 42, 0, 1, 16, 66, 116, 131, 132, 0, 1, 22, 127, 302, 407, 428, 429, 0, 1, 29, 225, 715, 1205, 1401, 1429, 1430] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 13, 7, 1, 0, 42, 41, 31, 11, 1, 0, 132, 131, 116, 66, 16, 1, 0, 429, 428, 407, 302, 127, 22, 1, 0, 1430, 1429, 1401, 1205, 715, 225, 29, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 7, 13, 14, 14, 1, 11, 31, 41, 42, 42, 1, 16, 66, 116, 131, 132, 132, 1, 22, 127, 302, 407, 428, 429, 429, 1, 29, 225, 715, 1205, 1401, 1429, 1430, 1430] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 3, 0, 1, 6, 1, 0, 1, 10, 6] | +Narayana Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 3, 1], [0, 1, 6, 6, 1], [0, 1, 10, 20, 10, 1], [0, 1, 15, 50, 50, 15, 1], [0, 1, 21, 105, 175, 105, 21, 1], [0, 1, 28, 196, 490, 490, 196, 28, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 3, 1, 0], [1, 6, 6, 1, 0], [1, 10, 20, 10, 1, 0], [1, 15, 50, 50, 15, 1, 0], [1, 21, 105, 175, 105, 21, 1, 0], [1, 28, 196, 490, 490, 196, 28, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 3], [0, 1, 6, 1], [0, 1, 10, 6], [0, 1, 15, 20, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 4, 5], [0, 1, 7, 13, 14], [0, 1, 11, 31, 41, 42], [0, 1, 16, 66, 116, 131, 132], [0, 1, 22, 127, 302, 407, 428, 429], [0, 1, 29, 225, 715, 1205, 1401, 1429, 1430]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [5, 4, 1, 0], [14, 13, 7, 1, 0], [42, 41, 31, 11, 1, 0], [132, 131, 116, 66, 16, 1, 0], [429, 428, 407, 302, 127, 22, 1, 0], [1430, 1429, 1401, 1205, 715, 225, 29, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 5, 5], [1, 7, 13, 14, 14], [1, 11, 31, 41, 42, 42], [1, 16, 66, 116, 131, 132, 132], [1, 22, 127, 302, 407, 428, 429, 429], [1, 29, 225, 715, 1205, 1401, 1429, 1430, 1430]] | Narayana Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| EvenSum | [1, 0, 1, 3, 7, 20, 66, 217, 715] | -| OddSum | [0, 1, 1, 2, 7, 22, 66, 212, 715] | -| AltSum | [1, -1, 0, 1, 0, -2, 0, 5, 0] | -| AccSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| AccRevSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| RevAccSum | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | -| DiagSum | [1, 0, 1, 1, 2, 4, 8, 17] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| EvenSum | [1, 0, 1, 3, 7, 20, 66, 217, 715] | +| OddSum | [0, 1, 1, 2, 7, 22, 66, 212, 715] | +| AltSum | [1, -1, 0, 1, 0, -2, 0, 5, 0] | +| AccSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| AccRevSum | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | +| AntiDiagSum | [1, 0, 1, 1, 2, 4, 8, 17, 37] | Narayana Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 20, 150, 525, 980] | -| RowGcd | [1, 1, 1, 3, 6, 10, 5, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 20, 50, 175, 490] | -| Middle | [1, 0, 1, 1, 6, 10, 50, 105, 490] | -| Central | [1, 1, 6, 50, 490] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | -| NegHalf | [1, 1, -1, -1, 5, -3, -21, 51, 41] | -| Bin | [1, 1, 3, 13, 65, 356, 2072, 12601, 79221] | -| InvBin | [1, 1, -1, -5, 9, 56, -120, -825, 1925] | -| TransSqrs | [0, 1, 5, 22, 95, 406, 1722, 7260, 30459] | -| TransNat0 | [0, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| TransNat1 | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 20, 150, 525, 980] | +| RowGcd | [1, 1, 1, 3, 6, 10, 5, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 20, 50, 175, 490] | +| ColMiddle | [1, 0, 1, 1, 6, 10, 50, 105, 490] | +| ColECenter | [1, 1, 6, 50, 490] | +| ColOCenter | [0, 1, 10, 105] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 22, 95, 406, 1722, 7260, 30459] | +| TransNat0 | [0, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| TransNat1 | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | Narayana Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Narayana Polynomial values as columns | PolyCol7 | [1, 7, 56, 497, 4760, 48174, 507696, 5516133, 61363736] | | PolyCol8 | [1, 8, 72, 712, 7560, 84616, 985032, 11814728, 145043208] | -# NarayanaInv -True +# Narayana:Inv +[] -NarayanaInv Triangle view +Narayana:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ NarayanaInv Triangle view | Row7 | [0, 3681, -6741, 4095, -1225, 210, -21, 1] | | Row8 | [0, -56197, 103068, -62916, 19110, -3430, 392, -28, 1] | -NarayanaInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -7, 12, -6, 1, 0, 39, -70, 40, -10, 1, 0, -321, 585, -350, 100, -15, 1, 0, 3681, -6741, 4095, -1225, 210, -21, 1, 0, -56197, 103068, -62916, 19110, -3430, 392, -28, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 12, -7, 0, 1, -10, 40, -70, 39, 0, 1, -15, 100, -350, 585, -321, 0, 1, -21, 210, -1225, 4095, -6741, 3681, 0, 1, -28, 392, -3430, 19110, -62916, 103068, -56197, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 2, -1, 0, 0, -7, 5, -1, 0, 0, 39, -31, 9, -1, 0, 0, -321, 264, -86, 14, -1, 0, 0, 3681, -3060, 1035, -190, 20, -1, 0, 0, -56197, 46871, -16045, 3065, -365, 27, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, 0, -1, 2, 0, 0, -1, 5, -7, 0, 0, -1, 9, -31, 39, 0, 0, -1, 14, -86, 264, -321, 0, 0, -1, 20, -190, 1035, -3060, 3681, 0, 0, -1, 27, -365, 3065, -16045, 46871, -56197, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -2, 0, 0, 1, -5, 7, 0, 0, 1, -9, 31, -39, 0, 0, 1, -14, 86, -264, 321, 0, 0, 1, -20, 190, -1035, 3060, -3681, 0, 0, 1, -27, 365, -3065, 16045, -46871, 56197, 0, 0] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 2, 1, 0, -7, -3, 0, 39, 12, 1, 0, -321, -70, -6] | - -NarayanaInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 0, 1, -3, 13, -80, 686, -7987, 122571] | -| OddSum | [0, 1, -1, 3, -13, 80, -686, 7987, -122571] | -| AltSum | [1, -1, 2, -6, 26, -160, 1372, -15974, 245142] | -| AccSum | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | -| AccRevSum | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | -| RevAccSum | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | -| DiagSum | [1, 0, 1, -1, 3, -10, 52, -397] | - -NarayanaInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 84, 10920, 8763300, 12544663950, 3947845887989640] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 12, 70, 585, 6741, 103068] | -| Middle | [1, 0, -1, 2, 12, -70, -350, 4095, 19110] | -| Central | [1, -1, 12, -350, 19110] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, 3, -19, 205, -3341, 76391, -2330343] | -| NegHalf | [1, 1, 3, 15, 117, 1365, 22863, 527499, 16137705] | -| Bin | [1, 1, -1, -2, 21, -154, 1260, -11080, 69405] | -| InvBin | [1, 1, 3, 16, 125, 1346, 19292, 358086, 8399757] | -| TransSqrs | [0, 1, 3, -1, 3, -16, 130, -1485, 22645] | -| TransNat0 | [0, 1, 1, -1, 3, -16, 130, -1485, 22645] | -| TransNat1 | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | - -NarayanaInv Diagonals as rows +Narayana:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 2, -3, 1], [0, -7, 12, -6, 1], [0, 39, -70, 40, -10, 1], [0, -321, 585, -350, 100, -15, 1], [0, 3681, -6741, 4095, -1225, 210, -21, 1], [0, -56197, 103068, -62916, 19110, -3430, 392, -28, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -3, 2, 0], [1, -6, 12, -7, 0], [1, -10, 40, -70, 39, 0], [1, -15, 100, -350, 585, -321, 0], [1, -21, 210, -1225, 4095, -6741, 3681, 0], [1, -28, 392, -3430, 19110, -62916, 103068, -56197, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 2, 1], [0, -7, -3], [0, 39, 12, 1], [0, -321, -70, -6], [0, 3681, 585, 40, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 2, -1, 0], [0, -7, 5, -1, 0], [0, 39, -31, 9, -1, 0], [0, -321, 264, -86, 14, -1, 0], [0, 3681, -3060, 1035, -190, 20, -1, 0], [0, -56197, 46871, -16045, 3065, -365, 27, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 2, 0], [0, -1, 5, -7, 0], [0, -1, 9, -31, 39, 0], [0, -1, 14, -86, 264, -321, 0], [0, -1, 20, -190, 1035, -3060, 3681, 0], [0, -1, 27, -365, 3065, -16045, 46871, -56197, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -2, 0, 0], [1, -5, 7, 0, 0], [1, -9, 31, -39, 0, 0], [1, -14, 86, -264, 321, 0, 0], [1, -20, 190, -1035, 3060, -3681, 0, 0], [1, -27, 365, -3065, 16045, -46871, 56197, 0, 0]] | + +Narayana:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -3, 13, -80, 686, -7987, 122571] | +| OddSum | [0, 1, -1, 3, -13, 80, -686, 7987, -122571] | +| AltSum | [1, -1, 2, -6, 26, -160, 1372, -15974, 245142] | +| AccSum | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | +| AccRevSum | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | +| AntiDiagSum | [1, 0, 1, -1, 3, -10, 52, -397, 4307] | + +Narayana:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 84, 10920, 8763300, 12544663950, 3947845887989640] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 12, 70, 585, 6741, 103068] | +| ColMiddle | [1, 0, -1, 2, 12, -70, -350, 4095, 19110] | +| ColECenter | [1, -1, 12, -350, 19110] | +| ColOCenter | [0, 2, -70, 4095] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -1, 3, -16, 130, -1485, 22645] | +| TransNat0 | [0, 1, 1, -1, 3, -16, 130, -1485, 22645] | +| TransNat1 | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | + +Narayana:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ NarayanaInv Diagonals as rows | DiagRow7 | [0, -56197, -2023092, -30346380, -278175150, -1835955990, -9546971148, -41370208308, -155138281155]| | DiagRow8 | [0, 1102571, 49615695, 909621075, 10005831825, 78045488235, 473475961959, 2367379809795, 10145913470550]| -NarayanaInv Diagonals as columns +Narayana:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ NarayanaInv Diagonals as columns | DiagCol7 | [1, -28, 672, -17640, 540540, -19575864, 833790672, -41370208308, 2367379809795] | | DiagCol8 | [1, -36, 1080, -34650, 1274130, -54532764, 2709819684, -155138281155, 10145913470550] | -NarayanaInv Polynomial values as rows +Narayana:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ NarayanaInv Polynomial values as rows | PolyRow7 | [0, 0, -938, -378, 756, 2380, 9450, 41202, 150472] | | PolyRow8 | [0, 0, 14310, 5802, -10068, -15660, 8850, 135870, 631512] | -NarayanaInv Polynomial values as columns +Narayana:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ NarayanaInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 210, 882, 3360, 12012, 41202, 135870] | | PolyCol8 | [1, 8, 56, 336, 1736, 8120, 35896, 150472, 631512] | -# NarayanaRev -True +# Narayana:Rev +[] -NarayanaRev Triangle view +Narayana:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ NarayanaRev Triangle view | Row7 | [1, 21, 105, 175, 105, 21, 1, 0] | | Row8 | [1, 28, 196, 490, 490, 196, 28, 1, 0] | -NarayanaRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 7, 13, 14, 14, 1, 11, 31, 41, 42, 42, 1, 16, 66, 116, 131, 132, 132, 1, 22, 127, 302, 407, 428, 429, 429, 1, 29, 225, 715, 1205, 1401, 1429, 1430, 1430] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 14, 14, 13, 7, 1, 42, 42, 41, 31, 11, 1, 132, 132, 131, 116, 66, 16, 1, 429, 429, 428, 407, 302, 127, 22, 1, 1430, 1430, 1429, 1401, 1205, 715, 225, 29, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 13, 14, 0, 1, 11, 31, 41, 42, 0, 1, 16, 66, 116, 131, 132, 0, 1, 22, 127, 302, 407, 428, 429, 0, 1, 29, 225, 715, 1205, 1401, 1429, 1430] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 6, 1, 1, 10, 6, 0, 1, 15, 20, 1] | - -NarayanaRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | -| EvenSum | [1, 1, 1, 2, 7, 22, 66, 212, 715] | -| OddSum | [0, 0, 1, 3, 7, 20, 66, 217, 715] | -| AltSum | [1, 1, 0, -1, 0, 2, 0, -5, 0] | -| AccSum | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | -| AccRevSum | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | -| RevAccSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | -| DiagSum | [1, 1, 1, 2, 4, 8, 17, 37] | - -NarayanaRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 6, 20, 150, 525, 980] | -| RowGcd | [1, 1, 1, 3, 6, 10, 5, 7, 14] | -| RowMax | [1, 1, 1, 3, 6, 20, 50, 175, 490] | -| Middle | [1, 1, 1, 3, 6, 20, 50, 175, 490] | -| Central | [1, 1, 6, 50, 490] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 22, 90, 394, 1806, 8558, 41586] | -| NegHalf | [1, -2, 2, 2, -10, 6, 42, -102, -82] | -| Bin | [1, 1, 3, 13, 65, 356, 2072, 12601, 79221] | -| InvBin | [1, -1, -1, 5, 9, -56, -120, 825, 1925] | -| TransSqrs | [0, 0, 1, 7, 39, 196, 930, 4257, 19019] | -| TransNat0 | [0, 0, 1, 5, 21, 84, 330, 1287, 5005] | -| TransNat1 | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | - -NarayanaRev Diagonals as rows +Narayana:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 3, 1, 0], [1, 6, 6, 1, 0], [1, 10, 20, 10, 1, 0], [1, 15, 50, 50, 15, 1, 0], [1, 21, 105, 175, 105, 21, 1, 0], [1, 28, 196, 490, 490, 196, 28, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 3, 1], [0, 1, 6, 6, 1], [0, 1, 10, 20, 10, 1], [0, 1, 15, 50, 50, 15, 1], [0, 1, 21, 105, 175, 105, 21, 1], [0, 1, 28, 196, 490, 490, 196, 28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 3, 0], [1, 6, 1], [1, 10, 6, 0], [1, 15, 20, 1], [1, 21, 50, 10, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 5, 5], [1, 7, 13, 14, 14], [1, 11, 31, 41, 42, 42], [1, 16, 66, 116, 131, 132, 132], [1, 22, 127, 302, 407, 428, 429, 429], [1, 29, 225, 715, 1205, 1401, 1429, 1430, 1430]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [5, 5, 4, 1], [14, 14, 13, 7, 1], [42, 42, 41, 31, 11, 1], [132, 132, 131, 116, 66, 16, 1], [429, 429, 428, 407, 302, 127, 22, 1], [1430, 1430, 1429, 1401, 1205, 715, 225, 29, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 4, 5], [0, 1, 7, 13, 14], [0, 1, 11, 31, 41, 42], [0, 1, 16, 66, 116, 131, 132], [0, 1, 22, 127, 302, 407, 428, 429], [0, 1, 29, 225, 715, 1205, 1401, 1429, 1430]] | + +Narayana:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 14, 42, 132, 429, 1430] | +| EvenSum | [1, 1, 1, 2, 7, 22, 66, 212, 715] | +| OddSum | [0, 0, 1, 3, 7, 20, 66, 217, 715] | +| AltSum | [1, 1, 0, -1, 0, 2, 0, -5, 0] | +| AccSum | [1, 2, 5, 15, 49, 168, 594, 2145, 7865] | +| AccRevSum | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | +| AntiDiagSum | [1, 1, 1, 2, 4, 8, 17, 37, 82] | + +Narayana:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 6, 20, 150, 525, 980] | +| RowGcd | [1, 1, 1, 3, 6, 10, 5, 7, 14] | +| RowMax | [1, 1, 1, 3, 6, 20, 50, 175, 490] | +| ColMiddle | [1, 1, 1, 3, 6, 20, 50, 175, 490] | +| ColECenter | [1, 1, 6, 50, 490] | +| ColOCenter | [1, 3, 20, 175] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 7, 39, 196, 930, 4257, 19019] | +| TransNat0 | [0, 0, 1, 5, 21, 84, 330, 1287, 5005] | +| TransNat1 | [1, 1, 3, 10, 35, 126, 462, 1716, 6435] | + +Narayana:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ NarayanaRev Diagonals as rows | DiagRow7 | [1, 28, 336, 2520, 13860, 60984, 226512, 736164, 2147145]| | DiagRow8 | [1, 36, 540, 4950, 32670, 169884, 736164, 2760615, 9202050]| -NarayanaRev Diagonals as columns +Narayana:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ NarayanaRev Diagonals as columns | DiagCol7 | [0, 1, 36, 540, 4950, 32670, 169884, 736164, 2760615] | | DiagCol8 | [0, 1, 45, 825, 9075, 70785, 429429, 2147145, 9202050] | -NarayanaRev Polynomial values as rows +Narayana:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ NarayanaRev Polynomial values as rows | PolyRow7 | [1, 429, 4279, 20071, 65445, 171481, 387739, 788019, 1476841] | | PolyRow8 | [1, 1430, 20793, 124996, 491825, 1500666, 3858505, 8766248, 18130401] | -NarayanaRev Polynomial values as columns +Narayana:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ NarayanaRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 71, 680, 6882, 72528, 788019, 8766248] | | PolyCol8 | [1, 1, 9, 89, 945, 10577, 123129, 1476841, 18130401] | -# NarayanaInvRev -True +# Narayana:Inv:Rev +[] -NarayanaInvRev Triangle view +Narayana:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ NarayanaInvRev Triangle view | Row7 | [1, -21, 210, -1225, 4095, -6741, 3681, 0] | | Row8 | [1, -28, 392, -3430, 19110, -62916, 103068, -56197, 0] | -NarayanaInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 12, -7, 0, 1, -10, 40, -70, 39, 0, 1, -15, 100, -350, 585, -321, 0, 1, -21, 210, -1225, 4095, -6741, 3681, 0, 1, -28, 392, -3430, 19110, -62916, 103068, -56197, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -7, 12, -6, 1, 0, 39, -70, 40, -10, 1, 0, -321, 585, -350, 100, -15, 1, 0, 3681, -6741, 4095, -1225, 210, -21, 1, 0, -56197, 103068, -62916, 19110, -3430, 392, -28, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -2, 0, 0, 1, -5, 7, 0, 0, 1, -9, 31, -39, 0, 0, 1, -14, 86, -264, 321, 0, 0, 1, -20, 190, -1035, 3060, -3681, 0, 0, 1, -27, 365, -3065, 16045, -46871, 56197, 0, 0] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, 0, 0, -2, 1, 0, 0, 7, -5, 1, 0, 0, -39, 31, -9, 1, 0, 0, 321, -264, 86, -14, 1, 0, 0, -3681, 3060, -1035, 190, -20, 1, 0, 0, 56197, -46871, 16045, -3065, 365, -27, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 2, -1, 0, 0, -7, 5, -1, 0, 0, 39, -31, 9, -1, 0, 0, -321, 264, -86, 14, -1, 0, 0, 3681, -3060, 1035, -190, 20, -1, 0, 0, -56197, 46871, -16045, 3065, -365, 27, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -3, 0, 1, -6, 2, 1, -10, 12, 0, 1, -15, 40, -7] | - -NarayanaInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 3, 13, 80, 686, 7987, 122571] | -| OddSum | [0, 0, -1, -3, -13, -80, -686, -7987, -122571] | -| AltSum | [1, 1, 2, 6, 26, 160, 1372, 15974, 245142] | -| AccSum | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | -| AccRevSum | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | -| RevAccSum | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | -| DiagSum | [1, 1, 1, 0, -2, -3, 3, 19] | - -NarayanaInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 84, 10920, 8763300, 12544663950, 3947845887989640] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 12, 70, 585, 6741, 103068] | -| Middle | [1, 1, -1, -3, 12, 40, -350, -1225, 19110] | -| Central | [1, -1, 12, -350, 19110] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, 0, 2, -10, 82, -938, 14310] | -| NegHalf | [1, -2, 6, -24, 126, -870, 7926, -94878, 1472442] | -| Bin | [1, 1, -1, -2, 21, -154, 1260, -11080, 69405] | -| InvBin | [1, -1, 3, -16, 125, -1346, 19292, -358086, 8399757] | -| TransSqrs | [0, 0, -1, 5, -21, 144, -1430, 19305, -339675] | -| TransNat0 | [0, 0, -1, 1, -3, 16, -130, 1485, -22645] | -| TransNat1 | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | - -NarayanaInvRev Diagonals as rows +Narayana:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -3, 2, 0], [1, -6, 12, -7, 0], [1, -10, 40, -70, 39, 0], [1, -15, 100, -350, 585, -321, 0], [1, -21, 210, -1225, 4095, -6741, 3681, 0], [1, -28, 392, -3430, 19110, -62916, 103068, -56197, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 2, -3, 1], [0, -7, 12, -6, 1], [0, 39, -70, 40, -10, 1], [0, -321, 585, -350, 100, -15, 1], [0, 3681, -6741, 4095, -1225, 210, -21, 1], [0, -56197, 103068, -62916, 19110, -3430, 392, -28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -3, 0], [1, -6, 2], [1, -10, 12, 0], [1, -15, 40, -7], [1, -21, 100, -70, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -2, 0, 0], [1, -5, 7, 0, 0], [1, -9, 31, -39, 0, 0], [1, -14, 86, -264, 321, 0, 0], [1, -20, 190, -1035, 3060, -3681, 0, 0], [1, -27, 365, -3065, 16045, -46871, 56197, 0, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [0, 0, -2, 1], [0, 0, 7, -5, 1], [0, 0, -39, 31, -9, 1], [0, 0, 321, -264, 86, -14, 1], [0, 0, -3681, 3060, -1035, 190, -20, 1], [0, 0, 56197, -46871, 16045, -3065, 365, -27, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 2, -1, 0], [0, -7, 5, -1, 0], [0, 39, -31, 9, -1, 0], [0, -321, 264, -86, 14, -1, 0], [0, 3681, -3060, 1035, -190, 20, -1, 0], [0, -56197, 46871, -16045, 3065, -365, 27, -1, 0]] | + +Narayana:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 3, 13, 80, 686, 7987, 122571] | +| OddSum | [0, 0, -1, -3, -13, -80, -686, -7987, -122571] | +| AltSum | [1, 1, 2, 6, 26, 160, 1372, 15974, 245142] | +| AccSum | [1, 2, 1, -1, 3, -16, 130, -1485, 22645] | +| AccRevSum | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | +| AntiDiagSum | [1, 1, 1, 0, -2, -3, 3, 19, 10] | + +Narayana:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 84, 10920, 8763300, 12544663950, 3947845887989640] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 12, 70, 585, 6741, 103068] | +| ColMiddle | [1, 1, -1, -3, 12, 40, -350, -1225, 19110] | +| ColECenter | [1, -1, 12, -350, 19110] | +| ColOCenter | [1, -3, 40, -1225] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, 5, -21, 144, -1430, 19305, -339675] | +| TransNat0 | [0, 0, -1, 1, -3, 16, -130, 1485, -22645] | +| TransNat1 | [1, 1, -1, 1, -3, 16, -130, 1485, -22645] | + +Narayana:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ NarayanaInvRev Diagonals as rows | DiagRow7 | [1, -28, 672, -17640, 540540, -19575864, 833790672, -41370208308, 2367379809795]| | DiagRow8 | [1, -36, 1080, -34650, 1274130, -54532764, 2709819684, -155138281155, 10145913470550]| -NarayanaInvRev Diagonals as columns +Narayana:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ NarayanaInvRev Diagonals as columns | DiagCol7 | [0, -56197, -2023092, -30346380, -278175150, -1835955990, -9546971148, -41370208308, -155138281155] | | DiagCol8 | [0, 1102571, 49615695, 909621075, 10005831825, 78045488235, 473475961959, 2367379809795, 10145913470550] | -NarayanaInvRev Polynomial values as rows +Narayana:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ NarayanaInvRev Polynomial values as rows | PolyRow7 | [1, 0, 76391, 1345834, 9147789, 38861396, 124372675, 329192046, 760222169] | | PolyRow8 | [1, 0, -2330343, -61596110, -558312303, -2965040964, -11388017975, -35167502418, -92820081375] | -NarayanaInvRev Polynomial values as columns +Narayana:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Nicomachus.md b/data/md/Nicomachus.md index 86d6e28..d6c234d 100644 --- a/data/md/Nicomachus.md +++ b/data/md/Nicomachus.md @@ -14,45 +14,41 @@ Nicomachus Triangle view | Row7 | [128, 192, 288, 432, 648, 972, 1458, 2187] | | Row8 | [256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561] | -Nicomachus Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561] | -| TriRev | [1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256] | -| TriAcc | [1, 2, 5, 4, 10, 19, 8, 20, 38, 65, 16, 40, 76, 130, 211, 32, 80, 152, 260, 422, 665, 64, 160, 304, 520, 844, 1330, 2059, 128, 320, 608, 1040, 1688, 2660, 4118, 6305, 256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171] | -| TriRevAcc | [1, 5, 2, 19, 10, 4, 65, 38, 20, 8, 211, 130, 76, 40, 16, 665, 422, 260, 152, 80, 32, 2059, 1330, 844, 520, 304, 160, 64, 6305, 4118, 2660, 1688, 1040, 608, 320, 128, 19171, 12610, 8236, 5320, 3376, 2080, 1216, 640, 256] | -| TriAccRev | [1, 3, 5, 9, 15, 19, 27, 45, 57, 65, 81, 135, 171, 195, 211, 243, 405, 513, 585, 633, 665, 729, 1215, 1539, 1755, 1899, 1995, 2059, 2187, 3645, 4617, 5265, 5697, 5985, 6177, 6305, 6561, 10935, 13851, 15795, 17091, 17955, 18531, 18915, 19171] | -| TriDiag | [1, 2, 4, 3, 8, 6, 16, 12, 9, 32, 24, 18, 64, 48, 36, 27, 128, 96, 72, 54] | +Nicomachus Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 3], [4, 6, 9], [8, 12, 18, 27], [16, 24, 36, 54, 81], [32, 48, 72, 108, 162, 243], [64, 96, 144, 216, 324, 486, 729], [128, 192, 288, 432, 648, 972, 1458, 2187], [256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561]] | +| RevTabl | [[1], [3, 2], [9, 6, 4], [27, 18, 12, 8], [81, 54, 36, 24, 16], [243, 162, 108, 72, 48, 32], [729, 486, 324, 216, 144, 96, 64], [2187, 1458, 972, 648, 432, 288, 192, 128], [6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256]] | +| AntiDiag | [[1], [2], [4, 3], [8, 6], [16, 12, 9], [32, 24, 18], [64, 48, 36, 27], [128, 96, 72, 54], [256, 192, 144, 108, 81]] | +| AccTabl | [[1], [2, 5], [4, 10, 19], [8, 20, 38, 65], [16, 40, 76, 130, 211], [32, 80, 152, 260, 422, 665], [64, 160, 304, 520, 844, 1330, 2059], [128, 320, 608, 1040, 1688, 2660, 4118, 6305], [256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171]] | +| RevAccTabl | [[1], [5, 2], [19, 10, 4], [65, 38, 20, 8], [211, 130, 76, 40, 16], [665, 422, 260, 152, 80, 32], [2059, 1330, 844, 520, 304, 160, 64], [6305, 4118, 2660, 1688, 1040, 608, 320, 128], [19171, 12610, 8236, 5320, 3376, 2080, 1216, 640, 256]] | +| AccRevTabl | [[1], [3, 5], [9, 15, 19], [27, 45, 57, 65], [81, 135, 171, 195, 211], [243, 405, 513, 585, 633, 665], [729, 1215, 1539, 1755, 1899, 1995, 2059], [2187, 3645, 4617, 5265, 5697, 5985, 6177, 6305], [6561, 10935, 13851, 15795, 17091, 17955, 18531, 18915, 19171]] | Nicomachus Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 5, 19, 65, 211, 665, 2059, 6305, 19171] | -| EvenSum | [1, 2, 13, 26, 133, 266, 1261, 2522, 11605] | -| OddSum | [0, 3, 6, 39, 78, 399, 798, 3783, 7566] | -| AltSum | [1, -1, 7, -13, 55, -133, 463, -1261, 4039] | -| AccSum | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | -| AccRevSum | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | -| RevAccSum | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | -| DiagSum | [1, 2, 7, 14, 37, 74, 175, 350] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 5, 19, 65, 211, 665, 2059, 6305, 19171] | +| EvenSum | [1, 2, 13, 26, 133, 266, 1261, 2522, 11605] | +| OddSum | [0, 3, 6, 39, 78, 399, 798, 3783, 7566] | +| AltSum | [1, -1, 7, -13, 55, -133, 463, -1261, 4039] | +| AccSum | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | +| AccRevSum | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | +| AntiDiagSum | [1, 2, 7, 14, 37, 74, 175, 350, 781] | Nicomachus Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| Middle | [1, 2, 6, 12, 36, 72, 216, 432, 1296] | -| Central | [1, 6, 36, 216, 1296] | -| LeftSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RightSide | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| PosHalf | [1, 7, 37, 175, 781, 3367, 14197, 58975, 242461] | -| NegHalf | [1, -1, 13, -25, 181, -481, 2653, -8425, 40261] | -| Bin | [1, 5, 25, 125, 625, 3125, 15625, 78125, 390625] | -| InvBin | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TransSqrs | [0, 3, 42, 327, 1950, 9975, 46194, 199551, 819006] | -| TransNat0 | [0, 3, 24, 129, 582, 2379, 9132, 33573, 119634] | -| TransNat1 | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| ColMiddle | [1, 2, 6, 12, 36, 72, 216, 432, 1296] | +| ColECenter | [1, 6, 36, 216, 1296] | +| ColOCenter | [2, 12, 72, 432] | +| ColLeft | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| ColRight | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| TransSqrs | [0, 3, 42, 327, 1950, 9975, 46194, 199551, 819006] | +| TransNat0 | [0, 3, 24, 129, 582, 2379, 9132, 33573, 119634] | +| TransNat1 | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | Nicomachus Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Nicomachus Polynomial values as columns | PolyCol7 | [1, 23, 487, 10235, 214951, 4514003, 94794127, 1990676795, 41804212951] | | PolyCol8 | [1, 26, 628, 15080, 361936, 8686496, 208475968, 5003423360, 120082160896] | -# NicomachusRev -True +# Nicomachus:Rev +[] -NicomachusRev Triangle view +Nicomachus:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ NicomachusRev Triangle view | Row7 | [2187, 1458, 972, 648, 432, 288, 192, 128] | | Row8 | [6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256] | -NicomachusRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256] | -| TriRev | [1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561] | -| TriAcc | [1, 3, 5, 9, 15, 19, 27, 45, 57, 65, 81, 135, 171, 195, 211, 243, 405, 513, 585, 633, 665, 729, 1215, 1539, 1755, 1899, 1995, 2059, 2187, 3645, 4617, 5265, 5697, 5985, 6177, 6305, 6561, 10935, 13851, 15795, 17091, 17955, 18531, 18915, 19171] | -| TriRevAcc | [1, 5, 3, 19, 15, 9, 65, 57, 45, 27, 211, 195, 171, 135, 81, 665, 633, 585, 513, 405, 243, 2059, 1995, 1899, 1755, 1539, 1215, 729, 6305, 6177, 5985, 5697, 5265, 4617, 3645, 2187, 19171, 18915, 18531, 17955, 17091, 15795, 13851, 10935, 6561] | -| TriAccRev | [1, 2, 5, 4, 10, 19, 8, 20, 38, 65, 16, 40, 76, 130, 211, 32, 80, 152, 260, 422, 665, 64, 160, 304, 520, 844, 1330, 2059, 128, 320, 608, 1040, 1688, 2660, 4118, 6305, 256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171] | -| TriDiag | [1, 3, 9, 2, 27, 6, 81, 18, 4, 243, 54, 12, 729, 162, 36, 8, 2187, 486, 108, 24] | - -NicomachusRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 5, 19, 65, 211, 665, 2059, 6305, 19171] | -| EvenSum | [1, 3, 13, 39, 133, 399, 1261, 3783, 11605] | -| OddSum | [0, 2, 6, 26, 78, 266, 798, 2522, 7566] | -| AltSum | [1, 1, 7, 13, 55, 133, 463, 1261, 4039] | -| AccSum | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | -| AccRevSum | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | -| RevAccSum | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | -| DiagSum | [1, 3, 11, 33, 103, 309, 935, 2805] | - -NicomachusRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| Middle | [1, 3, 6, 18, 36, 108, 216, 648, 1296] | -| Central | [1, 6, 36, 216, 1296] | -| LeftSide | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | -| RightSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| PosHalf | [1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296] | -| NegHalf | [1, -4, 28, -160, 976, -5824, 35008, -209920, 1259776] | -| Bin | [1, 5, 25, 125, 625, 3125, 15625, 78125, 390625] | -| InvBin | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| TransSqrs | [0, 2, 22, 138, 670, 2810, 10734, 38474, 131806] | -| TransNat0 | [0, 2, 14, 66, 262, 946, 3222, 10562, 33734] | -| TransNat1 | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | - -NicomachusRev Diagonals as rows +Nicomachus:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [3, 2], [9, 6, 4], [27, 18, 12, 8], [81, 54, 36, 24, 16], [243, 162, 108, 72, 48, 32], [729, 486, 324, 216, 144, 96, 64], [2187, 1458, 972, 648, 432, 288, 192, 128], [6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256]] | +| RevTabl | [[1], [2, 3], [4, 6, 9], [8, 12, 18, 27], [16, 24, 36, 54, 81], [32, 48, 72, 108, 162, 243], [64, 96, 144, 216, 324, 486, 729], [128, 192, 288, 432, 648, 972, 1458, 2187], [256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561]] | +| AntiDiag | [[1], [3], [9, 2], [27, 6], [81, 18, 4], [243, 54, 12], [729, 162, 36, 8], [2187, 486, 108, 24], [6561, 1458, 324, 72, 16]] | +| AccTabl | [[1], [3, 5], [9, 15, 19], [27, 45, 57, 65], [81, 135, 171, 195, 211], [243, 405, 513, 585, 633, 665], [729, 1215, 1539, 1755, 1899, 1995, 2059], [2187, 3645, 4617, 5265, 5697, 5985, 6177, 6305], [6561, 10935, 13851, 15795, 17091, 17955, 18531, 18915, 19171]] | +| RevAccTabl | [[1], [5, 3], [19, 15, 9], [65, 57, 45, 27], [211, 195, 171, 135, 81], [665, 633, 585, 513, 405, 243], [2059, 1995, 1899, 1755, 1539, 1215, 729], [6305, 6177, 5985, 5697, 5265, 4617, 3645, 2187], [19171, 18915, 18531, 17955, 17091, 15795, 13851, 10935, 6561]] | +| AccRevTabl | [[1], [2, 5], [4, 10, 19], [8, 20, 38, 65], [16, 40, 76, 130, 211], [32, 80, 152, 260, 422, 665], [64, 160, 304, 520, 844, 1330, 2059], [128, 320, 608, 1040, 1688, 2660, 4118, 6305], [256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171]] | + +Nicomachus:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 5, 19, 65, 211, 665, 2059, 6305, 19171] | +| EvenSum | [1, 3, 13, 39, 133, 399, 1261, 3783, 11605] | +| OddSum | [0, 2, 6, 26, 78, 266, 798, 2522, 7566] | +| AltSum | [1, 1, 7, 13, 55, 133, 463, 1261, 4039] | +| AccSum | [1, 8, 43, 194, 793, 3044, 11191, 39878, 138805] | +| AccRevSum | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | +| AntiDiagSum | [1, 3, 11, 33, 103, 309, 935, 2805, 8431] | + +Nicomachus:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| ColMiddle | [1, 3, 6, 18, 36, 108, 216, 648, 1296] | +| ColECenter | [1, 6, 36, 216, 1296] | +| ColOCenter | [3, 18, 108, 648] | +| ColLeft | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | +| ColRight | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| TransSqrs | [0, 2, 22, 138, 670, 2810, 10734, 38474, 131806] | +| TransNat0 | [0, 2, 14, 66, 262, 946, 3222, 10562, 33734] | +| TransNat1 | [1, 7, 33, 131, 473, 1611, 5281, 16867, 52905] | + +Nicomachus:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 2, 4, 8, 16, 32, 64, 128, 256]| @@ -175,7 +167,7 @@ NicomachusRev Diagonals as rows | DiagRow7 | [2187, 4374, 8748, 17496, 34992, 69984, 139968, 279936, 559872]| | DiagRow8 | [6561, 13122, 26244, 52488, 104976, 209952, 419904, 839808, 1679616]| -NicomachusRev Diagonals as columns +Nicomachus:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | @@ -188,7 +180,7 @@ NicomachusRev Diagonals as columns | DiagCol7 | [128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 839808] | | DiagCol8 | [256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616] | -NicomachusRev Polynomial values as rows +Nicomachus:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ NicomachusRev Polynomial values as rows | PolyRow7 | [2187, 6305, 58975, 557685, 3354131, 14284777, 47775015, 134162045, 330381595] | | PolyRow8 | [6561, 19171, 242461, 3352671, 26839609, 142854331, 573306741, 1878275191, 5286112081] | -NicomachusRev Polynomial values as columns +Nicomachus:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 3, 9, 27, 81, 243, 729, 2187, 6561] | diff --git a/data/md/One.md b/data/md/One.md index 7f40be3..b455d62 100644 --- a/data/md/One.md +++ b/data/md/One.md @@ -14,45 +14,41 @@ One Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -One Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriRev | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriAcc | [1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9] | -| TriRevAcc | [1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1] | -| TriAccRev | [1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9] | -| TriDiag | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | +One Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]] | +| RevTabl | [[1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1]] | +| AccTabl | [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 9]] | +| RevAccTabl | [[1], [2, 1], [3, 2, 1], [4, 3, 2, 1], [5, 4, 3, 2, 1], [6, 5, 4, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1], [9, 8, 7, 6, 5, 4, 3, 2, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 9]] | One Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | -| OddSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | -| AltSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -| AccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| AccRevSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| RevAccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| DiagSum | [1, 1, 2, 2, 3, 3, 4, 4] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | +| OddSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | +| AltSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | +| AccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | +| AccRevSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | +| AntiDiagSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | One Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Central | [1, 1, 1, 1, 1] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 7, 15, 31, 63, 127, 255, 511] | -| NegHalf | [1, -1, 3, -5, 11, -21, 43, -85, 171] | -| Bin | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| InvBin | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 1, 5, 14, 30, 55, 91, 140, 204] | -| TransNat0 | [0, 1, 3, 6, 10, 15, 21, 28, 36] | -| TransNat1 | [1, 3, 6, 10, 15, 21, 28, 36, 45] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColECenter | [1, 1, 1, 1, 1] | +| ColOCenter | [1, 1, 1, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 14, 30, 55, 91, 140, 204] | +| TransNat0 | [0, 1, 3, 6, 10, 15, 21, 28, 36] | +| TransNat1 | [1, 3, 6, 10, 15, 21, 28, 36, 45] | One Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ One Polynomial values as columns | PolyCol7 | [1, 8, 57, 400, 2801, 19608, 137257, 960800, 6725601] | | PolyCol8 | [1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961] | -# OneInv -True +# One:Inv +[] -OneInv Triangle view +One:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ OneInv Triangle view | Row7 | [0, 0, 0, 0, 0, 0, -1, 1] | | Row8 | [0, 0, 0, 0, 0, 0, 0, -1, 1] | -OneInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1] | -| TriRev | [1, 1, -1, 1, -1, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0] | -| TriAcc | [1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0] | -| TriAccRev | [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] | -| TriDiag | [1, -1, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1] | - -OneInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| OddSum | [0, 1, -1, 1, -1, 1, -1, 1, -1] | -| AltSum | [1, -2, 2, -2, 2, -2, 2, -2, 2] | -| AccSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | -| AccRevSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | -| RevAccSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| DiagSum | [1, -1, 1, -1, 1, -1, 1, -1] | - -OneInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, -1, -1, 0, 0, 0, 0, 0, 0] | -| Central | [1, -1, 0, 0, 0] | -| LeftSide | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -1, -1, -1, -1, -1, -1, -1] | -| NegHalf | [1, 3, 3, 3, 3, 3, 3, 3, 3] | -| Bin | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| InvBin | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| TransSqrs | [0, 1, 3, 5, 7, 9, 11, 13, 15] | -| TransNat0 | [0, 1, 1, 1, 1, 1, 1, 1, 1] | -| TransNat1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | - -OneInv Diagonals as rows +One:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -1, 1], [0, 0, -1, 1], [0, 0, 0, -1, 1], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 1]] | +| RevTabl | [[1], [1, -1], [1, -1, 0], [1, -1, 0, 0], [1, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [0, 1], [0, -1], [0, 0, 1], [0, 0, -1], [0, 0, 0, 1], [0, 0, 0, -1], [0, 0, 0, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -1, 0], [0, 0, -1, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 0], [0, -1, 0, 0], [0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0]] | + +One:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| OddSum | [0, 1, -1, 1, -1, 1, -1, 1, -1] | +| AltSum | [1, -2, 2, -2, 2, -2, 2, -2, 2] | +| AccSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | +| AccRevSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AntiDiagSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | + +One:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, -1, 0, 0, 0, 0, 0, 0] | +| ColECenter | [1, -1, 0, 0, 0] | +| ColOCenter | [-1, 0, 0, 0] | +| ColLeft | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 5, 7, 9, 11, 13, 15] | +| TransNat0 | [0, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransNat1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | + +One:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ OneInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| -OneInv Diagonals as columns +One:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ OneInv Diagonals as columns | DiagCol7 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -OneInv Polynomial values as rows +One:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ OneInv Polynomial values as rows | PolyRow7 | [0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008] | | PolyRow8 | [0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064] | -OneInv Polynomial values as columns +One:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ OneInv Polynomial values as columns | PolyCol7 | [1, 6, 42, 294, 2058, 14406, 100842, 705894, 4941258] | | PolyCol8 | [1, 7, 56, 448, 3584, 28672, 229376, 1835008, 14680064] | -# OneRev -True +# One:Rev +[] -OneRev Triangle view +One:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ OneRev Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -OneRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriRev | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriAcc | [1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9] | -| TriRevAcc | [1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1] | -| TriAccRev | [1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9] | -| TriDiag | [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] | - -OneRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | -| OddSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | -| AltSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | -| AccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| AccRevSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| RevAccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | -| DiagSum | [1, 1, 2, 2, 3, 3, 4, 4] | - -OneRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Central | [1, 1, 1, 1, 1] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 7, 15, 31, 63, 127, 255, 511] | -| NegHalf | [1, -1, 3, -5, 11, -21, 43, -85, 171] | -| Bin | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| InvBin | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 1, 5, 14, 30, 55, 91, 140, 204] | -| TransNat0 | [0, 1, 3, 6, 10, 15, 21, 28, 36] | -| TransNat1 | [1, 3, 6, 10, 15, 21, 28, 36, 45] | - -OneRev Diagonals as rows +One:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]] | +| RevTabl | [[1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1, 1]] | +| AccTabl | [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 9]] | +| RevAccTabl | [[1], [2, 1], [3, 2, 1], [4, 3, 2, 1], [5, 4, 3, 2, 1], [6, 5, 4, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1], [9, 8, 7, 6, 5, 4, 3, 2, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 9]] | + +One:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| EvenSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | +| OddSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | +| AltSum | [1, 0, 1, 0, 1, 0, 1, 0, 1] | +| AccSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | +| AccRevSum | [1, 3, 6, 10, 15, 21, 28, 36, 45] | +| AntiDiagSum | [1, 1, 2, 2, 3, 3, 4, 4, 5] | + +One:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColECenter | [1, 1, 1, 1, 1] | +| ColOCenter | [1, 1, 1, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 14, 30, 55, 91, 140, 204] | +| TransNat0 | [0, 1, 3, 6, 10, 15, 21, 28, 36] | +| TransNat1 | [1, 3, 6, 10, 15, 21, 28, 36, 45] | + +One:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ OneRev Diagonals as rows | DiagRow7 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| | DiagRow8 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| -OneRev Diagonals as columns +One:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ OneRev Diagonals as columns | DiagCol7 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | | DiagCol8 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -OneRev Polynomial values as rows +One:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ OneRev Polynomial values as rows | PolyRow7 | [1, 8, 255, 3280, 21845, 97656, 335923, 960800, 2396745] | | PolyRow8 | [1, 9, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961] | -OneRev Polynomial values as columns +One:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ OneRev Polynomial values as columns | PolyCol7 | [1, 8, 57, 400, 2801, 19608, 137257, 960800, 6725601] | | PolyCol8 | [1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961] | -# OneInvRev -True +# One:Inv:Rev +[] -OneInvRev Triangle view +One:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ OneInvRev Triangle view | Row7 | [1, -1, 0, 0, 0, 0, 0, 0] | | Row8 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -OneInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -1, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0] | -| TriRev | [1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1] | -| TriAcc | [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] | -| TriRevAcc | [1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriAccRev | [1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, 0, 0] | - -OneInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| OddSum | [0, -1, -1, -1, -1, -1, -1, -1, -1] | -| AltSum | [1, 2, 2, 2, 2, 2, 2, 2, 2] | -| AccSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| AccRevSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RevAccSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | -| DiagSum | [1, 1, 0, 0, 0, 0, 0, 0] | - -OneInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, -1, -1, 0, 0, 0, 0, 0] | -| Central | [1, -1, 0, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 1, 2, 4, 8, 16, 32, 64, 128] | -| NegHalf | [1, -3, 6, -12, 24, -48, 96, -192, 384] | -| Bin | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| InvBin | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| TransSqrs | [0, -1, -1, -1, -1, -1, -1, -1, -1] | -| TransNat0 | [0, -1, -1, -1, -1, -1, -1, -1, -1] | -| TransNat1 | [1, -1, -1, -1, -1, -1, -1, -1, -1] | - -OneInvRev Diagonals as rows +One:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -1, 0], [1, -1, 0, 0], [1, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0]] | +| RevTabl | [[1], [-1, 1], [0, -1, 1], [0, 0, -1, 1], [0, 0, 0, -1, 1], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -1], [1, -1, 0], [1, -1, 0], [1, -1, 0, 0], [1, -1, 0, 0], [1, -1, 0, 0, 0]] | +| AccTabl | [[1], [1, 0], [1, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0]] | +| RevAccTabl | [[1], [0, 1], [0, 0, 1], [0, 0, 0, 1], [0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1]] | +| AccRevTabl | [[1], [-1, 0], [0, -1, 0], [0, 0, -1, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0]] | + +One:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| OddSum | [0, -1, -1, -1, -1, -1, -1, -1, -1] | +| AltSum | [1, 2, 2, 2, 2, 2, 2, 2, 2] | +| AccSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AccRevSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | +| AntiDiagSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | + +One:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, -1, -1, 0, 0, 0, 0, 0] | +| ColECenter | [1, -1, 0, 0, 0] | +| ColOCenter | [1, -1, 0, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, -1, -1, -1, -1, -1, -1, -1, -1] | +| TransNat0 | [0, -1, -1, -1, -1, -1, -1, -1, -1] | +| TransNat1 | [1, -1, -1, -1, -1, -1, -1, -1, -1] | + +One:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ OneInvRev Diagonals as rows | DiagRow7 | [1, -1, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, -1, 0, 0, 0, 0, 0, 0, 0]| -OneInvRev Diagonals as columns +One:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ OneInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -OneInvRev Polynomial values as rows +One:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ OneInvRev Polynomial values as rows | PolyRow7 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | | PolyRow8 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -OneInvRev Polynomial values as columns +One:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ OneInvRev Polynomial values as columns | PolyCol7 | [1, -6, -6, -6, -6, -6, -6, -6, -6] | | PolyCol8 | [1, -7, -7, -7, -7, -7, -7, -7, -7] | +# One:Rev:Inv +[] + +One:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [0, -1, 1] | +| Row3 | [0, 0, -1, 1] | +| Row4 | [0, 0, 0, -1, 1] | +| Row5 | [0, 0, 0, 0, -1, 1] | +| Row6 | [0, 0, 0, 0, 0, -1, 1] | +| Row7 | [0, 0, 0, 0, 0, 0, -1, 1] | +| Row8 | [0, 0, 0, 0, 0, 0, 0, -1, 1] | + +One:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -1, 1], [0, 0, -1, 1], [0, 0, 0, -1, 1], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 1]] | +| RevTabl | [[1], [1, -1], [1, -1, 0], [1, -1, 0, 0], [1, -1, 0, 0, 0], [1, -1, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [0, 1], [0, -1], [0, 0, 1], [0, 0, -1], [0, 0, 0, 1], [0, 0, 0, -1], [0, 0, 0, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -1, 0], [0, 0, -1, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, 0], [0, -1, 0, 0], [0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0]] | + +One:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| OddSum | [0, 1, -1, 1, -1, 1, -1, 1, -1] | +| AltSum | [1, -2, 2, -2, 2, -2, 2, -2, 2] | +| AccSum | [1, -1, -1, -1, -1, -1, -1, -1, -1] | +| AccRevSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AntiDiagSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | + +One:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, -1, 0, 0, 0, 0, 0, 0] | +| ColECenter | [1, -1, 0, 0, 0] | +| ColOCenter | [-1, 0, 0, 0] | +| ColLeft | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 5, 7, 9, 11, 13, 15] | +| TransNat0 | [0, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransNat1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | + +One:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -1, -1, -1, -1, -1, -1, -1, -1]| +| DiagRow2 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow3 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow4 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow5 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow6 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| + +One:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol2 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol3 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol4 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol5 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol6 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol7 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol8 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | + +One:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [0, 0, 2, 6, 12, 20, 30, 42, 56] | +| PolyRow3 | [0, 0, 4, 18, 48, 100, 180, 294, 448] | +| PolyRow4 | [0, 0, 8, 54, 192, 500, 1080, 2058, 3584] | +| PolyRow5 | [0, 0, 16, 162, 768, 2500, 6480, 14406, 28672] | +| PolyRow6 | [0, 0, 32, 486, 3072, 12500, 38880, 100842, 229376] | +| PolyRow7 | [0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008] | +| PolyRow8 | [0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064] | + +One:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 2, 4, 8, 16, 32, 64, 128] | +| PolyCol3 | [1, 2, 6, 18, 54, 162, 486, 1458, 4374] | +| PolyCol4 | [1, 3, 12, 48, 192, 768, 3072, 12288, 49152] | +| PolyCol5 | [1, 4, 20, 100, 500, 2500, 12500, 62500, 312500] | +| PolyCol6 | [1, 5, 30, 180, 1080, 6480, 38880, 233280, 1399680] | +| PolyCol7 | [1, 6, 42, 294, 2058, 14406, 100842, 705894, 4941258] | +| PolyCol8 | [1, 7, 56, 448, 3584, 28672, 229376, 1835008, 14680064] | + diff --git a/data/md/OrderedCycle.md b/data/md/OrderedCycle.md new file mode 100644 index 0000000..52c0744 --- /dev/null +++ b/data/md/OrderedCycle.md @@ -0,0 +1,208 @@ +# OrderedCycle +['A225479', 'A048594', 'A075181'] + +OrderedCycle Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [0, 1] | +| Row2 | [0, 1, 2] | +| Row3 | [0, 2, 6, 6] | +| Row4 | [0, 6, 22, 36, 24] | +| Row5 | [0, 24, 100, 210, 240, 120] | +| Row6 | [0, 120, 548, 1350, 2040, 1800, 720] | +| Row7 | [0, 720, 3528, 9744, 17640, 21000, 15120, 5040] | +| Row8 | [0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320] | + +OrderedCycle Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 2, 6, 6], [0, 6, 22, 36, 24], [0, 24, 100, 210, 240, 120], [0, 120, 548, 1350, 2040, 1800, 720], [0, 720, 3528, 9744, 17640, 21000, 15120, 5040], [0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [6, 6, 2, 0], [24, 36, 22, 6, 0], [120, 240, 210, 100, 24, 0], [720, 1800, 2040, 1350, 548, 120, 0], [5040, 15120, 21000, 17640, 9744, 3528, 720, 0], [40320, 141120, 231840, 235200, 162456, 78792, 26136, 5040, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 2, 2], [0, 6, 6], [0, 24, 22, 6], [0, 120, 100, 36], [0, 720, 548, 210, 24]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 2, 8, 14], [0, 6, 28, 64, 88], [0, 24, 124, 334, 574, 694], [0, 120, 668, 2018, 4058, 5858, 6578], [0, 720, 4248, 13992, 31632, 52632, 67752, 72792], [0, 5040, 31176, 109968, 272424, 507624, 739464, 880584, 920904]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [14, 8, 2, 0], [88, 64, 28, 6, 0], [694, 574, 334, 124, 24, 0], [6578, 5858, 4058, 2018, 668, 120, 0], [72792, 67752, 52632, 31632, 13992, 4248, 720, 0], [920904, 880584, 739464, 507624, 272424, 109968, 31176, 5040, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [6, 12, 14, 14], [24, 60, 82, 88, 88], [120, 360, 570, 670, 694, 694], [720, 2520, 4560, 5910, 6458, 6578, 6578], [5040, 20160, 41160, 58800, 68544, 72072, 72792, 72792], [40320, 181440, 413280, 648480, 810936, 889728, 915864, 920904, 920904]] | + +OrderedCycle Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 14, 88, 694, 6578, 72792, 920904] | +| EvenSum | [1, 0, 2, 6, 46, 340, 3308, 36288, 460752] | +| OddSum | [0, 1, 1, 8, 42, 354, 3270, 36504, 460152] | +| AltSum | [1, -1, 1, -2, 4, -14, 38, -216, 600] | +| AccSum | [1, 1, 4, 24, 186, 1750, 19300, 243768, 3467184] | +| AccRevSum | [1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856] | +| AntiDiagSum | [1, 0, 1, 1, 4, 12, 52, 256, 1502] | + +OrderedCycle Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 792, 8400, 25153200, 76734000, 763351047043200] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 24, 24] | +| RowMax | [1, 1, 2, 6, 36, 240, 2040, 21000, 235200] | +| ColMiddle | [1, 0, 1, 2, 22, 100, 1350, 9744, 162456] | +| ColECenter | [1, 1, 22, 1350, 162456] | +| ColOCenter | [0, 2, 100, 9744] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 9, 80, 802, 9154, 118022, 1701048, 27139608] | +| TransNat0 | [0, 1, 5, 32, 254, 2414, 26746, 338568, 4820952] | +| TransNat1 | [1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856] | + +OrderedCycle Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| +| DiagRow1 | [0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520]| +| DiagRow2 | [0, 2, 22, 210, 2040, 21000, 231840, 2751840, 35078400]| +| DiagRow3 | [0, 6, 100, 1350, 17640, 235200, 3265920, 47628000, 731808000]| +| DiagRow4 | [0, 24, 548, 9744, 162456, 2693880, 45556560, 795175920, 14411295360]| +| DiagRow5 | [0, 120, 3528, 78792, 1614816, 32319000, 649479600, 13293292320, 279281882880]| +| DiagRow6 | [0, 720, 26136, 708744, 17368320, 410031600, 9604465200, 226750764240, 5444670591360]| +| DiagRow7 | [0, 5040, 219168, 7036200, 201828000, 5519487600, 148370508000, 3986353491120, 108116216208000]| +| DiagRow8 | [0, 40320, 2053152, 76521456, 2526193824, 78864820320, 2402005525920, 72622987557120, 2202727143576960]| + +OrderedCycle Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| DiagCol1 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| DiagCol2 | [2, 6, 22, 100, 548, 3528, 26136, 219168, 2053152] | +| DiagCol3 | [6, 36, 210, 1350, 9744, 78792, 708744, 7036200, 76521456] | +| DiagCol4 | [24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824] | +| DiagCol5 | [120, 1800, 21000, 235200, 2693880, 32319000, 410031600, 5519487600, 78864820320] | +| DiagCol6 | [720, 15120, 231840, 3265920, 45556560, 649479600, 9604465200, 148370508000, 2402005525920] | +| DiagCol7 | [5040, 141120, 2751840, 47628000, 795175920, 13293292320, 226750764240, 3986353491120, 72622987557120] | +| DiagCol8 | [40320, 1451520, 35078400, 731808000, 14411295360, 279281882880, 5444670591360, 108116216208000, 2202727143576960] | + +OrderedCycle Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| PolyRow2 | [0, 3, 10, 21, 36, 55, 78, 105, 136] | +| PolyRow3 | [0, 14, 76, 222, 488, 910, 1524, 2366, 3472] | +| PolyRow4 | [0, 88, 772, 3132, 8824, 20080, 39708, 71092, 118192] | +| PolyRow5 | [0, 694, 9808, 55242, 199456, 553870, 1293264, 2670178, 5029312] | +| PolyRow6 | [0, 6578, 149552, 1169262, 5410208, 18333050, 50545008, 120348662, 256809152] | +| PolyRow7 | [0, 72792, 2660544, 28873800, 171209664, 707959800, 2304709632, 6328330344, 15298865280] | +| PolyRow8 | [0, 920904, 54093696, 814870584, 6192052800, 31244562600, 120100860864, 380302313496, 1041597412224] | + +OrderedCycle Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol1 | [1, 1, 3, 14, 88, 694, 6578, 72792, 920904] | +| PolyCol2 | [1, 2, 10, 76, 772, 9808, 149552, 2660544, 54093696] | +| PolyCol3 | [1, 3, 21, 222, 3132, 55242, 1169262, 28873800, 814870584] | +| PolyCol4 | [1, 4, 36, 488, 8824, 199456, 5410208, 171209664, 6192052800] | +| PolyCol5 | [1, 5, 55, 910, 20080, 553870, 18333050, 707959800, 31244562600] | +| PolyCol6 | [1, 6, 78, 1524, 39708, 1293264, 50545008, 2304709632, 120100860864] | +| PolyCol7 | [1, 7, 105, 2366, 71092, 2670178, 120348662, 6328330344, 380302313496] | +| PolyCol8 | [1, 8, 136, 3472, 118192, 5029312, 256809152, 15298865280, 1041597412224] | + +# OrderedCycle:Rev +[] + +OrderedCycle:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 0] | +| Row2 | [2, 1, 0] | +| Row3 | [6, 6, 2, 0] | +| Row4 | [24, 36, 22, 6, 0] | +| Row5 | [120, 240, 210, 100, 24, 0] | +| Row6 | [720, 1800, 2040, 1350, 548, 120, 0] | +| Row7 | [5040, 15120, 21000, 17640, 9744, 3528, 720, 0] | +| Row8 | [40320, 141120, 231840, 235200, 162456, 78792, 26136, 5040, 0] | + +OrderedCycle:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [6, 6, 2, 0], [24, 36, 22, 6, 0], [120, 240, 210, 100, 24, 0], [720, 1800, 2040, 1350, 548, 120, 0], [5040, 15120, 21000, 17640, 9744, 3528, 720, 0], [40320, 141120, 231840, 235200, 162456, 78792, 26136, 5040, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 2, 6, 6], [0, 6, 22, 36, 24], [0, 24, 100, 210, 240, 120], [0, 120, 548, 1350, 2040, 1800, 720], [0, 720, 3528, 9744, 17640, 21000, 15120, 5040], [0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320]] | +| AntiDiag | [[1], [1], [2, 0], [6, 1], [24, 6, 0], [120, 36, 2], [720, 240, 22, 0], [5040, 1800, 210, 6], [40320, 15120, 2040, 100, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [6, 12, 14, 14], [24, 60, 82, 88, 88], [120, 360, 570, 670, 694, 694], [720, 2520, 4560, 5910, 6458, 6578, 6578], [5040, 20160, 41160, 58800, 68544, 72072, 72792, 72792], [40320, 181440, 413280, 648480, 810936, 889728, 915864, 920904, 920904]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [14, 14, 12, 6], [88, 88, 82, 60, 24], [694, 694, 670, 570, 360, 120], [6578, 6578, 6458, 5910, 4560, 2520, 720], [72792, 72792, 72072, 68544, 58800, 41160, 20160, 5040], [920904, 920904, 915864, 889728, 810936, 648480, 413280, 181440, 40320]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 2, 8, 14], [0, 6, 28, 64, 88], [0, 24, 124, 334, 574, 694], [0, 120, 668, 2018, 4058, 5858, 6578], [0, 720, 4248, 13992, 31632, 52632, 67752, 72792], [0, 5040, 31176, 109968, 272424, 507624, 739464, 880584, 920904]] | + +OrderedCycle:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 14, 88, 694, 6578, 72792, 920904] | +| EvenSum | [1, 1, 2, 8, 46, 354, 3308, 36504, 460752] | +| OddSum | [0, 0, 1, 6, 42, 340, 3270, 36288, 460152] | +| AltSum | [1, 1, 1, 2, 4, 14, 38, 216, 600] | +| AccSum | [1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856] | +| AccRevSum | [1, 1, 4, 24, 186, 1750, 19300, 243768, 3467184] | +| AntiDiagSum | [1, 1, 2, 7, 30, 158, 982, 7056, 57580] | + +OrderedCycle:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 792, 8400, 25153200, 76734000, 763351047043200] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 24, 24] | +| RowMax | [1, 1, 2, 6, 36, 240, 2040, 21000, 235200] | +| ColMiddle | [1, 1, 1, 6, 22, 210, 1350, 17640, 162456] | +| ColECenter | [1, 1, 22, 1350, 162456] | +| ColOCenter | [1, 6, 210, 17640] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 14, 178, 2364, 33878, 527904, 8942232] | +| TransNat0 | [0, 0, 1, 10, 98, 1056, 12722, 170976, 2546280] | +| TransNat1 | [1, 1, 4, 24, 186, 1750, 19300, 243768, 3467184] | + +OrderedCycle:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| +| DiagRow1 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| +| DiagRow2 | [2, 6, 22, 100, 548, 3528, 26136, 219168, 2053152]| +| DiagRow3 | [6, 36, 210, 1350, 9744, 78792, 708744, 7036200, 76521456]| +| DiagRow4 | [24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824]| +| DiagRow5 | [120, 1800, 21000, 235200, 2693880, 32319000, 410031600, 5519487600, 78864820320]| +| DiagRow6 | [720, 15120, 231840, 3265920, 45556560, 649479600, 9604465200, 148370508000, 2402005525920]| +| DiagRow7 | [5040, 141120, 2751840, 47628000, 795175920, 13293292320, 226750764240, 3986353491120, 72622987557120]| +| DiagRow8 | [40320, 1451520, 35078400, 731808000, 14411295360, 279281882880, 5444670591360, 108116216208000, 2202727143576960]| + +OrderedCycle:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| DiagCol1 | [0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520] | +| DiagCol2 | [0, 2, 22, 210, 2040, 21000, 231840, 2751840, 35078400] | +| DiagCol3 | [0, 6, 100, 1350, 17640, 235200, 3265920, 47628000, 731808000] | +| DiagCol4 | [0, 24, 548, 9744, 162456, 2693880, 45556560, 795175920, 14411295360] | +| DiagCol5 | [0, 120, 3528, 78792, 1614816, 32319000, 649479600, 13293292320, 279281882880] | +| DiagCol6 | [0, 720, 26136, 708744, 17368320, 410031600, 9604465200, 226750764240, 5444670591360] | +| DiagCol7 | [0, 5040, 219168, 7036200, 201828000, 5519487600, 148370508000, 3986353491120, 108116216208000] | +| DiagCol8 | [0, 40320, 2053152, 76521456, 2526193824, 78864820320, 2402005525920, 72622987557120, 2202727143576960] | + +OrderedCycle:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow2 | [2, 3, 4, 5, 6, 7, 8, 9, 10] | +| PolyRow3 | [6, 14, 26, 42, 62, 86, 114, 146, 182] | +| PolyRow4 | [24, 88, 232, 492, 904, 1504, 2328, 3412, 4792] | +| PolyRow5 | [120, 694, 2624, 7374, 16984, 34070, 61824, 104014, 164984] | +| PolyRow6 | [720, 6578, 35888, 134478, 390128, 946970, 2019888, 3908918, 7013648] | +| PolyRow7 | [5040, 72792, 575280, 2887128, 10586736, 31175640, 78316272, 174588120, 354762288] | +| PolyRow8 | [40320, 920904, 10569984, 71281656, 331267200, 1185826920, 3513544704, 9032906904, 20804747136] | + +OrderedCycle:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| PolyCol1 | [1, 1, 3, 14, 88, 694, 6578, 72792, 920904] | +| PolyCol2 | [1, 1, 4, 26, 232, 2624, 35888, 575280, 10569984] | +| PolyCol3 | [1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656] | +| PolyCol4 | [1, 1, 6, 62, 904, 16984, 390128, 10586736, 331267200] | +| PolyCol5 | [1, 1, 7, 86, 1504, 34070, 946970, 31175640, 1185826920] | +| PolyCol6 | [1, 1, 8, 114, 2328, 61824, 2019888, 78316272, 3513544704] | +| PolyCol7 | [1, 1, 9, 146, 3412, 104014, 3908918, 174588120, 9032906904] | +| PolyCol8 | [1, 1, 10, 182, 4792, 164984, 7013648, 354762288, 20804747136] | + diff --git a/data/md/Ordinals.md b/data/md/Ordinals.md index a08eebc..2dff62a 100644 --- a/data/md/Ordinals.md +++ b/data/md/Ordinals.md @@ -1,5 +1,5 @@ # Ordinals -['A002260', 'A002262', 'A004736', 'A025581'] +['A002262', 'A002260', 'A004736', 'A025581'] Ordinals Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Ordinals Triangle view | Row7 | [0, 1, 2, 3, 4, 5, 6, 7] | | Row8 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -Ordinals Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8] | -| TriRev | [0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0] | -| TriAcc | [0, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 3, 6, 10, 0, 1, 3, 6, 10, 15, 0, 1, 3, 6, 10, 15, 21, 0, 1, 3, 6, 10, 15, 21, 28, 0, 1, 3, 6, 10, 15, 21, 28, 36] | -| TriRevAcc | [0, 1, 0, 3, 1, 0, 6, 3, 1, 0, 10, 6, 3, 1, 0, 15, 10, 6, 3, 1, 0, 21, 15, 10, 6, 3, 1, 0, 28, 21, 15, 10, 6, 3, 1, 0, 36, 28, 21, 15, 10, 6, 3, 1, 0] | -| TriAccRev | [0, 1, 1, 2, 3, 3, 3, 5, 6, 6, 4, 7, 9, 10, 10, 5, 9, 12, 14, 15, 15, 6, 11, 15, 18, 20, 21, 21, 7, 13, 18, 22, 25, 27, 28, 28, 8, 15, 21, 26, 30, 33, 35, 36, 36] | -| TriDiag | [0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3] | +Ordinals Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7, 8]] | +| RevTabl | [[0], [1, 0], [2, 1, 0], [3, 2, 1, 0], [4, 3, 2, 1, 0], [5, 4, 3, 2, 1, 0], [6, 5, 4, 3, 2, 1, 0], [7, 6, 5, 4, 3, 2, 1, 0], [8, 7, 6, 5, 4, 3, 2, 1, 0]] | +| AntiDiag | [[0], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] | +| AccTabl | [[0], [0, 1], [0, 1, 3], [0, 1, 3, 6], [0, 1, 3, 6, 10], [0, 1, 3, 6, 10, 15], [0, 1, 3, 6, 10, 15, 21], [0, 1, 3, 6, 10, 15, 21, 28], [0, 1, 3, 6, 10, 15, 21, 28, 36]] | +| RevAccTabl | [[0], [1, 0], [3, 1, 0], [6, 3, 1, 0], [10, 6, 3, 1, 0], [15, 10, 6, 3, 1, 0], [21, 15, 10, 6, 3, 1, 0], [28, 21, 15, 10, 6, 3, 1, 0], [36, 28, 21, 15, 10, 6, 3, 1, 0]] | +| AccRevTabl | [[0], [1, 1], [2, 3, 3], [3, 5, 6, 6], [4, 7, 9, 10, 10], [5, 9, 12, 14, 15, 15], [6, 11, 15, 18, 20, 21, 21], [7, 13, 18, 22, 25, 27, 28, 28], [8, 15, 21, 26, 30, 33, 35, 36, 36]] | Ordinals Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 3, 6, 10, 15, 21, 28, 36] | -| EvenSum | [0, 0, 2, 2, 6, 6, 12, 12, 20] | -| OddSum | [0, 1, 1, 4, 4, 9, 9, 16, 16] | -| AltSum | [0, -1, 1, -2, 2, -3, 3, -4, 4] | -| AccSum | [0, 1, 4, 10, 20, 35, 56, 84, 120] | -| AccRevSum | [0, 1, 4, 10, 20, 35, 56, 84, 120] | -| RevAccSum | [0, 2, 8, 20, 40, 70, 112, 168, 240] | -| DiagSum | [0, 0, 1, 1, 3, 3, 6, 6] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 3, 6, 10, 15, 21, 28, 36] | +| EvenSum | [0, 0, 2, 2, 6, 6, 12, 12, 20] | +| OddSum | [0, 1, 1, 4, 4, 9, 9, 16, 16] | +| AltSum | [0, -1, 1, -2, 2, -3, 3, -4, 4] | +| AccSum | [0, 1, 4, 10, 20, 35, 56, 84, 120] | +| AccRevSum | [0, 2, 8, 20, 40, 70, 112, 168, 240] | +| AntiDiagSum | [0, 0, 1, 1, 3, 3, 6, 6, 10] | Ordinals Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 12, 60, 60, 420, 840] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| Middle | [0, 0, 1, 1, 2, 2, 3, 3, 4] | -| Central | [0, 1, 2, 3, 4] | -| LeftSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| PosHalf | [0, 1, 4, 11, 26, 57, 120, 247, 502] | -| NegHalf | [0, 1, 0, 3, -2, 9, -12, 31, -54] | -| Bin | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | -| InvBin | [0, 1, 0, 0, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 1, 9, 36, 100, 225, 441, 784, 1296] | -| TransNat0 | [0, 1, 5, 14, 30, 55, 91, 140, 204] | -| TransNat1 | [0, 2, 8, 20, 40, 70, 112, 168, 240] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 12, 60, 60, 420, 840] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColMiddle | [0, 0, 1, 1, 2, 2, 3, 3, 4] | +| ColECenter | [0, 1, 2, 3, 4] | +| ColOCenter | [0, 1, 2, 3] | +| ColLeft | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| TransSqrs | [0, 1, 9, 36, 100, 225, 441, 784, 1296] | +| TransNat0 | [0, 1, 5, 14, 30, 55, 91, 140, 204] | +| TransNat1 | [0, 2, 8, 20, 40, 70, 112, 168, 240] | Ordinals Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Ordinals Polynomial values as columns | PolyCol7 | [0, 7, 105, 1134, 10738, 94773, 800667, 6565468, 52683876] | | PolyCol8 | [0, 8, 136, 1672, 18056, 181896, 1754760, 16434824, 150652552] | -# OrdinalsRev -True +# Ordinals:Rev +[] -OrdinalsRev Triangle view +Ordinals:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [0] | @@ -122,47 +118,43 @@ OrdinalsRev Triangle view | Row7 | [7, 6, 5, 4, 3, 2, 1, 0] | | Row8 | [8, 7, 6, 5, 4, 3, 2, 1, 0] | -OrdinalsRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0] | -| TriRev | [0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8] | -| TriAcc | [0, 1, 1, 2, 3, 3, 3, 5, 6, 6, 4, 7, 9, 10, 10, 5, 9, 12, 14, 15, 15, 6, 11, 15, 18, 20, 21, 21, 7, 13, 18, 22, 25, 27, 28, 28, 8, 15, 21, 26, 30, 33, 35, 36, 36] | -| TriRevAcc | [0, 1, 1, 3, 3, 2, 6, 6, 5, 3, 10, 10, 9, 7, 4, 15, 15, 14, 12, 9, 5, 21, 21, 20, 18, 15, 11, 6, 28, 28, 27, 25, 22, 18, 13, 7, 36, 36, 35, 33, 30, 26, 21, 15, 8] | -| TriAccRev | [0, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 3, 6, 10, 0, 1, 3, 6, 10, 15, 0, 1, 3, 6, 10, 15, 21, 0, 1, 3, 6, 10, 15, 21, 28, 0, 1, 3, 6, 10, 15, 21, 28, 36] | -| TriDiag | [0, 1, 2, 0, 3, 1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1] | - -OrdinalsRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 3, 6, 10, 15, 21, 28, 36] | -| EvenSum | [0, 1, 2, 4, 6, 9, 12, 16, 20] | -| OddSum | [0, 0, 1, 2, 4, 6, 9, 12, 16] | -| AltSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | -| AccSum | [0, 2, 8, 20, 40, 70, 112, 168, 240] | -| AccRevSum | [0, 2, 8, 20, 40, 70, 112, 168, 240] | -| RevAccSum | [0, 1, 4, 10, 20, 35, 56, 84, 120] | -| DiagSum | [0, 1, 2, 4, 6, 9, 12, 16] | - -OrdinalsRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 12, 60, 60, 420, 840] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| Middle | [0, 1, 1, 2, 2, 3, 3, 4, 4] | -| Central | [0, 1, 2, 3, 4] | -| LeftSide | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| RightSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [0, 2, 10, 34, 98, 258, 642, 1538, 3586] | -| NegHalf | [0, -2, 6, -18, 46, -114, 270, -626, 1422] | -| Bin | [0, 1, 4, 12, 32, 80, 192, 448, 1024] | -| InvBin | [0, -1, 0, 0, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 0, 1, 6, 20, 50, 105, 196, 336] | -| TransNat0 | [0, 0, 1, 4, 10, 20, 35, 56, 84] | -| TransNat1 | [0, 1, 4, 10, 20, 35, 56, 84, 120] | - -OrdinalsRev Diagonals as rows +Ordinals:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [1, 0], [2, 1, 0], [3, 2, 1, 0], [4, 3, 2, 1, 0], [5, 4, 3, 2, 1, 0], [6, 5, 4, 3, 2, 1, 0], [7, 6, 5, 4, 3, 2, 1, 0], [8, 7, 6, 5, 4, 3, 2, 1, 0]] | +| RevTabl | [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7, 8]] | +| AntiDiag | [[0], [1], [2, 0], [3, 1], [4, 2, 0], [5, 3, 1], [6, 4, 2, 0], [7, 5, 3, 1], [8, 6, 4, 2, 0]] | +| AccTabl | [[0], [1, 1], [2, 3, 3], [3, 5, 6, 6], [4, 7, 9, 10, 10], [5, 9, 12, 14, 15, 15], [6, 11, 15, 18, 20, 21, 21], [7, 13, 18, 22, 25, 27, 28, 28], [8, 15, 21, 26, 30, 33, 35, 36, 36]] | +| RevAccTabl | [[0], [1, 1], [3, 3, 2], [6, 6, 5, 3], [10, 10, 9, 7, 4], [15, 15, 14, 12, 9, 5], [21, 21, 20, 18, 15, 11, 6], [28, 28, 27, 25, 22, 18, 13, 7], [36, 36, 35, 33, 30, 26, 21, 15, 8]] | +| AccRevTabl | [[0], [0, 1], [0, 1, 3], [0, 1, 3, 6], [0, 1, 3, 6, 10], [0, 1, 3, 6, 10, 15], [0, 1, 3, 6, 10, 15, 21], [0, 1, 3, 6, 10, 15, 21, 28], [0, 1, 3, 6, 10, 15, 21, 28, 36]] | + +Ordinals:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 3, 6, 10, 15, 21, 28, 36] | +| EvenSum | [0, 1, 2, 4, 6, 9, 12, 16, 20] | +| OddSum | [0, 0, 1, 2, 4, 6, 9, 12, 16] | +| AltSum | [0, 1, 1, 2, 2, 3, 3, 4, 4] | +| AccSum | [0, 2, 8, 20, 40, 70, 112, 168, 240] | +| AccRevSum | [0, 1, 4, 10, 20, 35, 56, 84, 120] | +| AntiDiagSum | [0, 1, 2, 4, 6, 9, 12, 16, 20] | + +Ordinals:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 12, 60, 60, 420, 840] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColMiddle | [0, 1, 1, 2, 2, 3, 3, 4, 4] | +| ColECenter | [0, 1, 2, 3, 4] | +| ColOCenter | [1, 2, 3, 4] | +| ColLeft | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColRight | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 6, 20, 50, 105, 196, 336] | +| TransNat0 | [0, 0, 1, 4, 10, 20, 35, 56, 84] | +| TransNat1 | [0, 1, 4, 10, 20, 35, 56, 84, 120] | + +Ordinals:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ OrdinalsRev Diagonals as rows | DiagRow7 | [7, 7, 7, 7, 7, 7, 7, 7, 7]| | DiagRow8 | [8, 8, 8, 8, 8, 8, 8, 8, 8]| -OrdinalsRev Diagonals as columns +Ordinals:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | @@ -188,7 +180,7 @@ OrdinalsRev Diagonals as columns | DiagCol7 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | | DiagCol8 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -OrdinalsRev Polynomial values as rows +Ordinals:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -201,7 +193,7 @@ OrdinalsRev Polynomial values as rows | PolyRow7 | [7, 28, 247, 1636, 7279, 24412, 67183, 160132, 342391] | | PolyRow8 | [8, 36, 502, 4916, 29124, 122068, 403106, 1120932, 2739136] | -OrdinalsRev Polynomial values as columns +Ordinals:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | diff --git a/data/md/Partition.md b/data/md/Partition.md index 0c2bc6a..2620af1 100644 --- a/data/md/Partition.md +++ b/data/md/Partition.md @@ -1,5 +1,5 @@ # Partition -['A008284', 'A058398', 'A072233'] +['A072233', 'A008284', 'A058398'] Partition Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Partition Triangle view | Row7 | [0, 1, 3, 4, 3, 2, 1, 1] | | Row8 | [0, 1, 4, 5, 5, 3, 2, 1, 1] | -Partition Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 0, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 5, 4, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 5, 6, 7, 0, 1, 4, 7, 9, 10, 11, 0, 1, 4, 8, 11, 13, 14, 15, 0, 1, 5, 10, 15, 18, 20, 21, 22] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 3, 1, 0, 7, 6, 5, 3, 1, 0, 11, 10, 9, 7, 4, 1, 0, 15, 14, 13, 11, 8, 4, 1, 0, 22, 21, 20, 18, 15, 10, 5, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 5, 5, 1, 2, 4, 6, 7, 7, 1, 2, 4, 7, 10, 11, 11, 1, 2, 4, 7, 11, 14, 15, 15, 1, 2, 4, 7, 12, 17, 21, 22, 22] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 1] | +Partition Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 1, 1], [0, 1, 2, 1, 1], [0, 1, 2, 2, 1, 1], [0, 1, 3, 3, 2, 1, 1], [0, 1, 3, 4, 3, 2, 1, 1], [0, 1, 4, 5, 5, 3, 2, 1, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 1, 1, 0], [1, 1, 2, 1, 0], [1, 1, 2, 2, 1, 0], [1, 1, 2, 3, 3, 1, 0], [1, 1, 2, 3, 4, 3, 1, 0], [1, 1, 2, 3, 5, 5, 4, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 1], [0, 1, 2, 1], [0, 1, 2, 1], [0, 1, 3, 2, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3, 4, 5], [0, 1, 3, 5, 6, 7], [0, 1, 4, 7, 9, 10, 11], [0, 1, 4, 8, 11, 13, 14, 15], [0, 1, 5, 10, 15, 18, 20, 21, 22]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [3, 2, 1, 0], [5, 4, 3, 1, 0], [7, 6, 5, 3, 1, 0], [11, 10, 9, 7, 4, 1, 0], [15, 14, 13, 11, 8, 4, 1, 0], [22, 21, 20, 18, 15, 10, 5, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 2, 3, 3], [1, 2, 4, 5, 5], [1, 2, 4, 6, 7, 7], [1, 2, 4, 7, 10, 11, 11], [1, 2, 4, 7, 11, 14, 15, 15], [1, 2, 4, 7, 12, 17, 21, 22, 22]] | Partition Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| EvenSum | [1, 0, 1, 1, 3, 3, 6, 7, 12] | -| OddSum | [0, 1, 1, 2, 2, 4, 5, 8, 10] | -| AltSum | [1, -1, 0, -1, 1, -1, 1, -1, 2] | -| AccSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | -| AccRevSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | -| RevAccSum | [1, 2, 5, 9, 17, 27, 46, 69, 108] | -| DiagSum | [1, 0, 1, 1, 2, 2, 4, 4] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| EvenSum | [1, 0, 1, 1, 3, 3, 6, 7, 12] | +| OddSum | [0, 1, 1, 2, 2, 4, 5, 8, 10] | +| AltSum | [1, -1, 0, -1, 1, -1, 1, -1, 2] | +| AccSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | +| AccRevSum | [1, 2, 5, 9, 17, 27, 46, 69, 108] | +| AntiDiagSum | [1, 0, 1, 1, 2, 2, 4, 4, 7] | Partition Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 2, 2, 6, 12, 60] | -| RowGcd | [1, 1, 1, 1, 2, 2, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 2, 2, 3, 4, 5] | -| Middle | [1, 0, 1, 1, 2, 2, 3, 4, 5] | -| Central | [1, 1, 2, 3, 5] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 7, 19, 43, 115, 259, 659] | -| NegHalf | [1, 1, -1, 3, -1, 7, -1, 15, 31] | -| Bin | [1, 1, 3, 7, 21, 51, 148, 365, 983] | -| InvBin | [1, 1, -1, 1, 5, 1, 4, 15, 55] | -| TransSqrs | [0, 1, 5, 14, 34, 68, 133, 232, 402] | -| TransNat0 | [0, 1, 3, 6, 12, 20, 35, 54, 86] | -| TransNat1 | [1, 2, 5, 9, 17, 27, 46, 69, 108] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 2, 2, 6, 12, 60] | +| RowGcd | [1, 1, 1, 1, 2, 2, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 2, 2, 3, 4, 5] | +| ColMiddle | [1, 0, 1, 1, 2, 2, 3, 4, 5] | +| ColECenter | [1, 1, 2, 3, 5] | +| ColOCenter | [0, 1, 2, 4] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 14, 34, 68, 133, 232, 402] | +| TransNat0 | [0, 1, 3, 6, 12, 20, 35, 54, 86] | +| TransNat1 | [1, 2, 5, 9, 17, 27, 46, 69, 108] | Partition Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Partition Polynomial values as columns | PolyCol7 | [1, 7, 56, 399, 2849, 19999, 140441, 983535, 6887986] | | PolyCol8 | [1, 8, 72, 584, 4744, 38024, 304840, 2439368, 19520264] | -# PartitionInv -True +# Partition:Inv +[] -PartitionInv Triangle view +Partition:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ PartitionInv Triangle view | Row7 | [0, -1, 1, 1, 0, -1, -1, 1] | | Row8 | [0, -1, 0, 2, 0, 0, -1, -1, 1] | -PartitionInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 1, -1, -1, 1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, -1, 1, 0, -1, 0, 2, 0, 0, -1, -1, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 0, 2, 0, -1, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 1, 1, 1, 0, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, -1, 0, 1, 1, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, -1, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 1, 0, -1, -1, 0, 1, 0, 0, 1, 0, -1, -1, -1, 1, 1, 0, 0] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1] | - -PartitionInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 0, 1, -1, 0, 0, 1, 0, 0] | -| OddSum | [0, 1, -1, 1, 0, 0, -1, 0, 0] | -| AltSum | [1, -1, 2, -2, 0, 0, 2, 0, 0] | -| AccSum | [1, 1, -1, -1, 0, 0, 1, 0, 0] | -| AccRevSum | [1, 1, -1, -1, 0, 0, 1, 0, 0] | -| RevAccSum | [1, 2, 1, 1, 0, 0, -1, 0, 0] | -| DiagSum | [1, 0, 1, -1, 1, 0, 0, 0] | - -PartitionInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| Middle | [1, 0, -1, 0, -1, 1, 0, 1, 0] | -| Central | [1, -1, -1, 0, 0] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, -1, 3, 3, 11, -21, -69] | -| NegHalf | [1, 1, 3, 3, -9, -9, 15, -81, 63] | -| Bin | [1, 1, -1, -2, -5, -4, -5, 22, 69] | -| InvBin | [1, 1, 3, 4, -5, -14, 7, -6, -123] | -| TransSqrs | [0, 1, 3, 5, 4, 4, -1, 0, -4] | -| TransNat0 | [0, 1, 1, 1, 0, 0, -1, 0, 0] | -| TransNat1 | [1, 2, 1, 1, 0, 0, -1, 0, 0] | - -PartitionInv Diagonals as rows +Partition:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 0, -1, 1], [0, 1, -1, -1, 1], [0, 0, 1, -1, -1, 1], [0, 0, 1, 0, -1, -1, 1], [0, -1, 1, 1, 0, -1, -1, 1], [0, -1, 0, 2, 0, 0, -1, -1, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -1, 0, 0], [1, -1, -1, 1, 0], [1, -1, -1, 1, 0, 0], [1, -1, -1, 0, 1, 0, 0], [1, -1, -1, 0, 1, 1, -1, 0], [1, -1, -1, 0, 0, 2, 0, -1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 0, 1], [0, 1, -1], [0, 0, -1, 1], [0, 0, 1, -1], [0, -1, 1, -1, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 0, -1, 0], [0, 1, 0, -1, 0], [0, 0, 1, 0, -1, 0], [0, 0, 1, 1, 0, -1, 0], [0, -1, 0, 1, 1, 0, -1, 0], [0, -1, -1, 1, 1, 1, 0, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 0, 0], [0, -1, 0, 1, 0], [0, -1, 0, 1, 0, 0], [0, -1, 0, 1, 1, 0, 0], [0, -1, 0, 1, 1, 0, -1, 0], [0, -1, 0, 1, 1, 1, -1, -1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, 0, 0, 0], [1, 0, -1, 0, 0], [1, 0, -1, 0, 0, 0], [1, 0, -1, -1, 0, 0, 0], [1, 0, -1, -1, 0, 1, 0, 0], [1, 0, -1, -1, -1, 1, 1, 0, 0]] | + +Partition:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -1, 0, 0, 1, 0, 0] | +| OddSum | [0, 1, -1, 1, 0, 0, -1, 0, 0] | +| AltSum | [1, -1, 2, -2, 0, 0, 2, 0, 0] | +| AccSum | [1, 1, -1, -1, 0, 0, 1, 0, 0] | +| AccRevSum | [1, 2, 1, 1, 0, 0, -1, 0, 0] | +| AntiDiagSum | [1, 0, 1, -1, 1, 0, 0, 0, 0] | + +Partition:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| ColMiddle | [1, 0, -1, 0, -1, 1, 0, 1, 0] | +| ColECenter | [1, -1, -1, 0, 0] | +| ColOCenter | [0, 0, 1, 1] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, 5, 4, 4, -1, 0, -4] | +| TransNat0 | [0, 1, 1, 1, 0, 0, -1, 0, 0] | +| TransNat1 | [1, 2, 1, 1, 0, 0, -1, 0, 0] | + +Partition:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ PartitionInv Diagonals as rows | DiagRow7 | [0, -1, -1, 1, 1, 2, 2, 1, 1]| | DiagRow8 | [0, 0, -2, -1, 0, 0, 1, 1, 0]| -PartitionInv Diagonals as columns +Partition:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ PartitionInv Diagonals as columns | DiagCol7 | [1, -1, -1, 0, 0, 1, 0, 1, 1] | | DiagCol8 | [1, -1, -1, 0, 0, 1, 0, 1, 0] | -PartitionInv Polynomial values as rows +Partition:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ PartitionInv Polynomial values as rows | PolyRow7 | [0, 0, 42, 1248, 11340, 59520, 225750, 689472, 1802808] | | PolyRow8 | [0, 0, 78, 3696, 45180, 297120, 1353450, 4824288, 14418936] | -PartitionInv Polynomial values as columns +Partition:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ PartitionInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 294, 2016, 14112, 98490, 689472, 4824288] | | PolyCol8 | [1, 8, 56, 448, 3528, 28224, 225344, 1802808, 14418936] | -# PartitionRev -True +# Partition:Rev +[] -PartitionRev Triangle view +Partition:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ PartitionRev Triangle view | Row7 | [1, 1, 2, 3, 4, 3, 1, 0] | | Row8 | [1, 1, 2, 3, 5, 5, 4, 1, 0] | -PartitionRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 0, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 5, 4, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 5, 5, 1, 2, 4, 6, 7, 7, 1, 2, 4, 7, 10, 11, 11, 1, 2, 4, 7, 11, 14, 15, 15, 1, 2, 4, 7, 12, 17, 21, 22, 22] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 3, 3, 2, 1, 5, 5, 4, 2, 1, 7, 7, 6, 4, 2, 1, 11, 11, 10, 7, 4, 2, 1, 15, 15, 14, 11, 7, 4, 2, 1, 22, 22, 21, 17, 12, 7, 4, 2, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 5, 6, 7, 0, 1, 4, 7, 9, 10, 11, 0, 1, 4, 8, 11, 13, 14, 15, 0, 1, 5, 10, 15, 18, 20, 21, 22] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1] | - -PartitionRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| EvenSum | [1, 1, 1, 2, 3, 4, 6, 8, 12] | -| OddSum | [0, 0, 1, 1, 2, 3, 5, 7, 10] | -| AltSum | [1, 1, 0, 1, 1, 1, 1, 1, 2] | -| AccSum | [1, 2, 5, 9, 17, 27, 46, 69, 108] | -| AccRevSum | [1, 2, 5, 9, 17, 27, 46, 69, 108] | -| RevAccSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | -| DiagSum | [1, 1, 1, 2, 2, 3, 4, 5] | - -PartitionRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 2, 2, 6, 12, 60] | -| RowGcd | [1, 1, 1, 1, 2, 2, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 2, 2, 3, 4, 5] | -| Middle | [1, 1, 1, 1, 2, 2, 3, 3, 5] | -| Central | [1, 1, 2, 3, 5] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 14, 34, 74, 166, 350, 746] | -| NegHalf | [1, -2, 2, -6, 14, -26, 50, -102, 214] | -| Bin | [1, 1, 3, 7, 21, 51, 148, 365, 983] | -| InvBin | [1, -1, -1, -1, 5, -1, 4, -15, 55] | -| TransSqrs | [0, 0, 1, 5, 18, 43, 109, 211, 434] | -| TransNat0 | [0, 0, 1, 3, 8, 15, 31, 51, 90] | -| TransNat1 | [1, 1, 3, 6, 13, 22, 42, 66, 112] | - -PartitionRev Diagonals as rows +Partition:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 1, 1, 0], [1, 1, 2, 1, 0], [1, 1, 2, 2, 1, 0], [1, 1, 2, 3, 3, 1, 0], [1, 1, 2, 3, 4, 3, 1, 0], [1, 1, 2, 3, 5, 5, 4, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 1, 1], [0, 1, 2, 1, 1], [0, 1, 2, 2, 1, 1], [0, 1, 3, 3, 2, 1, 1], [0, 1, 3, 4, 3, 2, 1, 1], [0, 1, 4, 5, 5, 3, 2, 1, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 1, 0], [1, 1, 1], [1, 1, 2, 0], [1, 1, 2, 1], [1, 1, 2, 2, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 2, 3, 3], [1, 2, 4, 5, 5], [1, 2, 4, 6, 7, 7], [1, 2, 4, 7, 10, 11, 11], [1, 2, 4, 7, 11, 14, 15, 15], [1, 2, 4, 7, 12, 17, 21, 22, 22]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [3, 3, 2, 1], [5, 5, 4, 2, 1], [7, 7, 6, 4, 2, 1], [11, 11, 10, 7, 4, 2, 1], [15, 15, 14, 11, 7, 4, 2, 1], [22, 22, 21, 17, 12, 7, 4, 2, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3, 4, 5], [0, 1, 3, 5, 6, 7], [0, 1, 4, 7, 9, 10, 11], [0, 1, 4, 8, 11, 13, 14, 15], [0, 1, 5, 10, 15, 18, 20, 21, 22]] | + +Partition:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| EvenSum | [1, 1, 1, 2, 3, 4, 6, 8, 12] | +| OddSum | [0, 0, 1, 1, 2, 3, 5, 7, 10] | +| AltSum | [1, 1, 0, 1, 1, 1, 1, 1, 2] | +| AccSum | [1, 2, 5, 9, 17, 27, 46, 69, 108] | +| AccRevSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | +| AntiDiagSum | [1, 1, 1, 2, 2, 3, 4, 5, 6] | + +Partition:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 2, 2, 6, 12, 60] | +| RowGcd | [1, 1, 1, 1, 2, 2, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 2, 2, 3, 4, 5] | +| ColMiddle | [1, 1, 1, 1, 2, 2, 3, 3, 5] | +| ColECenter | [1, 1, 2, 3, 5] | +| ColOCenter | [1, 1, 2, 3] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 5, 18, 43, 109, 211, 434] | +| TransNat0 | [0, 0, 1, 3, 8, 15, 31, 51, 90] | +| TransNat1 | [1, 1, 3, 6, 13, 22, 42, 66, 112] | + +Partition:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ PartitionRev Diagonals as rows | DiagRow7 | [1, 1, 2, 3, 5, 7, 11, 15, 21]| | DiagRow8 | [1, 1, 2, 3, 5, 7, 11, 15, 22]| -PartitionRev Diagonals as columns +Partition:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ PartitionRev Diagonals as columns | DiagCol7 | [0, 1, 4, 8, 11, 13, 14, 15, 15] | | DiagCol8 | [0, 1, 5, 10, 15, 18, 20, 21, 22] | -PartitionRev Polynomial values as rows +Partition:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ PartitionRev Polynomial values as rows | PolyRow7 | [1, 15, 259, 1885, 8421, 27931, 75895, 178809, 378505] | | PolyRow8 | [1, 22, 659, 6826, 39397, 159806, 512647, 1391314, 3331721] | -PartitionRev Polynomial values as columns +Partition:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ PartitionRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 57, 449, 3193, 25145, 178809, 1391314] | | PolyCol8 | [1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721] | -# PartitionInvRev -True +# Partition:Inv:Rev +[] -PartitionInvRev Triangle view +Partition:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ PartitionInvRev Triangle view | Row7 | [1, -1, -1, 0, 1, 1, -1, 0] | | Row8 | [1, -1, -1, 0, 0, 2, 0, -1, 0] | -PartitionInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 0, 2, 0, -1, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 1, -1, -1, 1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, -1, 1, 0, -1, 0, 2, 0, 0, -1, -1, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 1, 0, -1, -1, 0, 1, 0, 0, 1, 0, -1, -1, -1, 1, 1, 0, 0] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, -1, 0, 1, 0, 0, 1, 0, -1, -1, 0, 1, 0, 0, 1, 1, -1, -1, -1, 0, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 1, 1, 1, 0, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 1] | - -PartitionInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 1, 0, 0, 1, 0, 0] | -| OddSum | [0, 0, -1, -1, 0, 0, -1, 0, 0] | -| AltSum | [1, 1, 2, 2, 0, 0, 2, 0, 0] | -| AccSum | [1, 2, 1, 1, 0, 0, -1, 0, 0] | -| AccRevSum | [1, 2, 1, 1, 0, 0, -1, 0, 0] | -| RevAccSum | [1, 1, -1, -1, 0, 0, 1, 0, 0] | -| DiagSum | [1, 1, 1, 0, 0, 0, -1, 0] | - -PartitionInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 2] | -| Middle | [1, 1, -1, -1, -1, -1, 0, 0, 0] | -| Central | [1, -1, -1, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, 4, 6, 12, 20, 42, 78] | -| NegHalf | [1, -2, 6, -12, 18, -36, 84, -162, 306] | -| Bin | [1, 1, -1, -2, -5, -4, -5, 22, 69] | -| InvBin | [1, -1, 3, -4, -5, 14, 7, 6, -123] | -| TransSqrs | [0, 0, -1, -1, 4, 4, 11, 0, -4] | -| TransNat0 | [0, 0, -1, -1, 0, 0, 1, 0, 0] | -| TransNat1 | [1, 1, -1, -1, 0, 0, 1, 0, 0] | - -PartitionInvRev Diagonals as rows +Partition:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -1, 0, 0], [1, -1, -1, 1, 0], [1, -1, -1, 1, 0, 0], [1, -1, -1, 0, 1, 0, 0], [1, -1, -1, 0, 1, 1, -1, 0], [1, -1, -1, 0, 0, 2, 0, -1, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 0, -1, 1], [0, 1, -1, -1, 1], [0, 0, 1, -1, -1, 1], [0, 0, 1, 0, -1, -1, 1], [0, -1, 1, 1, 0, -1, -1, 1], [0, -1, 0, 2, 0, 0, -1, -1, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -1, 0], [1, -1, 0], [1, -1, -1, 0], [1, -1, -1, 1], [1, -1, -1, 1, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, 0, 0, 0], [1, 0, -1, 0, 0], [1, 0, -1, 0, 0, 0], [1, 0, -1, -1, 0, 0, 0], [1, 0, -1, -1, 0, 1, 0, 0], [1, 0, -1, -1, -1, 1, 1, 0, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [0, 0, 0, 1], [0, 0, -1, 0, 1], [0, 0, 0, -1, 0, 1], [0, 0, 0, -1, -1, 0, 1], [0, 0, 1, 0, -1, -1, 0, 1], [0, 0, 1, 1, -1, -1, -1, 0, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 0, -1, 0], [0, 1, 0, -1, 0], [0, 0, 1, 0, -1, 0], [0, 0, 1, 1, 0, -1, 0], [0, -1, 0, 1, 1, 0, -1, 0], [0, -1, -1, 1, 1, 1, 0, -1, 0]] | + +Partition:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 1, 0, 0, 1, 0, 0] | +| OddSum | [0, 0, -1, -1, 0, 0, -1, 0, 0] | +| AltSum | [1, 1, 2, 2, 0, 0, 2, 0, 0] | +| AccSum | [1, 2, 1, 1, 0, 0, -1, 0, 0] | +| AccRevSum | [1, 1, -1, -1, 0, 0, 1, 0, 0] | +| AntiDiagSum | [1, 1, 1, 0, 0, 0, -1, 0, 0] | + +Partition:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 2] | +| ColMiddle | [1, 1, -1, -1, -1, -1, 0, 0, 0] | +| ColECenter | [1, -1, -1, 0, 0] | +| ColOCenter | [1, -1, -1, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, -1, 4, 4, 11, 0, -4] | +| TransNat0 | [0, 0, -1, -1, 0, 0, 1, 0, 0] | +| TransNat1 | [1, 1, -1, -1, 0, 0, 1, 0, 0] | + +Partition:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ PartitionInvRev Diagonals as rows | DiagRow7 | [1, -1, -1, 0, 0, 1, 0, 1, 1]| | DiagRow8 | [1, -1, -1, 0, 0, 1, 0, 1, 0]| -PartitionInvRev Diagonals as columns +Partition:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ PartitionInvRev Diagonals as columns | DiagCol7 | [0, -1, -1, 1, 1, 2, 2, 1, 1] | | DiagCol8 | [0, 0, -2, -1, 0, 0, 1, 1, 0] | -PartitionInvRev Polynomial values as rows +Partition:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ PartitionInvRev Polynomial values as rows | PolyRow7 | [1, 0, -21, -416, -2835, -11904, -37625, -98496, -225351] | | PolyRow8 | [1, 0, -69, -1712, -14355, -71904, -264425, -789984, -2031687] | -PartitionInvRev Polynomial values as columns +Partition:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/PartitionMax.md b/data/md/PartitionMax.md index 6bd47e6..1aea0d3 100644 --- a/data/md/PartitionMax.md +++ b/data/md/PartitionMax.md @@ -1,5 +1,5 @@ # PartitionMax -['A008284', 'A058398', 'A072233'] +['A026820'] PartitionMax Triangle view | Row | Seq | @@ -14,45 +14,41 @@ PartitionMax Triangle view | Row7 | [0, 1, 4, 8, 11, 13, 14, 15] | | Row8 | [0, 1, 5, 10, 15, 18, 20, 21, 22] | -PartitionMax Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 5, 6, 7, 0, 1, 4, 7, 9, 10, 11, 0, 1, 4, 8, 11, 13, 14, 15, 0, 1, 5, 10, 15, 18, 20, 21, 22] | -| TriRev | [1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 3, 1, 0, 7, 6, 5, 3, 1, 0, 11, 10, 9, 7, 4, 1, 0, 15, 14, 13, 11, 8, 4, 1, 0, 22, 21, 20, 18, 15, 10, 5, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 4, 8, 13, 0, 1, 4, 9, 15, 22, 0, 1, 5, 12, 21, 31, 42, 0, 1, 5, 13, 24, 37, 51, 66, 0, 1, 6, 16, 31, 49, 69, 90, 112] | -| TriRevAcc | [1, 1, 0, 3, 1, 0, 6, 3, 1, 0, 13, 8, 4, 1, 0, 22, 15, 9, 4, 1, 0, 42, 31, 21, 12, 5, 1, 0, 66, 51, 37, 24, 13, 5, 1, 0, 112, 90, 69, 49, 31, 16, 6, 1, 0] | -| TriAccRev | [1, 1, 1, 2, 3, 3, 3, 5, 6, 6, 5, 9, 12, 13, 13, 7, 13, 18, 21, 22, 22, 11, 21, 30, 37, 41, 42, 42, 15, 29, 42, 53, 61, 65, 66, 66, 22, 43, 63, 81, 96, 106, 111, 112, 112] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 3, 0, 1, 3, 4] | +PartitionMax Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3, 4, 5], [0, 1, 3, 5, 6, 7], [0, 1, 4, 7, 9, 10, 11], [0, 1, 4, 8, 11, 13, 14, 15], [0, 1, 5, 10, 15, 18, 20, 21, 22]] | +| RevTabl | [[1], [1, 0], [2, 1, 0], [3, 2, 1, 0], [5, 4, 3, 1, 0], [7, 6, 5, 3, 1, 0], [11, 10, 9, 7, 4, 1, 0], [15, 14, 13, 11, 8, 4, 1, 0], [22, 21, 20, 18, 15, 10, 5, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 2], [0, 1, 3, 3], [0, 1, 3, 4], [0, 1, 4, 5, 5]] | +| AccTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 3, 6], [0, 1, 4, 8, 13], [0, 1, 4, 9, 15, 22], [0, 1, 5, 12, 21, 31, 42], [0, 1, 5, 13, 24, 37, 51, 66], [0, 1, 6, 16, 31, 49, 69, 90, 112]] | +| RevAccTabl | [[1], [1, 0], [3, 1, 0], [6, 3, 1, 0], [13, 8, 4, 1, 0], [22, 15, 9, 4, 1, 0], [42, 31, 21, 12, 5, 1, 0], [66, 51, 37, 24, 13, 5, 1, 0], [112, 90, 69, 49, 31, 16, 6, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [2, 3, 3], [3, 5, 6, 6], [5, 9, 12, 13, 13], [7, 13, 18, 21, 22, 22], [11, 21, 30, 37, 41, 42, 42], [15, 29, 42, 53, 61, 65, 66, 66], [22, 43, 63, 81, 96, 106, 111, 112, 112]] | PartitionMax Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | -| EvenSum | [1, 0, 2, 2, 8, 9, 24, 29, 62] | -| OddSum | [0, 1, 1, 4, 5, 13, 18, 37, 50] | -| AltSum | [1, -1, 1, -2, 3, -4, 6, -8, 12] | -| AccSum | [1, 1, 4, 10, 26, 51, 112, 197, 374] | -| AccRevSum | [1, 1, 4, 10, 26, 51, 112, 197, 374] | -| RevAccSum | [1, 2, 8, 20, 52, 103, 224, 397, 746] | -| DiagSum | [1, 0, 1, 1, 3, 3, 7, 8] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | +| EvenSum | [1, 0, 2, 2, 8, 9, 24, 29, 62] | +| OddSum | [0, 1, 1, 4, 5, 13, 18, 37, 50] | +| AltSum | [1, -1, 1, -2, 3, -4, 6, -8, 12] | +| AccSum | [1, 1, 4, 10, 26, 51, 112, 197, 374] | +| AccRevSum | [1, 2, 8, 20, 52, 103, 224, 397, 746] | +| AntiDiagSum | [1, 0, 1, 1, 3, 3, 7, 8, 15] | PartitionMax Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 60, 210, 13860, 120120, 13860] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| Middle | [1, 0, 1, 1, 3, 3, 7, 8, 15] | -| Central | [1, 1, 3, 7, 15] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| PosHalf | [1, 1, 4, 11, 33, 79, 219, 503, 1296] | -| NegHalf | [1, 1, 0, 3, 1, 7, 3, 15, 28] | -| Bin | [1, 1, 4, 12, 43, 122, 412, 1142, 3516] | -| InvBin | [1, 1, 0, 0, 3, 2, 0, 8, 28] | -| TransSqrs | [0, 1, 9, 36, 129, 329, 870, 1829, 3958] | -| TransNat0 | [0, 1, 5, 14, 39, 81, 182, 331, 634] | -| TransNat1 | [1, 2, 8, 20, 52, 103, 224, 397, 746] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 60, 210, 13860, 120120, 13860] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| ColMiddle | [1, 0, 1, 1, 3, 3, 7, 8, 15] | +| ColECenter | [1, 1, 3, 7, 15] | +| ColOCenter | [0, 1, 3, 8] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| TransSqrs | [0, 1, 9, 36, 129, 329, 870, 1829, 3958] | +| TransNat0 | [0, 1, 5, 14, 39, 81, 182, 331, 634] | +| TransNat1 | [1, 2, 8, 20, 52, 103, 224, 397, 746] | PartitionMax Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ PartitionMax Polynomial values as columns | PolyCol7 | [1, 7, 105, 1134, 13531, 133924, 1486422, 14248080, 146815228] | | PolyCol8 | [1, 8, 136, 1672, 22728, 256712, 3251976, 35602696, 419038536] | -# PartitionMaxRev -True +# PartitionMax:Rev +[] -PartitionMaxRev Triangle view +PartitionMax:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ PartitionMaxRev Triangle view | Row7 | [15, 14, 13, 11, 8, 4, 1, 0] | | Row8 | [22, 21, 20, 18, 15, 10, 5, 1, 0] | -PartitionMaxRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 4, 3, 1, 0, 7, 6, 5, 3, 1, 0, 11, 10, 9, 7, 4, 1, 0, 15, 14, 13, 11, 8, 4, 1, 0, 22, 21, 20, 18, 15, 10, 5, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 1, 3, 5, 6, 7, 0, 1, 4, 7, 9, 10, 11, 0, 1, 4, 8, 11, 13, 14, 15, 0, 1, 5, 10, 15, 18, 20, 21, 22] | -| TriAcc | [1, 1, 1, 2, 3, 3, 3, 5, 6, 6, 5, 9, 12, 13, 13, 7, 13, 18, 21, 22, 22, 11, 21, 30, 37, 41, 42, 42, 15, 29, 42, 53, 61, 65, 66, 66, 22, 43, 63, 81, 96, 106, 111, 112, 112] | -| TriRevAcc | [1, 1, 1, 3, 3, 2, 6, 6, 5, 3, 13, 13, 12, 9, 5, 22, 22, 21, 18, 13, 7, 42, 42, 41, 37, 30, 21, 11, 66, 66, 65, 61, 53, 42, 29, 15, 112, 112, 111, 106, 96, 81, 63, 43, 22] | -| TriAccRev | [1, 0, 1, 0, 1, 3, 0, 1, 3, 6, 0, 1, 4, 8, 13, 0, 1, 4, 9, 15, 22, 0, 1, 5, 12, 21, 31, 42, 0, 1, 5, 13, 24, 37, 51, 66, 0, 1, 6, 16, 31, 49, 69, 90, 112] | -| TriDiag | [1, 1, 2, 0, 3, 1, 5, 2, 0, 7, 4, 1, 11, 6, 3, 0, 15, 10, 5, 1] | - -PartitionMaxRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | -| EvenSum | [1, 1, 2, 4, 8, 13, 24, 37, 62] | -| OddSum | [0, 0, 1, 2, 5, 9, 18, 29, 50] | -| AltSum | [1, 1, 1, 2, 3, 4, 6, 8, 12] | -| AccSum | [1, 2, 8, 20, 52, 103, 224, 397, 746] | -| AccRevSum | [1, 2, 8, 20, 52, 103, 224, 397, 746] | -| RevAccSum | [1, 1, 4, 10, 26, 51, 112, 197, 374] | -| DiagSum | [1, 1, 2, 4, 7, 12, 20, 31] | - -PartitionMaxRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 60, 210, 13860, 120120, 13860] | -| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| Middle | [1, 1, 1, 2, 3, 5, 7, 11, 15] | -| Central | [1, 1, 3, 7, 15] | -| LeftSide | [1, 1, 2, 3, 5, 7, 11, 15, 22] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 10, 34, 126, 374, 1242, 3490, 10518] | -| NegHalf | [1, -2, 6, -18, 58, -158, 486, -1314, 3826] | -| Bin | [1, 1, 4, 12, 43, 122, 412, 1142, 3516] | -| InvBin | [1, -1, 0, 0, 3, -2, 0, -8, 28] | -| TransSqrs | [0, 0, 1, 6, 25, 69, 198, 429, 982] | -| TransNat0 | [0, 0, 1, 4, 13, 29, 70, 131, 262] | -| TransNat1 | [1, 1, 4, 10, 26, 51, 112, 197, 374] | - -PartitionMaxRev Diagonals as rows +PartitionMax:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [2, 1, 0], [3, 2, 1, 0], [5, 4, 3, 1, 0], [7, 6, 5, 3, 1, 0], [11, 10, 9, 7, 4, 1, 0], [15, 14, 13, 11, 8, 4, 1, 0], [22, 21, 20, 18, 15, 10, 5, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3, 4, 5], [0, 1, 3, 5, 6, 7], [0, 1, 4, 7, 9, 10, 11], [0, 1, 4, 8, 11, 13, 14, 15], [0, 1, 5, 10, 15, 18, 20, 21, 22]] | +| AntiDiag | [[1], [1], [2, 0], [3, 1], [5, 2, 0], [7, 4, 1], [11, 6, 3, 0], [15, 10, 5, 1], [22, 14, 9, 3, 0]] | +| AccTabl | [[1], [1, 1], [2, 3, 3], [3, 5, 6, 6], [5, 9, 12, 13, 13], [7, 13, 18, 21, 22, 22], [11, 21, 30, 37, 41, 42, 42], [15, 29, 42, 53, 61, 65, 66, 66], [22, 43, 63, 81, 96, 106, 111, 112, 112]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 2], [6, 6, 5, 3], [13, 13, 12, 9, 5], [22, 22, 21, 18, 13, 7], [42, 42, 41, 37, 30, 21, 11], [66, 66, 65, 61, 53, 42, 29, 15], [112, 112, 111, 106, 96, 81, 63, 43, 22]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 3, 6], [0, 1, 4, 8, 13], [0, 1, 4, 9, 15, 22], [0, 1, 5, 12, 21, 31, 42], [0, 1, 5, 13, 24, 37, 51, 66], [0, 1, 6, 16, 31, 49, 69, 90, 112]] | + +PartitionMax:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 6, 13, 22, 42, 66, 112] | +| EvenSum | [1, 1, 2, 4, 8, 13, 24, 37, 62] | +| OddSum | [0, 0, 1, 2, 5, 9, 18, 29, 50] | +| AltSum | [1, 1, 1, 2, 3, 4, 6, 8, 12] | +| AccSum | [1, 2, 8, 20, 52, 103, 224, 397, 746] | +| AccRevSum | [1, 1, 4, 10, 26, 51, 112, 197, 374] | +| AntiDiagSum | [1, 1, 2, 4, 7, 12, 20, 31, 48] | + +PartitionMax:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 60, 210, 13860, 120120, 13860] | +| RowGcd | [1, 1, 2, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| ColMiddle | [1, 1, 1, 2, 3, 5, 7, 11, 15] | +| ColECenter | [1, 1, 3, 7, 15] | +| ColOCenter | [1, 2, 5, 11] | +| ColLeft | [1, 1, 2, 3, 5, 7, 11, 15, 22] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 6, 25, 69, 198, 429, 982] | +| TransNat0 | [0, 0, 1, 4, 13, 29, 70, 131, 262] | +| TransNat1 | [1, 1, 4, 10, 26, 51, 112, 197, 374] | + +PartitionMax:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ PartitionMaxRev Diagonals as rows | DiagRow7 | [15, 21, 28, 38, 49, 65, 82, 105, 131]| | DiagRow8 | [22, 29, 40, 52, 70, 89, 116, 146, 186]| -PartitionMaxRev Diagonals as columns +PartitionMax:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 3, 5, 7, 11, 15, 22] | @@ -188,7 +180,7 @@ PartitionMaxRev Diagonals as columns | DiagCol7 | [0, 1, 5, 14, 27, 47, 71, 105, 146] | | DiagCol8 | [0, 1, 6, 16, 34, 57, 90, 131, 186] | -PartitionMaxRev Polynomial values as rows +PartitionMax:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ PartitionMaxRev Polynomial values as rows | PolyRow7 | [15, 66, 503, 2820, 11223, 34910, 91071, 208608, 432575] | | PolyRow8 | [22, 112, 1296, 10228, 52522, 199752, 615172, 1623196, 3807678] | -PartitionMaxRev Polynomial values as columns +PartitionMax:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 3, 5, 7, 11, 15, 22] | diff --git a/data/md/Polygonal.md b/data/md/Polygonal.md index 45c31b6..c165ec9 100644 --- a/data/md/Polygonal.md +++ b/data/md/Polygonal.md @@ -1,5 +1,5 @@ # Polygonal -['A057145', 'A134394', 'A139600', 'A139601'] +['A139600', 'A057145', 'A134394', 'A139601'] Polygonal Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Polygonal Triangle view | Row7 | [0, 1, 7, 15, 22, 25, 21, 7] | | Row8 | [0, 1, 8, 18, 28, 35, 36, 28, 8] | -Polygonal Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8] | -| TriRev | [0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 9, 5, 1, 0, 6, 15, 16, 12, 6, 1, 0, 7, 21, 25, 22, 15, 7, 1, 0, 8, 28, 36, 35, 28, 18, 8, 1, 0] | -| TriAcc | [0, 0, 1, 0, 1, 3, 0, 1, 4, 7, 0, 1, 5, 11, 15, 0, 1, 6, 15, 25, 30, 0, 1, 7, 19, 35, 50, 56, 0, 1, 8, 23, 45, 70, 91, 98, 0, 1, 9, 27, 55, 90, 126, 154, 162] | -| TriRevAcc | [0, 1, 0, 3, 1, 0, 7, 4, 1, 0, 15, 11, 5, 1, 0, 30, 25, 15, 6, 1, 0, 56, 50, 35, 19, 7, 1, 0, 98, 91, 70, 45, 23, 8, 1, 0, 162, 154, 126, 90, 55, 27, 9, 1, 0] | -| TriAccRev | [0, 1, 1, 2, 3, 3, 3, 6, 7, 7, 4, 10, 14, 15, 15, 5, 15, 24, 29, 30, 30, 6, 21, 37, 49, 55, 56, 56, 7, 28, 53, 75, 90, 97, 98, 98, 8, 36, 72, 107, 135, 153, 161, 162, 162] | -| TriDiag | [0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 3, 0, 1, 5, 6] | +Polygonal Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [0, 1], [0, 1, 2], [0, 1, 3, 3], [0, 1, 4, 6, 4], [0, 1, 5, 9, 10, 5], [0, 1, 6, 12, 16, 15, 6], [0, 1, 7, 15, 22, 25, 21, 7], [0, 1, 8, 18, 28, 35, 36, 28, 8]] | +| RevTabl | [[0], [1, 0], [2, 1, 0], [3, 3, 1, 0], [4, 6, 4, 1, 0], [5, 10, 9, 5, 1, 0], [6, 15, 16, 12, 6, 1, 0], [7, 21, 25, 22, 15, 7, 1, 0], [8, 28, 36, 35, 28, 18, 8, 1, 0]] | +| AntiDiag | [[0], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 3], [0, 1, 4, 3], [0, 1, 5, 6], [0, 1, 6, 9, 4]] | +| AccTabl | [[0], [0, 1], [0, 1, 3], [0, 1, 4, 7], [0, 1, 5, 11, 15], [0, 1, 6, 15, 25, 30], [0, 1, 7, 19, 35, 50, 56], [0, 1, 8, 23, 45, 70, 91, 98], [0, 1, 9, 27, 55, 90, 126, 154, 162]] | +| RevAccTabl | [[0], [1, 0], [3, 1, 0], [7, 4, 1, 0], [15, 11, 5, 1, 0], [30, 25, 15, 6, 1, 0], [56, 50, 35, 19, 7, 1, 0], [98, 91, 70, 45, 23, 8, 1, 0], [162, 154, 126, 90, 55, 27, 9, 1, 0]] | +| AccRevTabl | [[0], [1, 1], [2, 3, 3], [3, 6, 7, 7], [4, 10, 14, 15, 15], [5, 15, 24, 29, 30, 30], [6, 21, 37, 49, 55, 56, 56], [7, 28, 53, 75, 90, 97, 98, 98], [8, 36, 72, 107, 135, 153, 161, 162, 162]] | Polygonal Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 3, 7, 15, 30, 56, 98, 162] | -| EvenSum | [0, 0, 2, 3, 8, 15, 28, 50, 80] | -| OddSum | [0, 1, 1, 4, 7, 15, 28, 48, 82] | -| AltSum | [0, -1, 1, -1, 1, 0, 0, 2, -2] | -| AccSum | [0, 1, 4, 12, 32, 77, 168, 336, 624] | -| AccRevSum | [0, 1, 4, 12, 32, 77, 168, 336, 624] | -| RevAccSum | [0, 2, 8, 23, 58, 133, 280, 546, 996] | -| DiagSum | [0, 0, 1, 1, 3, 4, 8, 12] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 3, 7, 15, 30, 56, 98, 162] | +| EvenSum | [0, 0, 2, 3, 8, 15, 28, 50, 80] | +| OddSum | [0, 1, 1, 4, 7, 15, 28, 48, 82] | +| AltSum | [0, -1, 1, -1, 1, 0, 0, 2, -2] | +| AccSum | [0, 1, 4, 12, 32, 77, 168, 336, 624] | +| AccRevSum | [0, 2, 8, 23, 58, 133, 280, 546, 996] | +| AntiDiagSum | [0, 0, 1, 1, 3, 4, 8, 12, 20] | Polygonal Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 3, 12, 90, 240, 11550, 2520] | -| RowGcd | [1, 1, 2, 3, 2, 1, 1, 1, 1] | -| RowMax | [0, 1, 2, 3, 6, 10, 16, 25, 36] | -| Middle | [0, 0, 1, 1, 4, 5, 12, 15, 28] | -| Central | [0, 1, 4, 12, 28] | -| LeftSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| PosHalf | [0, 1, 4, 13, 40, 117, 324, 853, 2152] | -| NegHalf | [0, 1, 0, 1, 0, -3, 8, -31, 72] | -| Bin | [0, 1, 4, 15, 56, 200, 672, 2128, 6400] | -| InvBin | [0, 1, 0, -3, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 1, 9, 40, 135, 387, 980, 2240, 4698] | -| TransNat0 | [0, 1, 5, 16, 43, 103, 224, 448, 834] | -| TransNat1 | [0, 2, 8, 23, 58, 133, 280, 546, 996] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 90, 240, 11550, 2520] | +| RowGcd | [1, 1, 2, 3, 2, 1, 1, 1, 1] | +| RowMax | [0, 1, 2, 3, 6, 10, 16, 25, 36] | +| ColMiddle | [0, 0, 1, 1, 4, 5, 12, 15, 28] | +| ColECenter | [0, 1, 4, 12, 28] | +| ColOCenter | [0, 1, 5, 15] | +| ColLeft | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| TransSqrs | [0, 1, 9, 40, 135, 387, 980, 2240, 4698] | +| TransNat0 | [0, 1, 5, 16, 43, 103, 224, 448, 834] | +| TransNat1 | [0, 2, 8, 23, 58, 133, 280, 546, 996] | Polygonal Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Polygonal Polynomial values as columns | PolyCol7 | [0, 7, 105, 1183, 11865, 111384, 1000832, 8713922, 74075022] | | PolyCol8 | [0, 8, 136, 1736, 19720, 209736, 2136456, 21102536, 203646472] | -# PolygonalRev -True +# Polygonal:Rev +[] -PolygonalRev Triangle view +Polygonal:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [0] | @@ -122,47 +118,43 @@ PolygonalRev Triangle view | Row7 | [7, 21, 25, 22, 15, 7, 1, 0] | | Row8 | [8, 28, 36, 35, 28, 18, 8, 1, 0] | -PolygonalRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 9, 5, 1, 0, 6, 15, 16, 12, 6, 1, 0, 7, 21, 25, 22, 15, 7, 1, 0, 8, 28, 36, 35, 28, 18, 8, 1, 0] | -| TriRev | [0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8] | -| TriAcc | [0, 1, 1, 2, 3, 3, 3, 6, 7, 7, 4, 10, 14, 15, 15, 5, 15, 24, 29, 30, 30, 6, 21, 37, 49, 55, 56, 56, 7, 28, 53, 75, 90, 97, 98, 98, 8, 36, 72, 107, 135, 153, 161, 162, 162] | -| TriRevAcc | [0, 1, 1, 3, 3, 2, 7, 7, 6, 3, 15, 15, 14, 10, 4, 30, 30, 29, 24, 15, 5, 56, 56, 55, 49, 37, 21, 6, 98, 98, 97, 90, 75, 53, 28, 7, 162, 162, 161, 153, 135, 107, 72, 36, 8] | -| TriAccRev | [0, 0, 1, 0, 1, 3, 0, 1, 4, 7, 0, 1, 5, 11, 15, 0, 1, 6, 15, 25, 30, 0, 1, 7, 19, 35, 50, 56, 0, 1, 8, 23, 45, 70, 91, 98, 0, 1, 9, 27, 55, 90, 126, 154, 162] | -| TriDiag | [0, 1, 2, 0, 3, 1, 4, 3, 0, 5, 6, 1, 6, 10, 4, 0, 7, 15, 9, 1] | - -PolygonalRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [0, 1, 3, 7, 15, 30, 56, 98, 162] | -| EvenSum | [0, 1, 2, 4, 8, 15, 28, 48, 80] | -| OddSum | [0, 0, 1, 3, 7, 15, 28, 50, 82] | -| AltSum | [0, 1, 1, 1, 1, 0, 0, -2, -2] | -| AccSum | [0, 2, 8, 23, 58, 133, 280, 546, 996] | -| AccRevSum | [0, 2, 8, 23, 58, 133, 280, 546, 996] | -| RevAccSum | [0, 1, 4, 12, 32, 77, 168, 336, 624] | -| DiagSum | [0, 1, 2, 4, 7, 12, 20, 32] | - -PolygonalRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 3, 12, 90, 240, 11550, 2520] | -| RowGcd | [1, 1, 2, 3, 2, 1, 1, 1, 1] | -| RowMax | [0, 1, 2, 3, 6, 10, 16, 25, 36] | -| Middle | [0, 1, 1, 3, 4, 9, 12, 22, 28] | -| Central | [0, 1, 4, 12, 28] | -| LeftSide | [0, 1, 2, 3, 4, 5, 6, 7, 8] | -| RightSide | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [0, 2, 10, 38, 130, 414, 1242, 3542, 9682] | -| NegHalf | [0, -2, 6, -14, 30, -54, 86, -94, -18] | -| Bin | [0, 1, 4, 15, 56, 200, 672, 2128, 6400] | -| InvBin | [0, -1, 0, 3, 0, 0, 0, 0, 0] | -| TransSqrs | [0, 0, 1, 7, 31, 107, 308, 770, 1722] | -| TransNat0 | [0, 0, 1, 5, 17, 47, 112, 238, 462] | -| TransNat1 | [0, 1, 4, 12, 32, 77, 168, 336, 624] | - -PolygonalRev Diagonals as rows +Polygonal:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[0], [1, 0], [2, 1, 0], [3, 3, 1, 0], [4, 6, 4, 1, 0], [5, 10, 9, 5, 1, 0], [6, 15, 16, 12, 6, 1, 0], [7, 21, 25, 22, 15, 7, 1, 0], [8, 28, 36, 35, 28, 18, 8, 1, 0]] | +| RevTabl | [[0], [0, 1], [0, 1, 2], [0, 1, 3, 3], [0, 1, 4, 6, 4], [0, 1, 5, 9, 10, 5], [0, 1, 6, 12, 16, 15, 6], [0, 1, 7, 15, 22, 25, 21, 7], [0, 1, 8, 18, 28, 35, 36, 28, 8]] | +| AntiDiag | [[0], [1], [2, 0], [3, 1], [4, 3, 0], [5, 6, 1], [6, 10, 4, 0], [7, 15, 9, 1], [8, 21, 16, 5, 0]] | +| AccTabl | [[0], [1, 1], [2, 3, 3], [3, 6, 7, 7], [4, 10, 14, 15, 15], [5, 15, 24, 29, 30, 30], [6, 21, 37, 49, 55, 56, 56], [7, 28, 53, 75, 90, 97, 98, 98], [8, 36, 72, 107, 135, 153, 161, 162, 162]] | +| RevAccTabl | [[0], [1, 1], [3, 3, 2], [7, 7, 6, 3], [15, 15, 14, 10, 4], [30, 30, 29, 24, 15, 5], [56, 56, 55, 49, 37, 21, 6], [98, 98, 97, 90, 75, 53, 28, 7], [162, 162, 161, 153, 135, 107, 72, 36, 8]] | +| AccRevTabl | [[0], [0, 1], [0, 1, 3], [0, 1, 4, 7], [0, 1, 5, 11, 15], [0, 1, 6, 15, 25, 30], [0, 1, 7, 19, 35, 50, 56], [0, 1, 8, 23, 45, 70, 91, 98], [0, 1, 9, 27, 55, 90, 126, 154, 162]] | + +Polygonal:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [0, 1, 3, 7, 15, 30, 56, 98, 162] | +| EvenSum | [0, 1, 2, 4, 8, 15, 28, 48, 80] | +| OddSum | [0, 0, 1, 3, 7, 15, 28, 50, 82] | +| AltSum | [0, 1, 1, 1, 1, 0, 0, -2, -2] | +| AccSum | [0, 2, 8, 23, 58, 133, 280, 546, 996] | +| AccRevSum | [0, 1, 4, 12, 32, 77, 168, 336, 624] | +| AntiDiagSum | [0, 1, 2, 4, 7, 12, 20, 32, 50] | + +Polygonal:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 12, 90, 240, 11550, 2520] | +| RowGcd | [1, 1, 2, 3, 2, 1, 1, 1, 1] | +| RowMax | [0, 1, 2, 3, 6, 10, 16, 25, 36] | +| ColMiddle | [0, 1, 1, 3, 4, 9, 12, 22, 28] | +| ColECenter | [0, 1, 4, 12, 28] | +| ColOCenter | [1, 3, 9, 22] | +| ColLeft | [0, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColRight | [0, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 7, 31, 107, 308, 770, 1722] | +| TransNat0 | [0, 0, 1, 5, 17, 47, 112, 238, 462] | +| TransNat1 | [0, 1, 4, 12, 32, 77, 168, 336, 624] | + +Polygonal:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ PolygonalRev Diagonals as rows | DiagRow7 | [7, 28, 49, 70, 91, 112, 133, 154, 175]| | DiagRow8 | [8, 36, 64, 92, 120, 148, 176, 204, 232]| -PolygonalRev Diagonals as columns +Polygonal:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | @@ -188,7 +180,7 @@ PolygonalRev Diagonals as columns | DiagCol7 | [0, 1, 9, 24, 46, 75, 111, 154, 204] | | DiagCol8 | [0, 1, 10, 27, 52, 85, 126, 175, 232] | -PolygonalRev Polynomial values as rows +Polygonal:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -201,7 +193,7 @@ PolygonalRev Polynomial values as rows | PolyRow7 | [7, 98, 853, 4534, 17003, 50362, 126313, 280238, 565999] | | PolyRow8 | [8, 162, 2152, 16022, 77688, 282298, 838472, 2148462, 4919272] | -PolygonalRev Polynomial values as columns +Polygonal:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [0, 1, 2, 3, 4, 5, 6, 7, 8] | diff --git a/data/md/PowLaguerre.md b/data/md/PowLaguerre.md index 26fd1da..3a22674 100644 --- a/data/md/PowLaguerre.md +++ b/data/md/PowLaguerre.md @@ -1,5 +1,5 @@ # PowLaguerre -['A021012', 'A196347'] +['A196347', 'A021012'] PowLaguerre Triangle view | Row | Seq | @@ -14,45 +14,41 @@ PowLaguerre Triangle view | Row7 | [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040] | | Row8 | [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -PowLaguerre Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -| TriRev | [1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -| TriAcc | [1, 1, 2, 2, 6, 8, 6, 24, 42, 48, 24, 120, 264, 360, 384, 120, 720, 1920, 3120, 3720, 3840, 720, 5040, 15840, 30240, 41040, 45360, 46080, 5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120, 40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 6, 2, 48, 42, 24, 6, 384, 360, 264, 120, 24, 3840, 3720, 3120, 1920, 720, 120, 46080, 45360, 41040, 30240, 15840, 5040, 720, 645120, 640080, 604800, 498960, 322560, 146160, 40320, 5040, 10321920, 10281600, 9959040, 8830080, 6572160, 3749760, 1491840, 362880, 40320] | -| TriAccRev | [1, 1, 2, 2, 6, 8, 6, 24, 42, 48, 24, 120, 264, 360, 384, 120, 720, 1920, 3120, 3720, 3840, 720, 5040, 15840, 30240, 41040, 45360, 46080, 5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120, 40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920] | -| TriDiag | [1, 1, 2, 1, 6, 4, 24, 18, 2, 120, 96, 18, 720, 600, 144, 6, 5040, 4320, 1200, 96] | +PowLaguerre Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 4, 2], [6, 18, 18, 6], [24, 96, 144, 96, 24], [120, 600, 1200, 1200, 600, 120], [720, 4320, 10800, 14400, 10800, 4320, 720], [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040], [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320]] | +| RevTabl | [[1], [1, 1], [2, 4, 2], [6, 18, 18, 6], [24, 96, 144, 96, 24], [120, 600, 1200, 1200, 600, 120], [720, 4320, 10800, 14400, 10800, 4320, 720], [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040], [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 4], [24, 18, 2], [120, 96, 18], [720, 600, 144, 6], [5040, 4320, 1200, 96], [40320, 35280, 10800, 1200, 24]] | +| AccTabl | [[1], [1, 2], [2, 6, 8], [6, 24, 42, 48], [24, 120, 264, 360, 384], [120, 720, 1920, 3120, 3720, 3840], [720, 5040, 15840, 30240, 41040, 45360, 46080], [5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120], [40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 6, 2], [48, 42, 24, 6], [384, 360, 264, 120, 24], [3840, 3720, 3120, 1920, 720, 120], [46080, 45360, 41040, 30240, 15840, 5040, 720], [645120, 640080, 604800, 498960, 322560, 146160, 40320, 5040], [10321920, 10281600, 9959040, 8830080, 6572160, 3749760, 1491840, 362880, 40320]] | +| AccRevTabl | [[1], [1, 2], [2, 6, 8], [6, 24, 42, 48], [24, 120, 264, 360, 384], [120, 720, 1920, 3120, 3720, 3840], [720, 5040, 15840, 30240, 41040, 45360, 46080], [5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120], [40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920]] | PowLaguerre Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| RevAccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| DiagSum | [1, 1, 3, 10, 44, 234, 1470, 10656] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AntiDiagSum | [1, 1, 3, 10, 44, 234, 1470, 10656, 87624] | PowLaguerre Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600] | -| RowGcd | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowMax | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | -| Middle | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | -| Central | [1, 4, 144, 14400, 2822400] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 18, 162, 1944, 29160, 524880, 11022480, 264539520] | -| NegHalf | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| Bin | [1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400] | -| InvBin | [1, 0, -4, 0, 144, 0, -14400, 0, 2822400] | -| TransSqrs | [0, 1, 12, 144, 1920, 28800, 483840, 9031680, 185794560] | -| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | -| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600] | +| RowGcd | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowMax | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | +| ColMiddle | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | +| ColECenter | [1, 4, 144, 14400, 2822400] | +| ColOCenter | [1, 18, 1200, 176400] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 12, 144, 1920, 28800, 483840, 9031680, 185794560] | +| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | +| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | PowLaguerre Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ PowLaguerre Polynomial values as columns | PolyCol7 | [1, 8, 128, 3072, 98304, 3932160, 188743680, 10569646080, 676457349120] | | PolyCol8 | [1, 9, 162, 4374, 157464, 7085880, 382637520, 24106163760, 1735643790720] | -# PowLaguerreRev -True +# PowLaguerre:Rev +[] -PowLaguerreRev Triangle view +PowLaguerre:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ PowLaguerreRev Triangle view | Row7 | [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040] | | Row8 | [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -PowLaguerreRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -| TriRev | [1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320] | -| TriAcc | [1, 1, 2, 2, 6, 8, 6, 24, 42, 48, 24, 120, 264, 360, 384, 120, 720, 1920, 3120, 3720, 3840, 720, 5040, 15840, 30240, 41040, 45360, 46080, 5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120, 40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 6, 2, 48, 42, 24, 6, 384, 360, 264, 120, 24, 3840, 3720, 3120, 1920, 720, 120, 46080, 45360, 41040, 30240, 15840, 5040, 720, 645120, 640080, 604800, 498960, 322560, 146160, 40320, 5040, 10321920, 10281600, 9959040, 8830080, 6572160, 3749760, 1491840, 362880, 40320] | -| TriAccRev | [1, 1, 2, 2, 6, 8, 6, 24, 42, 48, 24, 120, 264, 360, 384, 120, 720, 1920, 3120, 3720, 3840, 720, 5040, 15840, 30240, 41040, 45360, 46080, 5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120, 40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920] | -| TriDiag | [1, 1, 2, 1, 6, 4, 24, 18, 2, 120, 96, 18, 720, 600, 144, 6, 5040, 4320, 1200, 96] | - -PowLaguerreRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| RevAccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | -| DiagSum | [1, 1, 3, 10, 44, 234, 1470, 10656] | - -PowLaguerreRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600] | -| RowGcd | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowMax | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | -| Middle | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | -| Central | [1, 4, 144, 14400, 2822400] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 18, 162, 1944, 29160, 524880, 11022480, 264539520] | -| NegHalf | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| Bin | [1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400] | -| InvBin | [1, 0, -4, 0, 144, 0, -14400, 0, 2822400] | -| TransSqrs | [0, 1, 12, 144, 1920, 28800, 483840, 9031680, 185794560] | -| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | -| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | - -PowLaguerreRev Diagonals as rows +PowLaguerre:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 4, 2], [6, 18, 18, 6], [24, 96, 144, 96, 24], [120, 600, 1200, 1200, 600, 120], [720, 4320, 10800, 14400, 10800, 4320, 720], [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040], [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320]] | +| RevTabl | [[1], [1, 1], [2, 4, 2], [6, 18, 18, 6], [24, 96, 144, 96, 24], [120, 600, 1200, 1200, 600, 120], [720, 4320, 10800, 14400, 10800, 4320, 720], [5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040], [40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 4], [24, 18, 2], [120, 96, 18], [720, 600, 144, 6], [5040, 4320, 1200, 96], [40320, 35280, 10800, 1200, 24]] | +| AccTabl | [[1], [1, 2], [2, 6, 8], [6, 24, 42, 48], [24, 120, 264, 360, 384], [120, 720, 1920, 3120, 3720, 3840], [720, 5040, 15840, 30240, 41040, 45360, 46080], [5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120], [40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 6, 2], [48, 42, 24, 6], [384, 360, 264, 120, 24], [3840, 3720, 3120, 1920, 720, 120], [46080, 45360, 41040, 30240, 15840, 5040, 720], [645120, 640080, 604800, 498960, 322560, 146160, 40320, 5040], [10321920, 10281600, 9959040, 8830080, 6572160, 3749760, 1491840, 362880, 40320]] | +| AccRevTabl | [[1], [1, 2], [2, 6, 8], [6, 24, 42, 48], [24, 120, 264, 360, 384], [120, 720, 1920, 3120, 3720, 3840], [720, 5040, 15840, 30240, 41040, 45360, 46080], [5040, 40320, 146160, 322560, 498960, 604800, 640080, 645120], [40320, 362880, 1491840, 3749760, 6572160, 8830080, 9959040, 10281600, 10321920]] | + +PowLaguerre:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AccRevSum | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | +| AntiDiagSum | [1, 1, 3, 10, 44, 234, 1470, 10656, 87624] | + +PowLaguerre:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600] | +| RowGcd | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowMax | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | +| ColMiddle | [1, 1, 4, 18, 144, 1200, 14400, 176400, 2822400] | +| ColECenter | [1, 4, 144, 14400, 2822400] | +| ColOCenter | [1, 18, 1200, 176400] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 12, 144, 1920, 28800, 483840, 9031680, 185794560] | +| TransNat0 | [0, 1, 8, 72, 768, 9600, 138240, 2257920, 41287680] | +| TransNat1 | [1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600] | + +PowLaguerre:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| @@ -175,7 +167,7 @@ PowLaguerreRev Diagonals as rows | DiagRow7 | [5040, 322560, 13063680, 435456000, 13172544000, 379369267200, 10685567692800, 299195895398400, 8414884558080000]| | DiagRow8 | [40320, 3265920, 163296000, 6586272000, 237105792000, 8014175769600, 261796408473600, 8414884558080000, 269276305858560000]| -PowLaguerreRev Diagonals as columns +PowLaguerre:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -188,7 +180,7 @@ PowLaguerreRev Diagonals as columns | DiagCol7 | [5040, 322560, 13063680, 435456000, 13172544000, 379369267200, 10685567692800, 299195895398400, 8414884558080000] | | DiagCol8 | [40320, 3265920, 163296000, 6586272000, 237105792000, 8014175769600, 261796408473600, 8414884558080000, 269276305858560000] | -PowLaguerreRev Polynomial values as rows +PowLaguerre:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ PowLaguerreRev Polynomial values as rows | PolyRow7 | [5040, 645120, 11022480, 82575360, 393750000, 1410877440, 4150656720, 10569646080, 24106163760] | | PolyRow8 | [40320, 10321920, 264539520, 2642411520, 15750000000, 67722117120, 232436776320, 676457349120, 1735643790720] | -PowLaguerreRev Polynomial values as columns +PowLaguerre:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | diff --git a/data/md/Rencontres.md b/data/md/Rencontres.md index fd76d4f..649840c 100644 --- a/data/md/Rencontres.md +++ b/data/md/Rencontres.md @@ -14,45 +14,41 @@ Rencontres Triangle view | Row7 | [1854, 1855, 924, 315, 70, 21, 0, 1] | | Row8 | [14833, 14832, 7420, 2464, 630, 112, 28, 0, 1] | -Rencontres Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 9, 8, 6, 0, 1, 44, 45, 20, 10, 0, 1, 265, 264, 135, 40, 15, 0, 1, 1854, 1855, 924, 315, 70, 21, 0, 1, 14833, 14832, 7420, 2464, 630, 112, 28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 6, 8, 9, 1, 0, 10, 20, 45, 44, 1, 0, 15, 40, 135, 264, 265, 1, 0, 21, 70, 315, 924, 1855, 1854, 1, 0, 28, 112, 630, 2464, 7420, 14832, 14833] | -| TriAcc | [1, 0, 1, 1, 1, 2, 2, 5, 5, 6, 9, 17, 23, 23, 24, 44, 89, 109, 119, 119, 120, 265, 529, 664, 704, 719, 719, 720, 1854, 3709, 4633, 4948, 5018, 5039, 5039, 5040, 14833, 29665, 37085, 39549, 40179, 40291, 40319, 40319, 40320] | -| TriRevAcc | [1, 1, 0, 2, 1, 1, 6, 5, 5, 2, 24, 23, 23, 17, 9, 120, 119, 119, 109, 89, 44, 720, 719, 719, 704, 664, 529, 265, 5040, 5039, 5039, 5018, 4948, 4633, 3709, 1854, 40320, 40319, 40319, 40291, 40179, 39549, 37085, 29665, 14833] | -| TriAccRev | [1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 7, 15, 24, 1, 1, 11, 31, 76, 120, 1, 1, 16, 56, 191, 455, 720, 1, 1, 22, 92, 407, 1331, 3186, 5040, 1, 1, 29, 141, 771, 3235, 10655, 25487, 40320] | -| TriDiag | [1, 0, 1, 1, 2, 0, 9, 3, 1, 44, 8, 0, 265, 45, 6, 1, 1854, 264, 20, 0] | +Rencontres Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 0, 1], [2, 3, 0, 1], [9, 8, 6, 0, 1], [44, 45, 20, 10, 0, 1], [265, 264, 135, 40, 15, 0, 1], [1854, 1855, 924, 315, 70, 21, 0, 1], [14833, 14832, 7420, 2464, 630, 112, 28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 2], [1, 0, 6, 8, 9], [1, 0, 10, 20, 45, 44], [1, 0, 15, 40, 135, 264, 265], [1, 0, 21, 70, 315, 924, 1855, 1854], [1, 0, 28, 112, 630, 2464, 7420, 14832, 14833]] | +| AntiDiag | [[1], [0], [1, 1], [2, 0], [9, 3, 1], [44, 8, 0], [265, 45, 6, 1], [1854, 264, 20, 0], [14833, 1855, 135, 10, 1]] | +| AccTabl | [[1], [0, 1], [1, 1, 2], [2, 5, 5, 6], [9, 17, 23, 23, 24], [44, 89, 109, 119, 119, 120], [265, 529, 664, 704, 719, 719, 720], [1854, 3709, 4633, 4948, 5018, 5039, 5039, 5040], [14833, 29665, 37085, 39549, 40179, 40291, 40319, 40319, 40320]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 1], [6, 5, 5, 2], [24, 23, 23, 17, 9], [120, 119, 119, 109, 89, 44], [720, 719, 719, 704, 664, 529, 265], [5040, 5039, 5039, 5018, 4948, 4633, 3709, 1854], [40320, 40319, 40319, 40291, 40179, 39549, 37085, 29665, 14833]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 6], [1, 1, 7, 15, 24], [1, 1, 11, 31, 76, 120], [1, 1, 16, 56, 191, 455, 720], [1, 1, 22, 92, 407, 1331, 3186, 5040], [1, 1, 29, 141, 771, 3235, 10655, 25487, 40320]] | Rencontres Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 0, 2, 2, 16, 64, 416, 2848, 22912] | -| OddSum | [0, 1, 0, 4, 8, 56, 304, 2192, 17408] | -| AltSum | [1, -1, 2, -2, 8, 8, 112, 656, 5504] | -| AccSum | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | -| AccRevSum | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | -| RevAccSum | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | -| DiagSum | [1, 0, 2, 2, 13, 52, 317, 2138] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 0, 2, 2, 16, 64, 416, 2848, 22912] | +| OddSum | [0, 1, 0, 4, 8, 56, 304, 2192, 17408] | +| AltSum | [1, -1, 2, -2, 8, 8, 112, 656, 5504] | +| AccSum | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | +| AccRevSum | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | +| AntiDiagSum | [1, 0, 2, 2, 13, 52, 317, 2138, 16834] | Rencontres Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 72, 1980, 629640, 75661740, 1282617816480] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 9, 45, 265, 1855, 14833] | -| Middle | [1, 0, 0, 3, 6, 20, 40, 315, 630] | -| Central | [1, 0, 6, 40, 630] | -| LeftSide | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561] | -| NegHalf | [1, 1, 5, -3, 105, -807, 10413, -143595, 2304081] | -| Bin | [1, 1, 2, 12, 78, 570, 4900, 48160, 530390] | -| InvBin | [1, 1, 2, 8, 14, 82, 132, 744, 4566] | -| TransSqrs | [0, 1, 4, 12, 48, 240, 1440, 10080, 80640] | -| TransNat0 | [0, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| TransNat1 | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 72, 1980, 629640, 75661740, 1282617816480] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 9, 45, 265, 1855, 14833] | +| ColMiddle | [1, 0, 0, 3, 6, 20, 40, 315, 630] | +| ColECenter | [1, 0, 6, 40, 630] | +| ColOCenter | [0, 3, 20, 315] | +| ColLeft | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 12, 48, 240, 1440, 10080, 80640] | +| TransNat0 | [0, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransNat1 | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | Rencontres Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Rencontres Polynomial values as columns | PolyCol7 | [1, 7, 50, 366, 2760, 21576, 176112, 1512720, 13781376] | | PolyCol8 | [1, 8, 65, 538, 4553, 39572, 355081, 3309110, 32237681] | -# RencontresInv -True +# Rencontres:Inv +[] -RencontresInv Triangle view +Rencontres:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ RencontresInv Triangle view | Row7 | [-6, -35, -84, -105, -70, -21, 0, 1] | | Row8 | [-7, -48, -140, -224, -210, -112, -28, 0, 1] | -RencontresInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, -1, 0, 1, -2, -3, 0, 1, -3, -8, -6, 0, 1, -4, -15, -20, -10, 0, 1, -5, -24, -45, -40, -15, 0, 1, -6, -35, -84, -105, -70, -21, 0, 1, -7, -48, -140, -224, -210, -112, -28, 0, 1] | -| TriRev | [1, 1, 0, 1, 0, -1, 1, 0, -3, -2, 1, 0, -6, -8, -3, 1, 0, -10, -20, -15, -4, 1, 0, -15, -40, -45, -24, -5, 1, 0, -21, -70, -105, -84, -35, -6, 1, 0, -28, -112, -210, -224, -140, -48, -7] | -| TriAcc | [1, 0, 1, -1, -1, 0, -2, -5, -5, -4, -3, -11, -17, -17, -16, -4, -19, -39, -49, -49, -48, -5, -29, -74, -114, -129, -129, -128, -6, -41, -125, -230, -300, -321, -321, -320, -7, -55, -195, -419, -629, -741, -769, -769, -768] | -| TriRevAcc | [1, 1, 0, 0, -1, -1, -4, -5, -5, -2, -16, -17, -17, -11, -3, -48, -49, -49, -39, -19, -4, -128, -129, -129, -114, -74, -29, -5, -320, -321, -321, -300, -230, -125, -41, -6, -768, -769, -769, -741, -629, -419, -195, -55, -7] | -| TriAccRev | [1, 1, 1, 1, 1, 0, 1, 1, -2, -4, 1, 1, -5, -13, -16, 1, 1, -9, -29, -44, -48, 1, 1, -14, -54, -99, -123, -128, 1, 1, -20, -90, -195, -279, -314, -320, 1, 1, -27, -139, -349, -573, -713, -761, -768] | -| TriDiag | [1, 0, -1, 1, -2, 0, -3, -3, 1, -4, -8, 0, -5, -15, -6, 1, -6, -24, -20, 0] | - -RencontresInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -4, -16, -48, -128, -320, -768] | -| EvenSum | [1, 0, 0, -2, -8, -24, -64, -160, -384] | -| OddSum | [0, 1, 0, -2, -8, -24, -64, -160, -384] | -| AltSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | -| AccRevSum | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | -| RevAccSum | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | -| DiagSum | [1, 0, 0, -2, -5, -12, -25, -50] | - -RencontresInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 24, 60, 360, 420, 3360] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 8, 20, 45, 105, 224] | -| Middle | [1, 0, 0, -3, -6, -20, -40, -105, -210] | -| Central | [1, 0, -6, -40, -210] | -| LeftSide | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, -27, -135, -567, -2187, -8019, -28431] | -| NegHalf | [1, 1, -3, 5, -7, 9, -11, 13, -15] | -| Bin | [1, 1, 0, -10, -70, -378, -1848, -8580, -38610] | -| InvBin | [1, 1, 0, -6, -6, 30, 40, -140, -210] | -| TransSqrs | [0, 1, 4, 6, -16, -160, -768, -2912, -9728] | -| TransNat0 | [0, 1, 2, 0, -16, -80, -288, -896, -2560] | -| TransNat1 | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | - -RencontresInv Diagonals as rows +Rencontres:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [-1, 0, 1], [-2, -3, 0, 1], [-3, -8, -6, 0, 1], [-4, -15, -20, -10, 0, 1], [-5, -24, -45, -40, -15, 0, 1], [-6, -35, -84, -105, -70, -21, 0, 1], [-7, -48, -140, -224, -210, -112, -28, 0, 1]] | +| RevTabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, -2], [1, 0, -6, -8, -3], [1, 0, -10, -20, -15, -4], [1, 0, -15, -40, -45, -24, -5], [1, 0, -21, -70, -105, -84, -35, -6], [1, 0, -28, -112, -210, -224, -140, -48, -7]] | +| AntiDiag | [[1], [0], [-1, 1], [-2, 0], [-3, -3, 1], [-4, -8, 0], [-5, -15, -6, 1], [-6, -24, -20, 0], [-7, -35, -45, -10, 1]] | +| AccTabl | [[1], [0, 1], [-1, -1, 0], [-2, -5, -5, -4], [-3, -11, -17, -17, -16], [-4, -19, -39, -49, -49, -48], [-5, -29, -74, -114, -129, -129, -128], [-6, -41, -125, -230, -300, -321, -321, -320], [-7, -55, -195, -419, -629, -741, -769, -769, -768]] | +| RevAccTabl | [[1], [1, 0], [0, -1, -1], [-4, -5, -5, -2], [-16, -17, -17, -11, -3], [-48, -49, -49, -39, -19, -4], [-128, -129, -129, -114, -74, -29, -5], [-320, -321, -321, -300, -230, -125, -41, -6], [-768, -769, -769, -741, -629, -419, -195, -55, -7]] | +| AccRevTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -4], [1, 1, -5, -13, -16], [1, 1, -9, -29, -44, -48], [1, 1, -14, -54, -99, -123, -128], [1, 1, -20, -90, -195, -279, -314, -320], [1, 1, -27, -139, -349, -573, -713, -761, -768]] | + +Rencontres:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -4, -16, -48, -128, -320, -768] | +| EvenSum | [1, 0, 0, -2, -8, -24, -64, -160, -384] | +| OddSum | [0, 1, 0, -2, -8, -24, -64, -160, -384] | +| AltSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | +| AccRevSum | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | +| AntiDiagSum | [1, 0, 0, -2, -5, -12, -25, -50, -96] | + +Rencontres:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 24, 60, 360, 420, 3360] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 8, 20, 45, 105, 224] | +| ColMiddle | [1, 0, 0, -3, -6, -20, -40, -105, -210] | +| ColECenter | [1, 0, -6, -40, -210] | +| ColOCenter | [0, -3, -20, -105] | +| ColLeft | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 6, -16, -160, -768, -2912, -9728] | +| TransNat0 | [0, 1, 2, 0, -16, -80, -288, -896, -2560] | +| TransNat1 | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | + +Rencontres:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ RencontresInv Diagonals as rows | DiagRow7 | [-6, -48, -216, -720, -1980, -4752, -10296, -20592, -38610]| | DiagRow8 | [-7, -63, -315, -1155, -3465, -9009, -21021, -45045, -90090]| -RencontresInv Diagonals as columns +Rencontres:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | @@ -188,7 +180,7 @@ RencontresInv Diagonals as columns | DiagCol7 | [1, 0, -36, -240, -990, -3168, -8580, -20592, -45045] | | DiagCol8 | [1, 0, -45, -330, -1485, -5148, -15015, -38610, -90090] | -RencontresInv Polynomial values as rows +Rencontres:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ RencontresInv Polynomial values as rows | PolyRow7 | [-6, -320, -2916, -12288, -31250, -46656, 0, 262144, 1062882] | | PolyRow8 | [-7, -768, -10935, -65536, -234375, -559872, -823543, 0, 4782969] | -RencontresInv Polynomial values as columns +Rencontres:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | @@ -214,10 +206,10 @@ RencontresInv Polynomial values as columns | PolyCol7 | [1, 7, 48, 320, 2048, 12288, 65536, 262144, 0] | | PolyCol8 | [1, 8, 63, 486, 3645, 26244, 177147, 1062882, 4782969] | -# RencontresRev -True +# Rencontres:Rev +[] -RencontresRev Triangle view +Rencontres:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ RencontresRev Triangle view | Row7 | [1, 0, 21, 70, 315, 924, 1855, 1854] | | Row8 | [1, 0, 28, 112, 630, 2464, 7420, 14832, 14833] | -RencontresRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 6, 8, 9, 1, 0, 10, 20, 45, 44, 1, 0, 15, 40, 135, 264, 265, 1, 0, 21, 70, 315, 924, 1855, 1854, 1, 0, 28, 112, 630, 2464, 7420, 14832, 14833] | -| TriRev | [1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 9, 8, 6, 0, 1, 44, 45, 20, 10, 0, 1, 265, 264, 135, 40, 15, 0, 1, 1854, 1855, 924, 315, 70, 21, 0, 1, 14833, 14832, 7420, 2464, 630, 112, 28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 7, 15, 24, 1, 1, 11, 31, 76, 120, 1, 1, 16, 56, 191, 455, 720, 1, 1, 22, 92, 407, 1331, 3186, 5040, 1, 1, 29, 141, 771, 3235, 10655, 25487, 40320] | -| TriRevAcc | [1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 24, 15, 7, 1, 1, 120, 76, 31, 11, 1, 1, 720, 455, 191, 56, 16, 1, 1, 5040, 3186, 1331, 407, 92, 22, 1, 1, 40320, 25487, 10655, 3235, 771, 141, 29, 1, 1] | -| TriAccRev | [1, 0, 1, 1, 1, 2, 2, 5, 5, 6, 9, 17, 23, 23, 24, 44, 89, 109, 119, 119, 120, 265, 529, 664, 704, 719, 719, 720, 1854, 3709, 4633, 4948, 5018, 5039, 5039, 5040, 14833, 29665, 37085, 39549, 40179, 40291, 40319, 40319, 40320] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 3, 1, 0, 6, 2, 1, 0, 10, 8] | - -RencontresRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 1, 2, 4, 16, 56, 416, 2192, 22912] | -| OddSum | [0, 0, 0, 2, 8, 64, 304, 2848, 17408] | -| AltSum | [1, 1, 2, 2, 8, -8, 112, -656, 5504] | -| AccSum | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | -| AccRevSum | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | -| RevAccSum | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | -| DiagSum | [1, 1, 1, 1, 2, 4, 9, 19] | - -RencontresRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 72, 1980, 629640, 75661740, 1282617816480] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 9, 45, 265, 1855, 14833] | -| Middle | [1, 1, 0, 0, 6, 10, 40, 70, 630] | -| Central | [1, 0, 6, 40, 630] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| PosHalf | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| NegHalf | [1, -2, 5, -12, 33, -78, 261, -360, 3681] | -| Bin | [1, 1, 2, 12, 78, 570, 4900, 48160, 530390] | -| InvBin | [1, -1, 2, -8, 14, -82, 132, -744, 4566] | -| TransSqrs | [0, 0, 4, 30, 240, 2040, 18720, 186480, 2016000] | -| TransNat0 | [0, 0, 2, 12, 72, 480, 3600, 30240, 282240] | -| TransNat1 | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | - -RencontresRev Diagonals as rows +Rencontres:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, 1], [1, 0, 3, 2], [1, 0, 6, 8, 9], [1, 0, 10, 20, 45, 44], [1, 0, 15, 40, 135, 264, 265], [1, 0, 21, 70, 315, 924, 1855, 1854], [1, 0, 28, 112, 630, 2464, 7420, 14832, 14833]] | +| RevTabl | [[1], [0, 1], [1, 0, 1], [2, 3, 0, 1], [9, 8, 6, 0, 1], [44, 45, 20, 10, 0, 1], [265, 264, 135, 40, 15, 0, 1], [1854, 1855, 924, 315, 70, 21, 0, 1], [14833, 14832, 7420, 2464, 630, 112, 28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, 1], [1, 0, 3], [1, 0, 6, 2], [1, 0, 10, 8], [1, 0, 15, 20, 9]] | +| AccTabl | [[1], [1, 1], [1, 1, 2], [1, 1, 4, 6], [1, 1, 7, 15, 24], [1, 1, 11, 31, 76, 120], [1, 1, 16, 56, 191, 455, 720], [1, 1, 22, 92, 407, 1331, 3186, 5040], [1, 1, 29, 141, 771, 3235, 10655, 25487, 40320]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 1], [6, 4, 1, 1], [24, 15, 7, 1, 1], [120, 76, 31, 11, 1, 1], [720, 455, 191, 56, 16, 1, 1], [5040, 3186, 1331, 407, 92, 22, 1, 1], [40320, 25487, 10655, 3235, 771, 141, 29, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [1, 1, 2], [2, 5, 5, 6], [9, 17, 23, 23, 24], [44, 89, 109, 119, 119, 120], [265, 529, 664, 704, 719, 719, 720], [1854, 3709, 4633, 4948, 5018, 5039, 5039, 5040], [14833, 29665, 37085, 39549, 40179, 40291, 40319, 40319, 40320]] | + +Rencontres:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 1, 2, 4, 16, 56, 416, 2192, 22912] | +| OddSum | [0, 0, 0, 2, 8, 64, 304, 2848, 17408] | +| AltSum | [1, 1, 2, 2, 8, -8, 112, -656, 5504] | +| AccSum | [1, 2, 4, 12, 48, 240, 1440, 10080, 80640] | +| AccRevSum | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | +| AntiDiagSum | [1, 1, 1, 1, 2, 4, 9, 19, 45] | + +Rencontres:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 72, 1980, 629640, 75661740, 1282617816480] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 9, 45, 265, 1855, 14833] | +| ColMiddle | [1, 1, 0, 0, 6, 10, 40, 70, 630] | +| ColECenter | [1, 0, 6, 40, 630] | +| ColOCenter | [1, 0, 10, 70] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| TransSqrs | [0, 0, 4, 30, 240, 2040, 18720, 186480, 2016000] | +| TransNat0 | [0, 0, 2, 12, 72, 480, 3600, 30240, 282240] | +| TransNat1 | [1, 1, 4, 18, 96, 600, 4320, 35280, 322560] | + +Rencontres:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 2, 9, 44, 265, 1854, 14833]| @@ -283,7 +271,7 @@ RencontresRev Diagonals as rows | DiagRow7 | [1, 0, 36, 240, 2970, 34848, 454740, 6362928, 95450355]| | DiagRow8 | [1, 0, 45, 330, 4455, 56628, 795795, 11930490, 190900710]| -RencontresRev Diagonals as columns +Rencontres:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ RencontresRev Diagonals as columns | DiagCol7 | [1854, 14832, 66744, 222480, 611820, 1468368, 3181464, 6362928, 11930490] | | DiagCol8 | [14833, 133497, 667485, 2447445, 7342335, 19090071, 44543499, 95450355, 190900710] | -RencontresRev Polynomial values as rows +Rencontres:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ RencontresRev Polynomial values as rows | PolyRow7 | [1, 5040, 391285, 5659120, 39005649, 176921776, 613157365, 1761398640, 4406001985] | | PolyRow8 | [1, 40320, 6260561, 135819136, 1248187329, 7076936576, 29431944145, 98640003456, 281989891841] | -RencontresRev Polynomial values as columns +Rencontres:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ RencontresRev Polynomial values as columns | PolyCol7 | [1, 1, 50, 834, 24648, 854904, 35952624, 1761398640, 98640003456] | | PolyCol8 | [1, 1, 65, 1217, 41345, 1636993, 78693313, 4406001985, 281989891841] | -# RencontresInvRev -True +# Rencontres:Inv:Rev +[] -RencontresInvRev Triangle view +Rencontres:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ RencontresInvRev Triangle view | Row7 | [1, 0, -21, -70, -105, -84, -35, -6] | | Row8 | [1, 0, -28, -112, -210, -224, -140, -48, -7] | -RencontresInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 0, -1, 1, 0, -3, -2, 1, 0, -6, -8, -3, 1, 0, -10, -20, -15, -4, 1, 0, -15, -40, -45, -24, -5, 1, 0, -21, -70, -105, -84, -35, -6, 1, 0, -28, -112, -210, -224, -140, -48, -7] | -| TriRev | [1, 0, 1, -1, 0, 1, -2, -3, 0, 1, -3, -8, -6, 0, 1, -4, -15, -20, -10, 0, 1, -5, -24, -45, -40, -15, 0, 1, -6, -35, -84, -105, -70, -21, 0, 1, -7, -48, -140, -224, -210, -112, -28, 0, 1] | -| TriAcc | [1, 1, 1, 1, 1, 0, 1, 1, -2, -4, 1, 1, -5, -13, -16, 1, 1, -9, -29, -44, -48, 1, 1, -14, -54, -99, -123, -128, 1, 1, -20, -90, -195, -279, -314, -320, 1, 1, -27, -139, -349, -573, -713, -761, -768] | -| TriRevAcc | [1, 1, 1, 0, 1, 1, -4, -2, 1, 1, -16, -13, -5, 1, 1, -48, -44, -29, -9, 1, 1, -128, -123, -99, -54, -14, 1, 1, -320, -314, -279, -195, -90, -20, 1, 1, -768, -761, -713, -573, -349, -139, -27, 1, 1] | -| TriAccRev | [1, 0, 1, -1, -1, 0, -2, -5, -5, -4, -3, -11, -17, -17, -16, -4, -19, -39, -49, -49, -48, -5, -29, -74, -114, -129, -129, -128, -6, -41, -125, -230, -300, -321, -321, -320, -7, -55, -195, -419, -629, -741, -769, -769, -768] | -| TriDiag | [1, 1, 1, 0, 1, 0, 1, 0, -1, 1, 0, -3, 1, 0, -6, -2, 1, 0, -10, -8] | - -RencontresInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -4, -16, -48, -128, -320, -768] | -| EvenSum | [1, 1, 0, -2, -8, -24, -64, -160, -384] | -| OddSum | [0, 0, 0, -2, -8, -24, -64, -160, -384] | -| AltSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | -| AccRevSum | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | -| RevAccSum | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | -| DiagSum | [1, 1, 1, 1, 0, -2, -7, -17] | - -RencontresInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 24, 60, 360, 420, 3360] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 8, 20, 45, 105, 224] | -| Middle | [1, 1, 0, 0, -6, -10, -40, -70, -210] | -| Central | [1, 0, -6, -40, -210] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| PosHalf | [1, 2, 3, 0, -27, -162, -729, -2916, -10935] | -| NegHalf | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| Bin | [1, 1, 0, -10, -70, -378, -1848, -8580, -38610] | -| InvBin | [1, -1, 0, 6, -6, -30, 40, 140, -210] | -| TransSqrs | [0, 0, -4, -30, -144, -560, -1920, -6048, -17920] | -| TransNat0 | [0, 0, -2, -12, -48, -160, -480, -1344, -3584] | -| TransNat1 | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | - -RencontresInvRev Diagonals as rows +Rencontres:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 0, -1], [1, 0, -3, -2], [1, 0, -6, -8, -3], [1, 0, -10, -20, -15, -4], [1, 0, -15, -40, -45, -24, -5], [1, 0, -21, -70, -105, -84, -35, -6], [1, 0, -28, -112, -210, -224, -140, -48, -7]] | +| RevTabl | [[1], [0, 1], [-1, 0, 1], [-2, -3, 0, 1], [-3, -8, -6, 0, 1], [-4, -15, -20, -10, 0, 1], [-5, -24, -45, -40, -15, 0, 1], [-6, -35, -84, -105, -70, -21, 0, 1], [-7, -48, -140, -224, -210, -112, -28, 0, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 0], [1, 0, -1], [1, 0, -3], [1, 0, -6, -2], [1, 0, -10, -8], [1, 0, -15, -20, -3]] | +| AccTabl | [[1], [1, 1], [1, 1, 0], [1, 1, -2, -4], [1, 1, -5, -13, -16], [1, 1, -9, -29, -44, -48], [1, 1, -14, -54, -99, -123, -128], [1, 1, -20, -90, -195, -279, -314, -320], [1, 1, -27, -139, -349, -573, -713, -761, -768]] | +| RevAccTabl | [[1], [1, 1], [0, 1, 1], [-4, -2, 1, 1], [-16, -13, -5, 1, 1], [-48, -44, -29, -9, 1, 1], [-128, -123, -99, -54, -14, 1, 1], [-320, -314, -279, -195, -90, -20, 1, 1], [-768, -761, -713, -573, -349, -139, -27, 1, 1]] | +| AccRevTabl | [[1], [0, 1], [-1, -1, 0], [-2, -5, -5, -4], [-3, -11, -17, -17, -16], [-4, -19, -39, -49, -49, -48], [-5, -29, -74, -114, -129, -129, -128], [-6, -41, -125, -230, -300, -321, -321, -320], [-7, -55, -195, -419, -629, -741, -769, -769, -768]] | + +Rencontres:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -4, -16, -48, -128, -320, -768] | +| EvenSum | [1, 1, 0, -2, -8, -24, -64, -160, -384] | +| OddSum | [0, 0, 0, -2, -8, -24, -64, -160, -384] | +| AltSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 2, 2, -4, -32, -128, -416, -1216, -3328] | +| AccRevSum | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | +| AntiDiagSum | [1, 1, 1, 1, 0, -2, -7, -17, -37] | + +Rencontres:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 24, 60, 360, 420, 3360] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 8, 20, 45, 105, 224] | +| ColMiddle | [1, 1, 0, 0, -6, -10, -40, -70, -210] | +| ColECenter | [1, 0, -6, -40, -210] | +| ColOCenter | [1, 0, -10, -70] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| TransSqrs | [0, 0, -4, -30, -144, -560, -1920, -6048, -17920] | +| TransNat0 | [0, 0, -2, -12, -48, -160, -480, -1344, -3584] | +| TransNat1 | [1, 1, -2, -16, -64, -208, -608, -1664, -4352] | + +Rencontres:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, -1, -2, -3, -4, -5, -6, -7]| @@ -391,7 +375,7 @@ RencontresInvRev Diagonals as rows | DiagRow7 | [1, 0, -36, -240, -990, -3168, -8580, -20592, -45045]| | DiagRow8 | [1, 0, -45, -330, -1485, -5148, -15015, -38610, -90090]| -RencontresInvRev Diagonals as columns +Rencontres:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ RencontresInvRev Diagonals as columns | DiagCol7 | [-6, -48, -216, -720, -1980, -4752, -10296, -20592, -38610] | | DiagCol8 | [-7, -63, -315, -1155, -3465, -9009, -21021, -45045, -90090] | -RencontresInvRev Polynomial values as rows +Rencontres:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ RencontresInvRev Polynomial values as rows | PolyRow7 | [1, -320, -8019, -69632, -359375, -1353024, -4117715, -10747904, -24977727] | | PolyRow8 | [1, -768, -28431, -327680, -2109375, -9517824, -33765263, -100663296, -263063295] | -RencontresInvRev Polynomial values as columns +Rencontres:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/RisingFact.md b/data/md/RisingFact.md index f070a42..511a1dc 100644 --- a/data/md/RisingFact.md +++ b/data/md/RisingFact.md @@ -14,45 +14,41 @@ RisingFact Triangle view | Row7 | [5040, 5040, 2520, 840, 210, 42, 7, 1] | | Row8 | [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -RisingFact Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, 120, 120, 60, 20, 5, 1, 720, 720, 360, 120, 30, 6, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -| TriRev | [1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -| TriAcc | [1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601] | -| TriRevAcc | [1, 2, 1, 5, 4, 2, 16, 15, 12, 6, 65, 64, 60, 48, 24, 326, 325, 320, 300, 240, 120, 1957, 1956, 1950, 1920, 1800, 1440, 720, 13700, 13699, 13692, 13650, 13440, 12600, 10080, 5040, 109601, 109600, 109592, 109536, 109200, 107520, 100800, 80640, 40320] | -| TriAccRev | [1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601] | -| TriDiag | [1, 1, 2, 1, 6, 2, 24, 6, 1, 120, 24, 3, 720, 120, 12, 1, 5040, 720, 60, 4] | +RisingFact Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 2, 1], [6, 6, 3, 1], [24, 24, 12, 4, 1], [120, 120, 60, 20, 5, 1], [720, 720, 360, 120, 30, 6, 1], [5040, 5040, 2520, 840, 210, 42, 7, 1], [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1]] | +| RevTabl | [[1], [1, 1], [1, 2, 2], [1, 3, 6, 6], [1, 4, 12, 24, 24], [1, 5, 20, 60, 120, 120], [1, 6, 30, 120, 360, 720, 720], [1, 7, 42, 210, 840, 2520, 5040, 5040], [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 2], [24, 6, 1], [120, 24, 3], [720, 120, 12, 1], [5040, 720, 60, 4], [40320, 5040, 360, 20, 1]] | +| AccTabl | [[1], [1, 2], [2, 4, 5], [6, 12, 15, 16], [24, 48, 60, 64, 65], [120, 240, 300, 320, 325, 326], [720, 1440, 1800, 1920, 1950, 1956, 1957], [5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700], [40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601]] | +| RevAccTabl | [[1], [2, 1], [5, 4, 2], [16, 15, 12, 6], [65, 64, 60, 48, 24], [326, 325, 320, 300, 240, 120], [1957, 1956, 1950, 1920, 1800, 1440, 720], [13700, 13699, 13692, 13650, 13440, 12600, 10080, 5040], [109601, 109600, 109592, 109536, 109200, 107520, 100800, 80640, 40320]] | +| AccRevTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 10, 16], [1, 5, 17, 41, 65], [1, 6, 26, 86, 206, 326], [1, 7, 37, 157, 517, 1237, 1957], [1, 8, 50, 260, 1100, 3620, 8660, 13700], [1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601]] | RisingFact Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 1, 3, 9, 37, 185, 1111, 7777, 62217] | -| OddSum | [0, 1, 2, 7, 28, 141, 846, 5923, 47384] | -| AltSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| AccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| AccRevSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| RevAccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| DiagSum | [1, 1, 3, 8, 31, 147, 853, 5824] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 1, 3, 9, 37, 185, 1111, 7777, 62217] | +| OddSum | [0, 1, 2, 7, 28, 141, 846, 5923, 47384] | +| AltSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| AccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | +| AccRevSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | +| AntiDiagSum | [1, 1, 3, 8, 31, 147, 853, 5824, 45741] | RisingFact Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 1, 2, 6, 12, 60, 120, 840, 1680] | -| Central | [1, 2, 12, 120, 1680] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969] | -| NegHalf | [1, -1, 5, -29, 233, -2329, 27949, -391285, 6260561] | -| Bin | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| InvBin | [1, 0, -1, 4, -15, 56, -185, 204, 6209] | -| TransSqrs | [0, 1, 6, 27, 124, 645, 3906, 27391, 219192] | -| TransNat0 | [0, 1, 4, 15, 64, 325, 1956, 13699, 109600] | -| TransNat1 | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 1, 2, 6, 12, 60, 120, 840, 1680] | +| ColECenter | [1, 2, 12, 120, 1680] | +| ColOCenter | [1, 6, 60, 840] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 27, 124, 645, 3906, 27391, 219192] | +| TransNat0 | [0, 1, 4, 15, 64, 325, 1956, 13699, 109600] | +| TransNat1 | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | RisingFact Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ RisingFact Polynomial values as columns | PolyCol7 | [1, 8, 65, 538, 4553, 39572, 355081, 3309110, 32237681] | | PolyCol8 | [1, 9, 82, 758, 7128, 68408, 672592, 6805296, 71219584] | -# RisingFactInv -True +# RisingFact:Inv +[] -RisingFactInv Triangle view +RisingFact:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ RisingFactInv Triangle view | Row7 | [0, 0, 0, 0, 0, 0, -7, 1] | | Row8 | [0, 0, 0, 0, 0, 0, 0, -8, 1] | -RisingFactInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 0, -2, 1, 0, 0, -3, 1, 0, 0, 0, -4, 1, 0, 0, 0, 0, -5, 1, 0, 0, 0, 0, 0, -6, 1, 0, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 0, 0, 0, -8, 1] | -| TriRev | [1, 1, -1, 1, -2, 0, 1, -3, 0, 0, 1, -4, 0, 0, 0, 1, -5, 0, 0, 0, 0, 1, -6, 0, 0, 0, 0, 0, 1, -7, 0, 0, 0, 0, 0, 0, 1, -8, 0, 0, 0, 0, 0, 0, 0] | -| TriAcc | [1, -1, 0, 0, -2, -1, 0, 0, -3, -2, 0, 0, 0, -4, -3, 0, 0, 0, 0, -5, -4, 0, 0, 0, 0, 0, -6, -5, 0, 0, 0, 0, 0, 0, -7, -6, 0, 0, 0, 0, 0, 0, 0, -8, -7] | -| TriRevAcc | [1, 0, -1, -1, -2, 0, -2, -3, 0, 0, -3, -4, 0, 0, 0, -4, -5, 0, 0, 0, 0, -5, -6, 0, 0, 0, 0, 0, -6, -7, 0, 0, 0, 0, 0, 0, -7, -8, 0, 0, 0, 0, 0, 0, 0] | -| TriAccRev | [1, 1, 0, 1, -1, -1, 1, -2, -2, -2, 1, -3, -3, -3, -3, 1, -4, -4, -4, -4, -4, 1, -5, -5, -5, -5, -5, -5, 1, -6, -6, -6, -6, -6, -6, -6, 1, -7, -7, -7, -7, -7, -7, -7, -7] | -| TriDiag | [1, -1, 0, 1, 0, -2, 0, 0, 1, 0, 0, -3, 0, 0, 0, 1, 0, 0, 0, -4] | - -RisingFactInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| EvenSum | [1, -1, 1, -3, 1, -5, 1, -7, 1] | -| OddSum | [0, 1, -2, 1, -4, 1, -6, 1, -8] | -| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| AccSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | -| AccRevSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | -| RevAccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| DiagSum | [1, -1, 1, -2, 1, -3, 1, -4] | - -RisingFactInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, -3, -4, -5, -6, -7, -8] | -| RowGcd | [1, 1, -2, -3, -4, -5, -6, -7, -8] | -| RowMax | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| Middle | [1, -1, -2, 0, 0, 0, 0, 0, 0] | -| Central | [1, -2, 0, 0, 0] | -| LeftSide | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, -5, -7, -9, -11, -13, -15] | -| NegHalf | [1, 3, 5, 7, 9, 11, 13, 15, 17] | -| Bin | [1, 0, -3, -8, -15, -24, -35, -48, -63] | -| InvBin | [1, 2, 5, 10, 17, 26, 37, 50, 65] | -| TransSqrs | [0, 1, 2, -3, -20, -55, -114, -203, -328] | -| TransNat0 | [0, 1, 0, -3, -8, -15, -24, -35, -48] | -| TransNat1 | [1, 1, -1, -5, -11, -19, -29, -41, -55] | - -RisingFactInv Diagonals as rows +RisingFact:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [0, -2, 1], [0, 0, -3, 1], [0, 0, 0, -4, 1], [0, 0, 0, 0, -5, 1], [0, 0, 0, 0, 0, -6, 1], [0, 0, 0, 0, 0, 0, -7, 1], [0, 0, 0, 0, 0, 0, 0, -8, 1]] | +| RevTabl | [[1], [1, -1], [1, -2, 0], [1, -3, 0, 0], [1, -4, 0, 0, 0], [1, -5, 0, 0, 0, 0], [1, -6, 0, 0, 0, 0, 0], [1, -7, 0, 0, 0, 0, 0, 0], [1, -8, 0, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [0, 1], [0, -2], [0, 0, 1], [0, 0, -3], [0, 0, 0, 1], [0, 0, 0, -4], [0, 0, 0, 0, 1]] | +| AccTabl | [[1], [-1, 0], [0, -2, -1], [0, 0, -3, -2], [0, 0, 0, -4, -3], [0, 0, 0, 0, -5, -4], [0, 0, 0, 0, 0, -6, -5], [0, 0, 0, 0, 0, 0, -7, -6], [0, 0, 0, 0, 0, 0, 0, -8, -7]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 0], [-2, -3, 0, 0], [-3, -4, 0, 0, 0], [-4, -5, 0, 0, 0, 0], [-5, -6, 0, 0, 0, 0, 0], [-6, -7, 0, 0, 0, 0, 0, 0], [-7, -8, 0, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, -1, -1], [1, -2, -2, -2], [1, -3, -3, -3, -3], [1, -4, -4, -4, -4, -4], [1, -5, -5, -5, -5, -5, -5], [1, -6, -6, -6, -6, -6, -6, -6], [1, -7, -7, -7, -7, -7, -7, -7, -7]] | + +RisingFact:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| EvenSum | [1, -1, 1, -3, 1, -5, 1, -7, 1] | +| OddSum | [0, 1, -2, 1, -4, 1, -6, 1, -8] | +| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| AccSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | +| AccRevSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | +| AntiDiagSum | [1, -1, 1, -2, 1, -3, 1, -4, 1] | + +RisingFact:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColMiddle | [1, -1, -2, 0, 0, 0, 0, 0, 0] | +| ColECenter | [1, -2, 0, 0, 0] | +| ColOCenter | [-1, 0, 0, 0] | +| ColLeft | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -3, -20, -55, -114, -203, -328] | +| TransNat0 | [0, 1, 0, -3, -8, -15, -24, -35, -48] | +| TransNat1 | [1, 1, -1, -5, -11, -19, -29, -41, -55] | + +RisingFact:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ RisingFactInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| -RisingFactInv Diagonals as columns +RisingFact:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ RisingFactInv Diagonals as columns | DiagCol7 | [1, -8, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, -9, 0, 0, 0, 0, 0, 0, 0] | -RisingFactInv Polynomial values as rows +RisingFact:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ RisingFactInv Polynomial values as rows | PolyRow7 | [0, -6, -320, -2916, -12288, -31250, -46656, 0, 262144] | | PolyRow8 | [0, -7, -768, -10935, -65536, -234375, -559872, -823543, 0] | -RisingFactInv Polynomial values as columns +RisingFact:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ RisingFactInv Polynomial values as columns | PolyCol7 | [1, 6, 35, 196, 1029, 4802, 16807, 0, -823543] | | PolyCol8 | [1, 7, 48, 320, 2048, 12288, 65536, 262144, 0] | -# RisingFactRev -True +# RisingFact:Rev +[] -RisingFactRev Triangle view +RisingFact:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ RisingFactRev Triangle view | Row7 | [1, 7, 42, 210, 840, 2520, 5040, 5040] | | Row8 | [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -RisingFactRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -| TriRev | [1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, 120, 120, 60, 20, 5, 1, 720, 720, 360, 120, 30, 6, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1] | -| TriAcc | [1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601] | -| TriRevAcc | [1, 2, 1, 5, 3, 1, 16, 10, 4, 1, 65, 41, 17, 5, 1, 326, 206, 86, 26, 6, 1, 1957, 1237, 517, 157, 37, 7, 1, 13700, 8660, 3620, 1100, 260, 50, 8, 1, 109601, 69281, 28961, 8801, 2081, 401, 65, 9, 1] | -| TriAccRev | [1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601] | -| TriDiag | [1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 6, 1, 5, 12, 6, 1, 6, 20, 24] | - -RisingFactRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 1, 3, 7, 37, 141, 1111, 5923, 62217] | -| OddSum | [0, 1, 2, 9, 28, 185, 846, 7777, 47384] | -| AltSum | [1, 0, 1, -2, 9, -44, 265, -1854, 14833] | -| AccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| AccRevSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | -| RevAccSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | -| DiagSum | [1, 1, 2, 3, 6, 11, 24, 51] | - -RisingFactRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| Middle | [1, 1, 2, 3, 12, 20, 120, 210, 1680] | -| Central | [1, 2, 12, 120, 1680] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 10, 38, 168, 872, 5296, 37200, 297856] | -| NegHalf | [1, -1, 2, -2, 8, 8, 112, 656, 5504] | -| Bin | [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729] | -| InvBin | [1, 0, -1, -4, -15, -56, -185, -204, 6209] | -| TransSqrs | [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056] | -| TransNat0 | [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208] | -| TransNat1 | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | - -RisingFactRev Diagonals as rows +RisingFact:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 2, 2], [1, 3, 6, 6], [1, 4, 12, 24, 24], [1, 5, 20, 60, 120, 120], [1, 6, 30, 120, 360, 720, 720], [1, 7, 42, 210, 840, 2520, 5040, 5040], [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320]] | +| RevTabl | [[1], [1, 1], [2, 2, 1], [6, 6, 3, 1], [24, 24, 12, 4, 1], [120, 120, 60, 20, 5, 1], [720, 720, 360, 120, 30, 6, 1], [5040, 5040, 2520, 840, 210, 42, 7, 1], [40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 2], [1, 3, 2], [1, 4, 6], [1, 5, 12, 6], [1, 6, 20, 24], [1, 7, 30, 60, 24]] | +| AccTabl | [[1], [1, 2], [1, 3, 5], [1, 4, 10, 16], [1, 5, 17, 41, 65], [1, 6, 26, 86, 206, 326], [1, 7, 37, 157, 517, 1237, 1957], [1, 8, 50, 260, 1100, 3620, 8660, 13700], [1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601]] | +| RevAccTabl | [[1], [2, 1], [5, 3, 1], [16, 10, 4, 1], [65, 41, 17, 5, 1], [326, 206, 86, 26, 6, 1], [1957, 1237, 517, 157, 37, 7, 1], [13700, 8660, 3620, 1100, 260, 50, 8, 1], [109601, 69281, 28961, 8801, 2081, 401, 65, 9, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 4, 5], [6, 12, 15, 16], [24, 48, 60, 64, 65], [120, 240, 300, 320, 325, 326], [720, 1440, 1800, 1920, 1950, 1956, 1957], [5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700], [40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601]] | + +RisingFact:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 1, 3, 7, 37, 141, 1111, 5923, 62217] | +| OddSum | [0, 1, 2, 9, 28, 185, 846, 7777, 47384] | +| AltSum | [1, 0, 1, -2, 9, -44, 265, -1854, 14833] | +| AccSum | [1, 3, 9, 31, 129, 651, 3913, 27399, 219201] | +| AccRevSum | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | +| AntiDiagSum | [1, 1, 2, 3, 6, 11, 24, 51, 122] | + +RisingFact:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColMiddle | [1, 1, 2, 3, 12, 20, 120, 210, 1680] | +| ColECenter | [1, 2, 12, 120, 1680] | +| ColOCenter | [1, 3, 20, 210] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056] | +| TransNat0 | [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208] | +| TransNat1 | [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809] | + +RisingFact:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| @@ -283,7 +271,7 @@ RisingFactRev Diagonals as rows | DiagRow7 | [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]| | DiagRow8 | [1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400]| -RisingFactRev Diagonals as columns +RisingFact:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ RisingFactRev Diagonals as columns | DiagCol7 | [5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400] | | DiagCol8 | [40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200, 518918400] | -RisingFactRev Polynomial values as rows +RisingFact:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ RisingFactRev Polynomial values as rows | PolyRow7 | [1, 13700, 1063623, 15383110, 106028861, 480927336, 1666754755, 4788052298, 11976978105] | | PolyRow8 | [1, 109601, 17017969, 369194641, 3392923553, 19237093441, 80004228241, 268130928689, 766526598721] | -RisingFactRev Polynomial values as columns +RisingFact:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ RisingFactRev Polynomial values as columns | PolyCol7 | [1, 8, 113, 2374, 66473, 2326556, 97715353, 4788052298, 268130928689] | | PolyCol8 | [1, 9, 145, 3481, 111393, 4455721, 213874609, 11976978105, 766526598721] | -# RisingFactInvRev -True +# RisingFact:Inv:Rev +[] -RisingFactInvRev Triangle view +RisingFact:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ RisingFactInvRev Triangle view | Row7 | [1, -7, 0, 0, 0, 0, 0, 0] | | Row8 | [1, -8, 0, 0, 0, 0, 0, 0, 0] | -RisingFactInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -2, 0, 1, -3, 0, 0, 1, -4, 0, 0, 0, 1, -5, 0, 0, 0, 0, 1, -6, 0, 0, 0, 0, 0, 1, -7, 0, 0, 0, 0, 0, 0, 1, -8, 0, 0, 0, 0, 0, 0, 0] | -| TriRev | [1, -1, 1, 0, -2, 1, 0, 0, -3, 1, 0, 0, 0, -4, 1, 0, 0, 0, 0, -5, 1, 0, 0, 0, 0, 0, -6, 1, 0, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 0, 0, 0, -8, 1] | -| TriAcc | [1, 1, 0, 1, -1, -1, 1, -2, -2, -2, 1, -3, -3, -3, -3, 1, -4, -4, -4, -4, -4, 1, -5, -5, -5, -5, -5, -5, 1, -6, -6, -6, -6, -6, -6, -6, 1, -7, -7, -7, -7, -7, -7, -7, -7] | -| TriRevAcc | [1, 0, 1, -1, -1, 1, -2, -2, -2, 1, -3, -3, -3, -3, 1, -4, -4, -4, -4, -4, 1, -5, -5, -5, -5, -5, -5, 1, -6, -6, -6, -6, -6, -6, -6, 1, -7, -7, -7, -7, -7, -7, -7, -7, 1] | -| TriAccRev | [1, -1, 0, 0, -2, -1, 0, 0, -3, -2, 0, 0, 0, -4, -3, 0, 0, 0, 0, -5, -4, 0, 0, 0, 0, 0, -6, -5, 0, 0, 0, 0, 0, 0, -7, -6, 0, 0, 0, 0, 0, 0, 0, -8, -7] | -| TriDiag | [1, 1, 1, -1, 1, -2, 1, -3, 0, 1, -4, 0, 1, -5, 0, 0, 1, -6, 0, 0] | - -RisingFactInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| OddSum | [0, -1, -2, -3, -4, -5, -6, -7, -8] | -| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| AccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| AccRevSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | -| RevAccSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | -| DiagSum | [1, 1, 0, -1, -2, -3, -4, -5] | - -RisingFactInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, -3, -4, -5, -6, -7, -8] | -| RowGcd | [1, 1, -2, -3, -4, -5, -6, -7, -8] | -| RowMax | [1, 1, 2, 3, 4, 5, 6, 7, 8] | -| Middle | [1, 1, -2, -3, 0, 0, 0, 0, 0] | -| Central | [1, -2, 0, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 1, 0, -4, -16, -48, -128, -320, -768] | -| NegHalf | [1, -3, 8, -20, 48, -112, 256, -576, 1280] | -| Bin | [1, 0, -3, -8, -15, -24, -35, -48, -63] | -| InvBin | [1, -2, 5, -10, 17, -26, 37, -50, 65] | -| TransSqrs | [0, -1, -2, -3, -4, -5, -6, -7, -8] | -| TransNat0 | [0, -1, -2, -3, -4, -5, -6, -7, -8] | -| TransNat1 | [1, -1, -3, -5, -7, -9, -11, -13, -15] | - -RisingFactInvRev Diagonals as rows +RisingFact:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -2, 0], [1, -3, 0, 0], [1, -4, 0, 0, 0], [1, -5, 0, 0, 0, 0], [1, -6, 0, 0, 0, 0, 0], [1, -7, 0, 0, 0, 0, 0, 0], [1, -8, 0, 0, 0, 0, 0, 0, 0]] | +| RevTabl | [[1], [-1, 1], [0, -2, 1], [0, 0, -3, 1], [0, 0, 0, -4, 1], [0, 0, 0, 0, -5, 1], [0, 0, 0, 0, 0, -6, 1], [0, 0, 0, 0, 0, 0, -7, 1], [0, 0, 0, 0, 0, 0, 0, -8, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -2], [1, -3, 0], [1, -4, 0], [1, -5, 0, 0], [1, -6, 0, 0], [1, -7, 0, 0, 0]] | +| AccTabl | [[1], [1, 0], [1, -1, -1], [1, -2, -2, -2], [1, -3, -3, -3, -3], [1, -4, -4, -4, -4, -4], [1, -5, -5, -5, -5, -5, -5], [1, -6, -6, -6, -6, -6, -6, -6], [1, -7, -7, -7, -7, -7, -7, -7, -7]] | +| RevAccTabl | [[1], [0, 1], [-1, -1, 1], [-2, -2, -2, 1], [-3, -3, -3, -3, 1], [-4, -4, -4, -4, -4, 1], [-5, -5, -5, -5, -5, -5, 1], [-6, -6, -6, -6, -6, -6, -6, 1], [-7, -7, -7, -7, -7, -7, -7, -7, 1]] | +| AccRevTabl | [[1], [-1, 0], [0, -2, -1], [0, 0, -3, -2], [0, 0, 0, -4, -3], [0, 0, 0, 0, -5, -4], [0, 0, 0, 0, 0, -6, -5], [0, 0, 0, 0, 0, 0, -7, -6], [0, 0, 0, 0, 0, 0, 0, -8, -7]] | + +RisingFact:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| EvenSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| OddSum | [0, -1, -2, -3, -4, -5, -6, -7, -8] | +| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| AccSum | [1, 1, -1, -5, -11, -19, -29, -41, -55] | +| AccRevSum | [1, -1, -3, -5, -7, -9, -11, -13, -15] | +| AntiDiagSum | [1, 1, 0, -1, -2, -3, -4, -5, -6] | + +RisingFact:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowGcd | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| RowMax | [1, 1, 2, 3, 4, 5, 6, 7, 8] | +| ColMiddle | [1, 1, -2, -3, 0, 0, 0, 0, 0] | +| ColECenter | [1, -2, 0, 0, 0] | +| ColOCenter | [1, -3, 0, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, -1, -2, -3, -4, -5, -6, -7, -8] | +| TransNat0 | [0, -1, -2, -3, -4, -5, -6, -7, -8] | +| TransNat1 | [1, -1, -3, -5, -7, -9, -11, -13, -15] | + +RisingFact:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ RisingFactInvRev Diagonals as rows | DiagRow7 | [1, -8, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, -9, 0, 0, 0, 0, 0, 0, 0]| -RisingFactInvRev Diagonals as columns +RisingFact:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ RisingFactInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -RisingFactInvRev Polynomial values as rows +RisingFact:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ RisingFactInvRev Polynomial values as rows | PolyRow7 | [1, -6, -13, -20, -27, -34, -41, -48, -55] | | PolyRow8 | [1, -7, -15, -23, -31, -39, -47, -55, -63] | -RisingFactInvRev Polynomial values as columns +RisingFact:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Schroeder.md b/data/md/Schroeder.md index d8e5a9a..d328c47 100644 --- a/data/md/Schroeder.md +++ b/data/md/Schroeder.md @@ -1,5 +1,5 @@ # Schroeder -['A033877', 'A080245', 'A080247', 'A122538', 'A106579'] +['A122538', 'A033877', 'A080245', 'A080247', 'A106579'] Schroeder Triangle view | Row | Seq | @@ -14,45 +14,41 @@ Schroeder Triangle view | Row7 | [0, 1806, 1412, 714, 264, 70, 12, 1] | | Row8 | [0, 8558, 6752, 3534, 1408, 430, 96, 14, 1] | -Schroeder Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1] | -| TriRev | [1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, 8, 30, 68, 90, 0, 1, 10, 48, 146, 304, 394, 0, 1, 12, 70, 264, 714, 1412, 1806, 0, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 0] | -| TriAcc | [1, 0, 1, 0, 2, 3, 0, 6, 10, 11, 0, 22, 38, 44, 45, 0, 90, 158, 188, 196, 197, 0, 394, 698, 844, 892, 902, 903, 0, 1806, 3218, 3932, 4196, 4266, 4278, 4279, 0, 8558, 15310, 18844, 20252, 20682, 20778, 20792, 20793] | -| TriRevAcc | [1, 1, 0, 3, 2, 0, 11, 10, 6, 0, 45, 44, 38, 22, 0, 197, 196, 188, 158, 90, 0, 903, 902, 892, 844, 698, 394, 0, 4279, 4278, 4266, 4196, 3932, 3218, 1806, 0, 20793, 20792, 20778, 20682, 20252, 18844, 15310, 8558, 0] | -| TriAccRev | [1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793] | -| TriDiag | [1, 0, 0, 1, 0, 2, 0, 6, 1, 0, 22, 4, 0, 90, 16, 1, 0, 394, 68, 6] | +Schroeder Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 2, 1], [0, 6, 4, 1], [0, 22, 16, 6, 1], [0, 90, 68, 30, 8, 1], [0, 394, 304, 146, 48, 10, 1], [0, 1806, 1412, 714, 264, 70, 12, 1], [0, 8558, 6752, 3534, 1408, 430, 96, 14, 1]] | +| RevTabl | [[1], [1, 0], [1, 2, 0], [1, 4, 6, 0], [1, 6, 16, 22, 0], [1, 8, 30, 68, 90, 0], [1, 10, 48, 146, 304, 394, 0], [1, 12, 70, 264, 714, 1412, 1806, 0], [1, 14, 96, 430, 1408, 3534, 6752, 8558, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 2], [0, 6, 1], [0, 22, 4], [0, 90, 16, 1], [0, 394, 68, 6], [0, 1806, 304, 30, 1]] | +| AccTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 10, 11], [0, 22, 38, 44, 45], [0, 90, 158, 188, 196, 197], [0, 394, 698, 844, 892, 902, 903], [0, 1806, 3218, 3932, 4196, 4266, 4278, 4279], [0, 8558, 15310, 18844, 20252, 20682, 20778, 20792, 20793]] | +| RevAccTabl | [[1], [1, 0], [3, 2, 0], [11, 10, 6, 0], [45, 44, 38, 22, 0], [197, 196, 188, 158, 90, 0], [903, 902, 892, 844, 698, 394, 0], [4279, 4278, 4266, 4196, 3932, 3218, 1806, 0], [20793, 20792, 20778, 20682, 20252, 18844, 15310, 8558, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 11, 11], [1, 7, 23, 45, 45], [1, 9, 39, 107, 197, 197], [1, 11, 59, 205, 509, 903, 903], [1, 13, 83, 347, 1061, 2473, 4279, 4279], [1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793]] | Schroeder Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | -| EvenSum | [1, 0, 1, 4, 17, 76, 353, 1688, 8257] | -| OddSum | [0, 1, 2, 7, 28, 121, 550, 2591, 12536] | -| AltSum | [1, -1, -1, -3, -11, -45, -197, -903, -4279] | -| AccSum | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | -| AccRevSum | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | -| RevAccSum | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | -| DiagSum | [1, 0, 1, 2, 7, 26, 107, 468] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| EvenSum | [1, 0, 1, 4, 17, 76, 353, 1688, 8257] | +| OddSum | [0, 1, 2, 7, 28, 121, 550, 2591, 12536] | +| AltSum | [1, -1, -1, -3, -11, -45, -197, -903, -4279] | +| AccSum | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | +| AccRevSum | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | +| AntiDiagSum | [1, 0, 1, 2, 7, 26, 107, 468, 2141] | Schroeder Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 528, 6120, 65577360, 2384317320, 307331984910720] | -| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | -| RowMax | [1, 1, 2, 6, 22, 90, 394, 1806, 8558] | -| Middle | [1, 0, 2, 6, 16, 68, 146, 714, 1408] | -| Central | [1, 2, 16, 146, 1408] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 5, 33, 253, 2121, 18853, 174609, 1667021] | -| NegHalf | [1, 1, -3, 17, -123, 1001, -8739, 79969, -756939] | -| Bin | [1, 1, 5, 31, 209, 1471, 10625, 78079, 580865] | -| InvBin | [1, 1, -3, 7, -15, 31, -63, 127, -255] | -| TransSqrs | [0, 1, 6, 31, 156, 785, 3978, 20335, 104856] | -| TransNat0 | [0, 1, 4, 17, 76, 353, 1688, 8257, 41128] | -| TransNat1 | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 528, 6120, 65577360, 2384317320, 307331984910720] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | +| RowMax | [1, 1, 2, 6, 22, 90, 394, 1806, 8558] | +| ColMiddle | [1, 0, 2, 6, 16, 68, 146, 714, 1408] | +| ColECenter | [1, 2, 16, 146, 1408] | +| ColOCenter | [0, 6, 68, 714] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 6, 31, 156, 785, 3978, 20335, 104856] | +| TransNat0 | [0, 1, 4, 17, 76, 353, 1688, 8257, 41128] | +| TransNat1 | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | Schroeder Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Schroeder Polynomial values as columns | PolyCol7 | [1, 7, 63, 581, 5397, 50267, 468699, 4372417, 40799241] | | PolyCol8 | [1, 8, 80, 816, 8368, 85968, 883792, 9088368, 93470576] | -# SchroederInv -True +# Schroeder:Inv +[] -SchroederInv Triangle view +Schroeder:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SchroederInv Triangle view | Row7 | [0, 2, -20, 66, -88, 50, -12, 1] | | Row8 | [0, -2, 24, -102, 192, -170, 72, -14, 1] | -SchroederInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, -2, 8, -6, 1, 0, 2, -12, 18, -8, 1, 0, -2, 16, -38, 32, -10, 1, 0, 2, -20, 66, -88, 50, -12, 1, 0, -2, 24, -102, 192, -170, 72, -14, 1] | -| TriRev | [1, 1, 0, 1, -2, 0, 1, -4, 2, 0, 1, -6, 8, -2, 0, 1, -8, 18, -12, 2, 0, 1, -10, 32, -38, 16, -2, 0, 1, -12, 50, -88, 66, -20, 2, 0, 1, -14, 72, -170, 192, -102, 24, -2, 0] | -| TriAcc | [1, 0, 1, 0, -2, -1, 0, 2, -2, -1, 0, -2, 6, 0, 1, 0, 2, -10, 8, 0, 1, 0, -2, 14, -24, 8, -2, -1, 0, 2, -18, 48, -40, 10, -2, -1, 0, -2, 22, -80, 112, -58, 14, 0, 1] | -| TriRevAcc | [1, 1, 0, -1, -2, 0, -1, -2, 2, 0, 1, 0, 6, -2, 0, 1, 0, 8, -10, 2, 0, -1, -2, 8, -24, 14, -2, 0, -1, -2, 10, -40, 48, -18, 2, 0, 1, 0, 14, -58, 112, -80, 22, -2, 0] | -| TriAccRev | [1, 1, 1, 1, -1, -1, 1, -3, -1, -1, 1, -5, 3, 1, 1, 1, -7, 11, -1, 1, 1, 1, -9, 23, -15, 1, -1, -1, 1, -11, 39, -49, 17, -3, -1, -1, 1, -13, 59, -111, 81, -21, 3, 1, 1] | -| TriDiag | [1, 0, 0, 1, 0, -2, 0, 2, 1, 0, -2, -4, 0, 2, 8, 1, 0, -2, -12, -6] | - -SchroederInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -1, 1, 1, -1, -1, 1] | -| EvenSum | [1, 0, 1, -4, 9, -20, 49, -120, 289] | -| OddSum | [0, 1, -2, 3, -8, 21, -50, 119, -288] | -| AltSum | [1, -1, 3, -7, 17, -41, 99, -239, 577] | -| AccSum | [1, 1, -3, -1, 5, 1, -7, -1, 9] | -| AccRevSum | [1, 1, -3, -1, 5, 1, -7, -1, 9] | -| RevAccSum | [1, 2, -1, -4, 1, 6, -1, -8, 1] | -| DiagSum | [1, 0, 1, -2, 3, -6, 11, -20] | - -SchroederInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 4, 24, 72, 3040, 6600, 342720] | -| RowGcd | [1, 1, -2, 2, 2, 2, 2, 2, 2] | -| RowMax | [1, 1, 2, 4, 8, 18, 38, 88, 192] | -| Middle | [1, 0, -2, 2, 8, -12, -38, 66, 192] | -| Central | [1, -2, 8, -38, 192] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -3, 1, 5, -7, -3, 17, -11] | -| NegHalf | [1, 1, 5, 17, 61, 217, 773, 2753, 9805] | -| Bin | [1, 1, -3, -5, 17, 31, -111, -209, 769] | -| InvBin | [1, 1, 5, 19, 81, 351, 1553, 6959, 31489] | -| TransSqrs | [0, 1, 2, -5, -8, 13, 18, -25, -32] | -| TransNat0 | [0, 1, 0, -3, 0, 5, 0, -7, 0] | -| TransNat1 | [1, 2, -1, -4, 1, 6, -1, -8, 1] | - -SchroederInv Diagonals as rows +Schroeder:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -2, 1], [0, 2, -4, 1], [0, -2, 8, -6, 1], [0, 2, -12, 18, -8, 1], [0, -2, 16, -38, 32, -10, 1], [0, 2, -20, 66, -88, 50, -12, 1], [0, -2, 24, -102, 192, -170, 72, -14, 1]] | +| RevTabl | [[1], [1, 0], [1, -2, 0], [1, -4, 2, 0], [1, -6, 8, -2, 0], [1, -8, 18, -12, 2, 0], [1, -10, 32, -38, 16, -2, 0], [1, -12, 50, -88, 66, -20, 2, 0], [1, -14, 72, -170, 192, -102, 24, -2, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -2], [0, 2, 1], [0, -2, -4], [0, 2, 8, 1], [0, -2, -12, -6], [0, 2, 16, 18, 1]] | +| AccTabl | [[1], [0, 1], [0, -2, -1], [0, 2, -2, -1], [0, -2, 6, 0, 1], [0, 2, -10, 8, 0, 1], [0, -2, 14, -24, 8, -2, -1], [0, 2, -18, 48, -40, 10, -2, -1], [0, -2, 22, -80, 112, -58, 14, 0, 1]] | +| RevAccTabl | [[1], [1, 0], [-1, -2, 0], [-1, -2, 2, 0], [1, 0, 6, -2, 0], [1, 0, 8, -10, 2, 0], [-1, -2, 8, -24, 14, -2, 0], [-1, -2, 10, -40, 48, -18, 2, 0], [1, 0, 14, -58, 112, -80, 22, -2, 0]] | +| AccRevTabl | [[1], [1, 1], [1, -1, -1], [1, -3, -1, -1], [1, -5, 3, 1, 1], [1, -7, 11, -1, 1, 1], [1, -9, 23, -15, 1, -1, -1], [1, -11, 39, -49, 17, -3, -1, -1], [1, -13, 59, -111, 81, -21, 3, 1, 1]] | + +Schroeder:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -1, 1, 1, -1, -1, 1] | +| EvenSum | [1, 0, 1, -4, 9, -20, 49, -120, 289] | +| OddSum | [0, 1, -2, 3, -8, 21, -50, 119, -288] | +| AltSum | [1, -1, 3, -7, 17, -41, 99, -239, 577] | +| AccSum | [1, 1, -3, -1, 5, 1, -7, -1, 9] | +| AccRevSum | [1, 2, -1, -4, 1, 6, -1, -8, 1] | +| AntiDiagSum | [1, 0, 1, -2, 3, -6, 11, -20, 37] | + +Schroeder:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 4, 24, 72, 3040, 6600, 342720] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | +| RowMax | [1, 1, 2, 4, 8, 18, 38, 88, 192] | +| ColMiddle | [1, 0, -2, 2, 8, -12, -38, 66, 192] | +| ColECenter | [1, -2, 8, -38, 192] | +| ColOCenter | [0, 2, -12, 66] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 2, -5, -8, 13, 18, -25, -32] | +| TransNat0 | [0, 1, 0, -3, 0, 5, 0, -7, 0] | +| TransNat1 | [1, 2, -1, -4, 1, 6, -1, -8, 1] | + +Schroeder:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ SchroederInv Diagonals as rows | DiagRow7 | [0, -2, -28, -198, -952, -3530, -10836, -28814, -68464]| | DiagRow8 | [0, 2, 32, 258, 1408, 5890, 20256, 59906, 157184]| -SchroederInv Diagonals as columns +Schroeder:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ SchroederInv Diagonals as columns | DiagCol7 | [1, -14, 98, -462, 1666, -4942, 12642, -28814, 59906] | | DiagCol8 | [1, -16, 128, -688, 2816, -9424, 27008, -68464, 157184] | -SchroederInv Polynomial values as rows +Schroeder:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SchroederInv Polynomial values as rows | PolyRow7 | [0, -1, 4, 69, -184, -365, 8364, 62489, 261904] | | PolyRow8 | [0, 1, -20, 129, -8, -1535, 25476, 275905, 1456624] | -SchroederInv Polynomial values as columns +Schroeder:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ SchroederInv Polynomial values as columns | PolyCol7 | [1, 7, 35, 161, 721, 3199, 14147, 62489, 275905] | | PolyCol8 | [1, 8, 48, 272, 1520, 8464, 47088, 261904, 1456624] | -# SchroederRev -True +# Schroeder:Rev +[] -SchroederRev Triangle view +Schroeder:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ SchroederRev Triangle view | Row7 | [1, 12, 70, 264, 714, 1412, 1806, 0] | | Row8 | [1, 14, 96, 430, 1408, 3534, 6752, 8558, 0] | -SchroederRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, 8, 30, 68, 90, 0, 1, 10, 48, 146, 304, 394, 0, 1, 12, 70, 264, 714, 1412, 1806, 0, 1, 14, 96, 430, 1408, 3534, 6752, 8558, 0] | -| TriRev | [1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1] | -| TriAcc | [1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793] | -| TriRevAcc | [1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 45, 45, 23, 7, 1, 197, 197, 107, 39, 9, 1, 903, 903, 509, 205, 59, 11, 1, 4279, 4279, 2473, 1061, 347, 83, 13, 1, 20793, 20793, 12235, 5483, 1949, 541, 111, 15, 1] | -| TriAccRev | [1, 0, 1, 0, 2, 3, 0, 6, 10, 11, 0, 22, 38, 44, 45, 0, 90, 158, 188, 196, 197, 0, 394, 698, 844, 892, 902, 903, 0, 1806, 3218, 3932, 4196, 4266, 4278, 4279, 0, 8558, 15310, 18844, 20252, 20682, 20778, 20792, 20793] | -| TriDiag | [1, 1, 1, 0, 1, 2, 1, 4, 0, 1, 6, 6, 1, 8, 16, 0, 1, 10, 30, 22] | - -SchroederRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | -| EvenSum | [1, 1, 1, 7, 17, 121, 353, 2591, 8257] | -| OddSum | [0, 0, 2, 4, 28, 76, 550, 1688, 12536] | -| AltSum | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | -| AccSum | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | -| AccRevSum | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | -| RevAccSum | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | -| DiagSum | [1, 1, 1, 3, 5, 13, 25, 63] | - -SchroederRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 2, 12, 528, 6120, 65577360, 2384317320, 307331984910720] | -| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | -| RowMax | [1, 1, 2, 6, 22, 90, 394, 1806, 8558] | -| Middle | [1, 1, 2, 4, 16, 30, 146, 264, 1408] | -| Central | [1, 2, 16, 146, 1408] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 8, 36, 172, 852, 4324, 22332, 116876] | -| NegHalf | [1, -2, 0, -4, -12, -52, -228, -1052, -5004] | -| Bin | [1, 1, 5, 31, 209, 1471, 10625, 78079, 580865] | -| InvBin | [1, -1, -3, -7, -15, -31, -63, -127, -255] | -| TransSqrs | [0, 0, 2, 28, 268, 2180, 16230, 114408, 777560] | -| TransNat0 | [0, 0, 2, 16, 104, 632, 3730, 21696, 125216] | -| TransNat1 | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | - -SchroederRev Diagonals as rows +Schroeder:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 2, 0], [1, 4, 6, 0], [1, 6, 16, 22, 0], [1, 8, 30, 68, 90, 0], [1, 10, 48, 146, 304, 394, 0], [1, 12, 70, 264, 714, 1412, 1806, 0], [1, 14, 96, 430, 1408, 3534, 6752, 8558, 0]] | +| RevTabl | [[1], [0, 1], [0, 2, 1], [0, 6, 4, 1], [0, 22, 16, 6, 1], [0, 90, 68, 30, 8, 1], [0, 394, 304, 146, 48, 10, 1], [0, 1806, 1412, 714, 264, 70, 12, 1], [0, 8558, 6752, 3534, 1408, 430, 96, 14, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 2], [1, 4, 0], [1, 6, 6], [1, 8, 16, 0], [1, 10, 30, 22], [1, 12, 48, 68, 0]] | +| AccTabl | [[1], [1, 1], [1, 3, 3], [1, 5, 11, 11], [1, 7, 23, 45, 45], [1, 9, 39, 107, 197, 197], [1, 11, 59, 205, 509, 903, 903], [1, 13, 83, 347, 1061, 2473, 4279, 4279], [1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793]] | +| RevAccTabl | [[1], [1, 1], [3, 3, 1], [11, 11, 5, 1], [45, 45, 23, 7, 1], [197, 197, 107, 39, 9, 1], [903, 903, 509, 205, 59, 11, 1], [4279, 4279, 2473, 1061, 347, 83, 13, 1], [20793, 20793, 12235, 5483, 1949, 541, 111, 15, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 2, 3], [0, 6, 10, 11], [0, 22, 38, 44, 45], [0, 90, 158, 188, 196, 197], [0, 394, 698, 844, 892, 902, 903], [0, 1806, 3218, 3932, 4196, 4266, 4278, 4279], [0, 8558, 15310, 18844, 20252, 20682, 20778, 20792, 20793]] | + +Schroeder:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| EvenSum | [1, 1, 1, 7, 17, 121, 353, 2591, 8257] | +| OddSum | [0, 0, 2, 4, 28, 76, 550, 1688, 12536] | +| AltSum | [1, 1, -1, 3, -11, 45, -197, 903, -4279] | +| AccSum | [1, 2, 7, 28, 121, 550, 2591, 12536, 61921] | +| AccRevSum | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | +| AntiDiagSum | [1, 1, 1, 3, 5, 13, 25, 63, 129] | + +Schroeder:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 12, 528, 6120, 65577360, 2384317320, 307331984910720] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | +| RowMax | [1, 1, 2, 6, 22, 90, 394, 1806, 8558] | +| ColMiddle | [1, 1, 2, 4, 16, 30, 146, 264, 1408] | +| ColECenter | [1, 2, 16, 146, 1408] | +| ColOCenter | [1, 4, 30, 264] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 2, 28, 268, 2180, 16230, 114408, 777560] | +| TransNat0 | [0, 0, 2, 16, 104, 632, 3730, 21696, 125216] | +| TransNat1 | [1, 1, 5, 27, 149, 829, 4633, 25975, 146009] | + +Schroeder:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ SchroederRev Diagonals as rows | DiagRow7 | [1, 14, 126, 938, 6314, 40054, 244790, 1459810, 8561938]| | DiagRow8 | [1, 16, 160, 1296, 9344, 62736, 402080, 2496528, 15158272]| -SchroederRev Diagonals as columns +Schroeder:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ SchroederRev Diagonals as columns | DiagCol7 | [0, 8558, 33028, 89898, 206600, 426150, 812940, 1459810, 2496528] | | DiagCol8 | [0, 41586, 164512, 461010, 1093760, 2332850, 4605600, 8561938, 15158272] | -SchroederRev Polynomial values as rows +Schroeder:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ SchroederRev Polynomial values as rows | PolyRow7 | [1, 4279, 174609, 1725319, 9044113, 33112311, 96225409, 238013959, 522764769] | | PolyRow8 | [1, 20793, 1667021, 24623881, 171878841, 786073721, 2740115173, 7905175881, 19839218801] | -SchroederRev Polynomial values as columns +Schroeder:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ SchroederRev Polynomial values as columns | PolyCol7 | [1, 1, 15, 323, 8373, 240941, 7404363, 238013959, 7905175881] | | PolyCol8 | [1, 1, 17, 417, 12337, 405441, 14233681, 522764769, 19839218801] | -# SchroederInvRev -True +# Schroeder:Inv:Rev +[] -SchroederInvRev Triangle view +Schroeder:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ SchroederInvRev Triangle view | Row7 | [1, -12, 50, -88, 66, -20, 2, 0] | | Row8 | [1, -14, 72, -170, 192, -102, 24, -2, 0] | -SchroederInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -2, 0, 1, -4, 2, 0, 1, -6, 8, -2, 0, 1, -8, 18, -12, 2, 0, 1, -10, 32, -38, 16, -2, 0, 1, -12, 50, -88, 66, -20, 2, 0, 1, -14, 72, -170, 192, -102, 24, -2, 0] | -| TriRev | [1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, -2, 8, -6, 1, 0, 2, -12, 18, -8, 1, 0, -2, 16, -38, 32, -10, 1, 0, 2, -20, 66, -88, 50, -12, 1, 0, -2, 24, -102, 192, -170, 72, -14, 1] | -| TriAcc | [1, 1, 1, 1, -1, -1, 1, -3, -1, -1, 1, -5, 3, 1, 1, 1, -7, 11, -1, 1, 1, 1, -9, 23, -15, 1, -1, -1, 1, -11, 39, -49, 17, -3, -1, -1, 1, -13, 59, -111, 81, -21, 3, 1, 1] | -| TriRevAcc | [1, 1, 1, -1, -1, 1, -1, -1, -3, 1, 1, 1, 3, -5, 1, 1, 1, -1, 11, -7, 1, -1, -1, 1, -15, 23, -9, 1, -1, -1, -3, 17, -49, 39, -11, 1, 1, 1, 3, -21, 81, -111, 59, -13, 1] | -| TriAccRev | [1, 0, 1, 0, -2, -1, 0, 2, -2, -1, 0, -2, 6, 0, 1, 0, 2, -10, 8, 0, 1, 0, -2, 14, -24, 8, -2, -1, 0, 2, -18, 48, -40, 10, -2, -1, 0, -2, 22, -80, 112, -58, 14, 0, 1] | -| TriDiag | [1, 1, 1, 0, 1, -2, 1, -4, 0, 1, -6, 2, 1, -8, 8, 0, 1, -10, 18, -2] | - -SchroederInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, -1, -1, 1, 1, -1, -1, 1] | -| EvenSum | [1, 1, 1, 3, 9, 21, 49, 119, 289] | -| OddSum | [0, 0, -2, -4, -8, -20, -50, -120, -288] | -| AltSum | [1, 1, 3, 7, 17, 41, 99, 239, 577] | -| AccSum | [1, 2, -1, -4, 1, 6, -1, -8, 1] | -| AccRevSum | [1, 2, -1, -4, 1, 6, -1, -8, 1] | -| RevAccSum | [1, 1, -3, -1, 5, 1, -7, -1, 9] | -| DiagSum | [1, 1, 1, -1, -3, -3, 1, 7] | - -SchroederInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -2, 4, 24, 72, 3040, 6600, 342720] | -| RowGcd | [1, 1, -2, 2, 2, 2, 2, 2, 2] | -| RowMax | [1, 1, 2, 4, 8, 18, 38, 88, 192] | -| Middle | [1, 1, -2, -4, 8, 18, -38, -88, 192] | -| Central | [1, -2, 8, -38, 192] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 0, -4, -4, 4, 12, 4, -20] | -| NegHalf | [1, -2, 8, -28, 100, -356, 1268, -4516, 16084] | -| Bin | [1, 1, -3, -5, 17, 31, -111, -209, 769] | -| InvBin | [1, -1, 5, -19, 81, -351, 1553, -6959, 31489] | -| TransSqrs | [0, 0, -2, 4, 8, -12, -18, 24, 32] | -| TransNat0 | [0, 0, -2, 0, 4, 0, -6, 0, 8] | -| TransNat1 | [1, 1, -3, -1, 5, 1, -7, -1, 9] | - -SchroederInvRev Diagonals as rows +Schroeder:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -2, 0], [1, -4, 2, 0], [1, -6, 8, -2, 0], [1, -8, 18, -12, 2, 0], [1, -10, 32, -38, 16, -2, 0], [1, -12, 50, -88, 66, -20, 2, 0], [1, -14, 72, -170, 192, -102, 24, -2, 0]] | +| RevTabl | [[1], [0, 1], [0, -2, 1], [0, 2, -4, 1], [0, -2, 8, -6, 1], [0, 2, -12, 18, -8, 1], [0, -2, 16, -38, 32, -10, 1], [0, 2, -20, 66, -88, 50, -12, 1], [0, -2, 24, -102, 192, -170, 72, -14, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -2], [1, -4, 0], [1, -6, 2], [1, -8, 8, 0], [1, -10, 18, -2], [1, -12, 32, -12, 0]] | +| AccTabl | [[1], [1, 1], [1, -1, -1], [1, -3, -1, -1], [1, -5, 3, 1, 1], [1, -7, 11, -1, 1, 1], [1, -9, 23, -15, 1, -1, -1], [1, -11, 39, -49, 17, -3, -1, -1], [1, -13, 59, -111, 81, -21, 3, 1, 1]] | +| RevAccTabl | [[1], [1, 1], [-1, -1, 1], [-1, -1, -3, 1], [1, 1, 3, -5, 1], [1, 1, -1, 11, -7, 1], [-1, -1, 1, -15, 23, -9, 1], [-1, -1, -3, 17, -49, 39, -11, 1], [1, 1, 3, -21, 81, -111, 59, -13, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -2, -1], [0, 2, -2, -1], [0, -2, 6, 0, 1], [0, 2, -10, 8, 0, 1], [0, -2, 14, -24, 8, -2, -1], [0, 2, -18, 48, -40, 10, -2, -1], [0, -2, 22, -80, 112, -58, 14, 0, 1]] | + +Schroeder:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, -1, -1, 1, 1, -1, -1, 1] | +| EvenSum | [1, 1, 1, 3, 9, 21, 49, 119, 289] | +| OddSum | [0, 0, -2, -4, -8, -20, -50, -120, -288] | +| AltSum | [1, 1, 3, 7, 17, 41, 99, 239, 577] | +| AccSum | [1, 2, -1, -4, 1, 6, -1, -8, 1] | +| AccRevSum | [1, 1, -3, -1, 5, 1, -7, -1, 9] | +| AntiDiagSum | [1, 1, 1, -1, -3, -3, 1, 7, 9] | + +Schroeder:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 2, 4, 24, 72, 3040, 6600, 342720] | +| RowGcd | [1, 1, 2, 2, 2, 2, 2, 2, 2] | +| RowMax | [1, 1, 2, 4, 8, 18, 38, 88, 192] | +| ColMiddle | [1, 1, -2, -4, 8, 18, -38, -88, 192] | +| ColECenter | [1, -2, 8, -38, 192] | +| ColOCenter | [1, -4, 18, -88] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -2, 4, 8, -12, -18, 24, 32] | +| TransNat0 | [0, 0, -2, 0, 4, 0, -6, 0, 8] | +| TransNat1 | [1, 1, -3, -1, 5, 1, -7, -1, 9] | + +Schroeder:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ SchroederInvRev Diagonals as rows | DiagRow7 | [1, -14, 98, -462, 1666, -4942, 12642, -28814, 59906]| | DiagRow8 | [1, -16, 128, -688, 2816, -9424, 27008, -68464, 157184]| -SchroederInvRev Diagonals as columns +Schroeder:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ SchroederInvRev Diagonals as columns | DiagCol7 | [0, -2, -28, -198, -952, -3530, -10836, -28814, -68464] | | DiagCol8 | [0, 2, 32, 258, 1408, 5890, 20256, 59906, 157184] | -SchroederInvRev Polynomial values as rows +Schroeder:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ SchroederInvRev Polynomial values as rows | PolyRow7 | [1, -1, 17, -17, -271, 191, 6049, 29807, 97313] | | PolyRow8 | [1, 1, -11, -95, 457, 481, -18659, -131711, -541295] | -SchroederInvRev Polynomial values as columns +Schroeder:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/SchroederL.md b/data/md/SchroederL.md new file mode 100644 index 0000000..c73386f --- /dev/null +++ b/data/md/SchroederL.md @@ -0,0 +1,416 @@ +# SchroederL +['A172094'] + +SchroederL Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 1] | +| Row2 | [3, 4, 1] | +| Row3 | [11, 17, 7, 1] | +| Row4 | [45, 76, 40, 10, 1] | +| Row5 | [197, 353, 216, 72, 13, 1] | +| Row6 | [903, 1688, 1145, 458, 113, 16, 1] | +| Row7 | [4279, 8257, 6039, 2745, 829, 163, 19, 1] | +| Row8 | [20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1] | + +SchroederL Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [3, 4, 1], [11, 17, 7, 1], [45, 76, 40, 10, 1], [197, 353, 216, 72, 13, 1], [903, 1688, 1145, 458, 113, 16, 1], [4279, 8257, 6039, 2745, 829, 163, 19, 1], [20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1]] | +| RevTabl | [[1], [1, 1], [1, 4, 3], [1, 7, 17, 11], [1, 10, 40, 76, 45], [1, 13, 72, 216, 353, 197], [1, 16, 113, 458, 1145, 1688, 903], [1, 19, 163, 829, 2745, 6039, 8257, 4279], [1, 22, 222, 1356, 5558, 15932, 31864, 41128, 20793]] | +| AntiDiag | [[1], [1], [3, 1], [11, 4], [45, 17, 1], [197, 76, 7], [903, 353, 40, 1], [4279, 1688, 216, 10], [20793, 8257, 1145, 72, 1]] | +| AccTabl | [[1], [1, 2], [3, 7, 8], [11, 28, 35, 36], [45, 121, 161, 171, 172], [197, 550, 766, 838, 851, 852], [903, 2591, 3736, 4194, 4307, 4323, 4324], [4279, 12536, 18575, 21320, 22149, 22312, 22331, 22332], [20793, 61921, 93785, 109717, 115275, 116631, 116853, 116875, 116876]] | +| RevAccTabl | [[1], [2, 1], [8, 7, 3], [36, 35, 28, 11], [172, 171, 161, 121, 45], [852, 851, 838, 766, 550, 197], [4324, 4323, 4307, 4194, 3736, 2591, 903], [22332, 22331, 22312, 22149, 21320, 18575, 12536, 4279], [116876, 116875, 116853, 116631, 115275, 109717, 93785, 61921, 20793]] | +| AccRevTabl | [[1], [1, 2], [1, 5, 8], [1, 8, 25, 36], [1, 11, 51, 127, 172], [1, 14, 86, 302, 655, 852], [1, 17, 130, 588, 1733, 3421, 4324], [1, 20, 183, 1012, 3757, 9796, 18053, 22332], [1, 23, 245, 1601, 7159, 23091, 54955, 96083, 116876]] | + +SchroederL Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 36, 172, 852, 4324, 22332, 116876] | +| EvenSum | [1, 1, 4, 18, 86, 426, 2162, 11166, 58438] | +| OddSum | [0, 1, 4, 18, 86, 426, 2162, 11166, 58438] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 18, 110, 670, 4054, 24378, 145834, 868726] | +| AccRevSum | [1, 3, 14, 70, 362, 1910, 10214, 55154, 300034] | +| AntiDiagSum | [1, 1, 4, 15, 63, 280, 1297, 6193, 30268] | + +SchroederL Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1309, 6840, 195271128, 394433795280, 249001897016460555, 62191124612870563464] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 17, 76, 353, 1688, 8257, 41128] | +| ColMiddle | [1, 1, 4, 17, 40, 216, 458, 2745, 5558] | +| ColECenter | [1, 4, 40, 458, 5558] | +| ColOCenter | [1, 17, 216, 2745] | +| ColLeft | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 8, 54, 342, 2098, 12634, 75190, 443934] | +| TransNat0 | [0, 1, 6, 34, 190, 1058, 5890, 32822, 183158] | +| TransNat1 | [1, 3, 14, 70, 362, 1910, 10214, 55154, 300034] | + +SchroederL Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [1, 4, 7, 10, 13, 16, 19, 22, 25]| +| DiagRow2 | [3, 17, 40, 72, 113, 163, 222, 290, 367]| +| DiagRow3 | [11, 76, 216, 458, 829, 1356, 2066, 2986, 4143]| +| DiagRow4 | [45, 353, 1145, 2745, 5558, 10070, 16848, 26540, 39875]| +| DiagRow5 | [197, 1688, 6039, 15932, 35318, 69660, 126176, 214082, 344835]| +| DiagRow6 | [903, 8257, 31864, 90776, 216870, 459546, 891154, 1613150, 2762981]| +| DiagRow7 | [4279, 41128, 168584, 511548, 1301478, 2932468, 6034094, 11563214, 20916909]| +| DiagRow8 | [20793, 207905, 895209, 2863593, 7687119, 18266831, 39595413, 79811017, 151605142]| + +SchroederL Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| DiagCol1 | [1, 4, 17, 76, 353, 1688, 8257, 41128, 207905] | +| DiagCol2 | [1, 7, 40, 216, 1145, 6039, 31864, 168584, 895209] | +| DiagCol3 | [1, 10, 72, 458, 2745, 15932, 90776, 511548, 2863593] | +| DiagCol4 | [1, 13, 113, 829, 5558, 35318, 216870, 1301478, 7687119] | +| DiagCol5 | [1, 16, 163, 1356, 10070, 69660, 459546, 2932468, 18266831] | +| DiagCol6 | [1, 19, 222, 2066, 16848, 126176, 891154, 6034094, 39595413] | +| DiagCol7 | [1, 22, 290, 2986, 26540, 214082, 1613150, 11563214, 79811017] | +| DiagCol8 | [1, 25, 367, 4143, 39875, 344835, 2762981, 20916909, 151605142] | + +SchroederL Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| PolyRow2 | [3, 8, 15, 24, 35, 48, 63, 80, 99] | +| PolyRow3 | [11, 36, 81, 152, 255, 396, 581, 816, 1107] | +| PolyRow4 | [45, 172, 453, 984, 1885, 3300, 5397, 8368, 12429] | +| PolyRow5 | [197, 852, 2583, 6440, 14025, 27612, 50267, 85968, 139725] | +| PolyRow6 | [903, 4324, 14907, 42408, 104695, 231468, 468699, 883792, 1571463] | +| PolyRow7 | [4279, 22332, 86733, 280312, 782955, 1942164, 4372417, 9088368, 17676927] | +| PolyRow8 | [20793, 116876, 507561, 1857336, 5861465, 16303908, 40799241, 93470576, 198855801] | + +SchroederL Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| PolyCol1 | [1, 2, 8, 36, 172, 852, 4324, 22332, 116876] | +| PolyCol2 | [1, 3, 15, 81, 453, 2583, 14907, 86733, 507561] | +| PolyCol3 | [1, 4, 24, 152, 984, 6440, 42408, 280312, 1857336] | +| PolyCol4 | [1, 5, 35, 255, 1885, 14025, 104695, 782955, 5861465] | +| PolyCol5 | [1, 6, 48, 396, 3300, 27612, 231468, 1942164, 16303908] | +| PolyCol6 | [1, 7, 63, 581, 5397, 50267, 468699, 4372417, 40799241] | +| PolyCol7 | [1, 8, 80, 816, 8368, 85968, 883792, 9088368, 93470576] | +| PolyCol8 | [1, 9, 99, 1107, 12429, 139725, 1571463, 17676927, 198855801] | + +# SchroederL:Inv +[] + +SchroederL:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [1, -4, 1] | +| Row3 | [-1, 11, -7, 1] | +| Row4 | [1, -26, 30, -10, 1] | +| Row5 | [-1, 57, -102, 58, -13, 1] | +| Row6 | [1, -120, 303, -256, 95, -16, 1] | +| Row7 | [-1, 247, -825, 955, -515, 141, -19, 1] | +| Row8 | [1, -502, 2116, -3178, 2310, -906, 196, -22, 1] | + +SchroederL:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -4, 1], [-1, 11, -7, 1], [1, -26, 30, -10, 1], [-1, 57, -102, 58, -13, 1], [1, -120, 303, -256, 95, -16, 1], [-1, 247, -825, 955, -515, 141, -19, 1], [1, -502, 2116, -3178, 2310, -906, 196, -22, 1]] | +| RevTabl | [[1], [1, -1], [1, -4, 1], [1, -7, 11, -1], [1, -10, 30, -26, 1], [1, -13, 58, -102, 57, -1], [1, -16, 95, -256, 303, -120, 1], [1, -19, 141, -515, 955, -825, 247, -1], [1, -22, 196, -906, 2310, -3178, 2116, -502, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -4], [1, 11, 1], [-1, -26, -7], [1, 57, 30, 1], [-1, -120, -102, -10], [1, 247, 303, 58, 1]] | +| AccTabl | [[1], [-1, 0], [1, -3, -2], [-1, 10, 3, 4], [1, -25, 5, -5, -4], [-1, 56, -46, 12, -1, 0], [1, -119, 184, -72, 23, 7, 8], [-1, 246, -579, 376, -139, 2, -17, -16], [1, -501, 1615, -1563, 747, -159, 37, 15, 16]] | +| RevAccTabl | [[1], [0, -1], [-2, -3, 1], [4, 3, 10, -1], [-4, -5, 5, -25, 1], [0, -1, 12, -46, 56, -1], [8, 7, 23, -72, 184, -119, 1], [-16, -17, 2, -139, 376, -579, 246, -1], [16, 15, 37, -159, 747, -1563, 1615, -501, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -3, -2], [1, -6, 5, 4], [1, -9, 21, -5, -4], [1, -12, 46, -56, 1, 0], [1, -15, 80, -176, 127, 7, 8], [1, -18, 123, -392, 563, -262, -15, -16], [1, -21, 175, -731, 1579, -1599, 517, 15, 16]] | + +SchroederL:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 4, -4, 0, 8, -16, 16] | +| EvenSum | [1, -1, 2, -8, 32, -116, 400, -1360, 4624] | +| OddSum | [0, 1, -4, 12, -36, 116, -392, 1344, -4608] | +| AltSum | [1, -2, 6, -20, 68, -232, 792, -2704, 9232] | +| AccSum | [1, -1, -4, 16, -28, 20, 32, -128, 208] | +| AccRevSum | [1, 1, -4, 4, 4, -20, 32, -16, -48] | +| AntiDiagSum | [1, -1, 2, -5, 13, -34, 89, -233, 610] | + +SchroederL:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 77, 390, 730626, 7368960, 188416682025, 147187935422220] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 11, 30, 102, 303, 955, 3178] | +| ColMiddle | [1, -1, -4, 11, 30, -102, -256, 955, 2310] | +| ColECenter | [1, -4, 30, -256, 2310] | +| ColOCenter | [-1, 11, -102, 955] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 0, -8, 20, -12, -56, 192, -288] | +| TransNat0 | [0, 1, -2, 0, 8, -20, 24, 0, -64] | +| TransNat1 | [1, 1, -4, 4, 4, -20, 32, -16, -48] | + +SchroederL:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -4, -7, -10, -13, -16, -19, -22, -25]| +| DiagRow2 | [1, 11, 30, 58, 95, 141, 196, 260, 333]| +| DiagRow3 | [-1, -26, -102, -256, -515, -906, -1456, -2192, -3141]| +| DiagRow4 | [1, 57, 303, 955, 2310, 4746, 8722, 14778, 23535]| +| DiagRow5 | [-1, -120, -825, -3178, -9078, -21504, -44758, -84708, -149031]| +| DiagRow6 | [1, 247, 2116, 9740, 32354, 87374, 204204, 428772, 828795]| +| DiagRow7 | [-1, -502, -5200, -28064, -106970, -326084, -849180, -1966080, -4154403]| +| DiagRow8 | [1, 1013, 12381, 77093, 333295, 1136799, 3275931, 8316627, 19122246]| + +SchroederL:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| DiagCol1 | [1, -4, 11, -26, 57, -120, 247, -502, 1013] | +| DiagCol2 | [1, -7, 30, -102, 303, -825, 2116, -5200, 12381] | +| DiagCol3 | [1, -10, 58, -256, 955, -3178, 9740, -28064, 77093] | +| DiagCol4 | [1, -13, 95, -515, 2310, -9078, 32354, -106970, 333295] | +| DiagCol5 | [1, -16, 141, -906, 4746, -21504, 87374, -326084, 1136799] | +| DiagCol6 | [1, -19, 196, -1456, 8722, -44758, 204204, -849180, 3275931] | +| DiagCol7 | [1, -22, 260, -2192, 14778, -84708, 428772, -1966080, 8316627] | +| DiagCol8 | [1, -25, 333, -3141, 23535, -149031, 828795, -4154403, 19122246] | + +SchroederL:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [1, -2, -3, -2, 1, 6, 13, 22, 33] | +| PolyRow3 | [-1, 4, 1, -4, -5, 4, 29, 76, 151] | +| PolyRow4 | [1, -4, 5, 4, -7, -4, 61, 260, 689] | +| PolyRow5 | [-1, 0, -7, 8, 3, -16, 125, 888, 3143] | +| PolyRow6 | [1, 8, -3, -8, 17, -24, 253, 3032, 14337] | +| PolyRow7 | [-1, -16, 17, -16, 11, -16, 509, 10352, 65399] | +| PolyRow8 | [1, 16, -11, 16, -23, 16, 1021, 35344, 298321] | + +SchroederL:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| PolyCol1 | [1, 0, -2, 4, -4, 0, 8, -16, 16] | +| PolyCol2 | [1, 1, -3, 1, 5, -7, -3, 17, -11] | +| PolyCol3 | [1, 2, -2, -4, 4, 8, -8, -16, 16] | +| PolyCol4 | [1, 3, 1, -5, -7, 3, 17, 11, -23] | +| PolyCol5 | [1, 4, 6, 4, -4, -16, -24, -16, 16] | +| PolyCol6 | [1, 5, 13, 29, 61, 125, 253, 509, 1021] | +| PolyCol7 | [1, 6, 22, 76, 260, 888, 3032, 10352, 35344] | +| PolyCol8 | [1, 7, 33, 151, 689, 3143, 14337, 65399, 298321] | + +# SchroederL:Rev +[] + +SchroederL:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, 1] | +| Row2 | [1, 4, 3] | +| Row3 | [1, 7, 17, 11] | +| Row4 | [1, 10, 40, 76, 45] | +| Row5 | [1, 13, 72, 216, 353, 197] | +| Row6 | [1, 16, 113, 458, 1145, 1688, 903] | +| Row7 | [1, 19, 163, 829, 2745, 6039, 8257, 4279] | +| Row8 | [1, 22, 222, 1356, 5558, 15932, 31864, 41128, 20793] | + +SchroederL:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 4, 3], [1, 7, 17, 11], [1, 10, 40, 76, 45], [1, 13, 72, 216, 353, 197], [1, 16, 113, 458, 1145, 1688, 903], [1, 19, 163, 829, 2745, 6039, 8257, 4279], [1, 22, 222, 1356, 5558, 15932, 31864, 41128, 20793]] | +| RevTabl | [[1], [1, 1], [3, 4, 1], [11, 17, 7, 1], [45, 76, 40, 10, 1], [197, 353, 216, 72, 13, 1], [903, 1688, 1145, 458, 113, 16, 1], [4279, 8257, 6039, 2745, 829, 163, 19, 1], [20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 4], [1, 7, 3], [1, 10, 17], [1, 13, 40, 11], [1, 16, 72, 76], [1, 19, 113, 216, 45]] | +| AccTabl | [[1], [1, 2], [1, 5, 8], [1, 8, 25, 36], [1, 11, 51, 127, 172], [1, 14, 86, 302, 655, 852], [1, 17, 130, 588, 1733, 3421, 4324], [1, 20, 183, 1012, 3757, 9796, 18053, 22332], [1, 23, 245, 1601, 7159, 23091, 54955, 96083, 116876]] | +| RevAccTabl | [[1], [2, 1], [8, 5, 1], [36, 25, 8, 1], [172, 127, 51, 11, 1], [852, 655, 302, 86, 14, 1], [4324, 3421, 1733, 588, 130, 17, 1], [22332, 18053, 9796, 3757, 1012, 183, 20, 1], [116876, 96083, 54955, 23091, 7159, 1601, 245, 23, 1]] | +| AccRevTabl | [[1], [1, 2], [3, 7, 8], [11, 28, 35, 36], [45, 121, 161, 171, 172], [197, 550, 766, 838, 851, 852], [903, 2591, 3736, 4194, 4307, 4323, 4324], [4279, 12536, 18575, 21320, 22149, 22312, 22331, 22332], [20793, 61921, 93785, 109717, 115275, 116631, 116853, 116875, 116876]] | + +SchroederL:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 36, 172, 852, 4324, 22332, 116876] | +| EvenSum | [1, 1, 4, 18, 86, 426, 2162, 11166, 58438] | +| OddSum | [0, 1, 4, 18, 86, 426, 2162, 11166, 58438] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 14, 70, 362, 1910, 10214, 55154, 300034] | +| AccRevSum | [1, 3, 18, 110, 670, 4054, 24378, 145834, 868726] | +| AntiDiagSum | [1, 1, 2, 5, 11, 28, 65, 165, 394] | + +SchroederL:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1309, 6840, 195271128, 394433795280, 249001897016460555, 62191124612870563464] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 17, 76, 353, 1688, 8257, 41128] | +| ColMiddle | [1, 1, 4, 7, 40, 72, 458, 829, 5558] | +| ColECenter | [1, 4, 40, 458, 5558] | +| ColOCenter | [1, 7, 72, 829] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 3, 11, 45, 197, 903, 4279, 20793] | +| TransSqrs | [0, 1, 16, 174, 1574, 12818, 97618, 709950, 4993470] | +| TransNat0 | [0, 1, 10, 74, 498, 3202, 20054, 123502, 751850] | +| TransNat1 | [1, 3, 18, 110, 670, 4054, 24378, 145834, 868726] | + +SchroederL:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 3, 11, 45, 197, 903, 4279, 20793]| +| DiagRow1 | [1, 4, 17, 76, 353, 1688, 8257, 41128, 207905]| +| DiagRow2 | [1, 7, 40, 216, 1145, 6039, 31864, 168584, 895209]| +| DiagRow3 | [1, 10, 72, 458, 2745, 15932, 90776, 511548, 2863593]| +| DiagRow4 | [1, 13, 113, 829, 5558, 35318, 216870, 1301478, 7687119]| +| DiagRow5 | [1, 16, 163, 1356, 10070, 69660, 459546, 2932468, 18266831]| +| DiagRow6 | [1, 19, 222, 2066, 16848, 126176, 891154, 6034094, 39595413]| +| DiagRow7 | [1, 22, 290, 2986, 26540, 214082, 1613150, 11563214, 79811017]| +| DiagRow8 | [1, 25, 367, 4143, 39875, 344835, 2762981, 20916909, 151605142]| + +SchroederL:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [1, 4, 7, 10, 13, 16, 19, 22, 25] | +| DiagCol2 | [3, 17, 40, 72, 113, 163, 222, 290, 367] | +| DiagCol3 | [11, 76, 216, 458, 829, 1356, 2066, 2986, 4143] | +| DiagCol4 | [45, 353, 1145, 2745, 5558, 10070, 16848, 26540, 39875] | +| DiagCol5 | [197, 1688, 6039, 15932, 35318, 69660, 126176, 214082, 344835] | +| DiagCol6 | [903, 8257, 31864, 90776, 216870, 459546, 891154, 1613150, 2762981] | +| DiagCol7 | [4279, 41128, 168584, 511548, 1301478, 2932468, 6034094, 11563214, 20916909] | +| DiagCol8 | [20793, 207905, 895209, 2863593, 7687119, 18266831, 39595413, 79811017, 151605142] | + +SchroederL:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| PolyRow2 | [1, 8, 21, 40, 65, 96, 133, 176, 225] | +| PolyRow3 | [1, 36, 171, 472, 1005, 1836, 3031, 4656, 6777] | +| PolyRow4 | [1, 172, 1509, 6088, 17065, 38676, 76237, 136144, 225873] | +| PolyRow5 | [1, 852, 13995, 82984, 307125, 865116, 2038687, 4236240, 8016489] | +| PolyRow6 | [1, 4324, 134277, 1174648, 5751505, 20160156, 56843269, 137519152, 296960193] | +| PolyRow7 | [1, 22332, 1320651, 17091256, 110870205, 484007796, 1633786567, 4603748976, 11347800921] | +| PolyRow8 | [1, 116876, 13237221, 253959016, 2184873785, 11886702036, 48055501501, 157768541456, 443998810161] | + +SchroederL:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 2, 8, 36, 172, 852, 4324, 22332, 116876] | +| PolyCol2 | [1, 3, 21, 171, 1509, 13995, 134277, 1320651, 13237221] | +| PolyCol3 | [1, 4, 40, 472, 6088, 82984, 1174648, 17091256, 253959016] | +| PolyCol4 | [1, 5, 65, 1005, 17065, 307125, 5751505, 110870205, 2184873785] | +| PolyCol5 | [1, 6, 96, 1836, 38676, 865116, 20160156, 484007796, 11886702036] | +| PolyCol6 | [1, 7, 133, 3031, 76237, 2038687, 56843269, 1633786567, 48055501501] | +| PolyCol7 | [1, 8, 176, 4656, 136144, 4236240, 137519152, 4603748976, 157768541456] | +| PolyCol8 | [1, 9, 225, 6777, 225873, 8016489, 296960193, 11347800921, 443998810161] | + +# SchroederL:Inv:Rev +[] + +SchroederL:Inv:Rev Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [1, -1] | +| Row2 | [1, -4, 1] | +| Row3 | [1, -7, 11, -1] | +| Row4 | [1, -10, 30, -26, 1] | +| Row5 | [1, -13, 58, -102, 57, -1] | +| Row6 | [1, -16, 95, -256, 303, -120, 1] | +| Row7 | [1, -19, 141, -515, 955, -825, 247, -1] | +| Row8 | [1, -22, 196, -906, 2310, -3178, 2116, -502, 1] | + +SchroederL:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -4, 1], [1, -7, 11, -1], [1, -10, 30, -26, 1], [1, -13, 58, -102, 57, -1], [1, -16, 95, -256, 303, -120, 1], [1, -19, 141, -515, 955, -825, 247, -1], [1, -22, 196, -906, 2310, -3178, 2116, -502, 1]] | +| RevTabl | [[1], [-1, 1], [1, -4, 1], [-1, 11, -7, 1], [1, -26, 30, -10, 1], [-1, 57, -102, 58, -13, 1], [1, -120, 303, -256, 95, -16, 1], [-1, 247, -825, 955, -515, 141, -19, 1], [1, -502, 2116, -3178, 2310, -906, 196, -22, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -4], [1, -7, 1], [1, -10, 11], [1, -13, 30, -1], [1, -16, 58, -26], [1, -19, 95, -102, 1]] | +| AccTabl | [[1], [1, 0], [1, -3, -2], [1, -6, 5, 4], [1, -9, 21, -5, -4], [1, -12, 46, -56, 1, 0], [1, -15, 80, -176, 127, 7, 8], [1, -18, 123, -392, 563, -262, -15, -16], [1, -21, 175, -731, 1579, -1599, 517, 15, 16]] | +| RevAccTabl | [[1], [0, 1], [-2, -3, 1], [4, 5, -6, 1], [-4, -5, 21, -9, 1], [0, 1, -56, 46, -12, 1], [8, 7, 127, -176, 80, -15, 1], [-16, -15, -262, 563, -392, 123, -18, 1], [16, 15, 517, -1599, 1579, -731, 175, -21, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -3, -2], [-1, 10, 3, 4], [1, -25, 5, -5, -4], [-1, 56, -46, 12, -1, 0], [1, -119, 184, -72, 23, 7, 8], [-1, 246, -579, 376, -139, 2, -17, -16], [1, -501, 1615, -1563, 747, -159, 37, 15, 16]] | + +SchroederL:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 4, -4, 0, 8, -16, 16] | +| EvenSum | [1, 1, 2, 12, 32, 116, 400, 1344, 4624] | +| OddSum | [0, -1, -4, -8, -36, -116, -392, -1360, -4608] | +| AltSum | [1, 2, 6, 20, 68, 232, 792, 2704, 9232] | +| AccSum | [1, 1, -4, 4, 4, -20, 32, -16, -48] | +| AccRevSum | [1, -1, -4, 16, -28, 20, 32, -128, 208] | +| AntiDiagSum | [1, 1, 0, -3, -5, 2, 17, 17, -24] | + +SchroederL:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 77, 390, 730626, 7368960, 188416682025, 147187935422220] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 11, 30, 102, 303, 955, 3178] | +| ColMiddle | [1, 1, -4, -7, 30, 58, -256, -515, 2310] | +| ColECenter | [1, -4, 30, -256, 2310] | +| ColOCenter | [1, -7, 58, -515] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| TransSqrs | [0, -1, 0, 28, -108, 188, -56, -592, 1760] | +| TransNat0 | [0, -1, -2, 12, -24, 20, 24, -112, 192] | +| TransNat1 | [1, -1, -4, 16, -28, 20, 32, -128, 208] | + +SchroederL:Inv:Rev Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, -1, 1, -1, 1, -1, 1, -1, 1]| +| DiagRow1 | [1, -4, 11, -26, 57, -120, 247, -502, 1013]| +| DiagRow2 | [1, -7, 30, -102, 303, -825, 2116, -5200, 12381]| +| DiagRow3 | [1, -10, 58, -256, 955, -3178, 9740, -28064, 77093]| +| DiagRow4 | [1, -13, 95, -515, 2310, -9078, 32354, -106970, 333295]| +| DiagRow5 | [1, -16, 141, -906, 4746, -21504, 87374, -326084, 1136799]| +| DiagRow6 | [1, -19, 196, -1456, 8722, -44758, 204204, -849180, 3275931]| +| DiagRow7 | [1, -22, 260, -2192, 14778, -84708, 428772, -1966080, 8316627]| +| DiagRow8 | [1, -25, 333, -3141, 23535, -149031, 828795, -4154403, 19122246]| + +SchroederL:Inv:Rev Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| DiagCol1 | [-1, -4, -7, -10, -13, -16, -19, -22, -25] | +| DiagCol2 | [1, 11, 30, 58, 95, 141, 196, 260, 333] | +| DiagCol3 | [-1, -26, -102, -256, -515, -906, -1456, -2192, -3141] | +| DiagCol4 | [1, 57, 303, 955, 2310, 4746, 8722, 14778, 23535] | +| DiagCol5 | [-1, -120, -825, -3178, -9078, -21504, -44758, -84708, -149031] | +| DiagCol6 | [1, 247, 2116, 9740, 32354, 87374, 204204, 428772, 828795] | +| DiagCol7 | [-1, -502, -5200, -28064, -106970, -326084, -849180, -1966080, -4154403] | +| DiagCol8 | [1, 1013, 12381, 77093, 333295, 1136799, 3275931, 8316627, 19122246] | + +SchroederL:Inv:Rev Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| PolyRow2 | [1, -2, -3, -2, 1, 6, 13, 22, 33] | +| PolyRow3 | [1, 4, 23, 52, 85, 116, 139, 148, 137] | +| PolyRow4 | [1, -4, -91, -380, -967, -1924, -3299, -5116, -7375] | +| PolyRow5 | [1, 0, 271, 2104, 7917, 21136, 46075, 87816, 152089] | +| PolyRow6 | [1, 8, -627, -9992, -56143, -199704, -545747, -1254952, -2554047] | +| PolyRow7 | [1, -16, 967, 42064, 364229, 1739056, 5960299, 16493072, 39275689] | +| PolyRow8 | [1, 16, 181, -156656, -2209943, -14361584, -62031299, -206876144, -576422831] | + +SchroederL:Inv:Rev Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyCol1 | [1, 0, -2, 4, -4, 0, 8, -16, 16] | +| PolyCol2 | [1, -1, -3, 23, -91, 271, -627, 967, 181] | +| PolyCol3 | [1, -2, -2, 52, -380, 2104, -9992, 42064, -156656] | +| PolyCol4 | [1, -3, 1, 85, -967, 7917, -56143, 364229, -2209943] | +| PolyCol5 | [1, -4, 6, 116, -1924, 21136, -199704, 1739056, -14361584] | +| PolyCol6 | [1, -5, 13, 139, -3299, 46075, -545747, 5960299, -62031299] | +| PolyCol7 | [1, -6, 22, 148, -5116, 87816, -1254952, 16493072, -206876144] | +| PolyCol8 | [1, -7, 33, 137, -7375, 152089, -2554047, 39275689, -576422831] | + diff --git a/data/md/SchroederP.md b/data/md/SchroederP.md index 7f98197..7a75213 100644 --- a/data/md/SchroederP.md +++ b/data/md/SchroederP.md @@ -1,5 +1,5 @@ # SchroederP -['A063007', 'A104684'] +['A104684', 'A063007'] SchroederP Triangle view | Row | Seq | @@ -14,45 +14,41 @@ SchroederP Triangle view | Row7 | [3432, 12012, 16632, 11550, 4200, 756, 56, 1] | | Row8 | [12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1] | -SchroederP Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 1, 6, 6, 1, 20, 30, 12, 1, 70, 140, 90, 20, 1, 252, 630, 560, 210, 30, 1, 924, 2772, 3150, 1680, 420, 42, 1, 3432, 12012, 16632, 11550, 4200, 756, 56, 1, 12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1] | -| TriRev | [1, 1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, 70, 1, 30, 210, 560, 630, 252, 1, 42, 420, 1680, 3150, 2772, 924, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870] | -| TriAcc | [1, 2, 3, 6, 12, 13, 20, 50, 62, 63, 70, 210, 300, 320, 321, 252, 882, 1442, 1652, 1682, 1683, 924, 3696, 6846, 8526, 8946, 8988, 8989, 3432, 15444, 32076, 43626, 47826, 48582, 48638, 48639, 12870, 64350, 148434, 220506, 255156, 264396, 265656, 265728, 265729] | -| TriRevAcc | [1, 3, 2, 13, 12, 6, 63, 62, 50, 20, 321, 320, 300, 210, 70, 1683, 1682, 1652, 1442, 882, 252, 8989, 8988, 8946, 8526, 6846, 3696, 924, 48639, 48638, 48582, 47826, 43626, 32076, 15444, 3432, 265729, 265728, 265656, 264396, 255156, 220506, 148434, 64350, 12870] | -| TriAccRev | [1, 1, 3, 1, 7, 13, 1, 13, 43, 63, 1, 21, 111, 251, 321, 1, 31, 241, 801, 1431, 1683, 1, 43, 463, 2143, 5293, 8065, 8989, 1, 57, 813, 5013, 16563, 33195, 45207, 48639, 1, 73, 1333, 10573, 45223, 117295, 201379, 252859, 265729] | -| TriDiag | [1, 2, 6, 1, 20, 6, 70, 30, 1, 252, 140, 12, 924, 630, 90, 1, 3432, 2772, 560, 20] | +SchroederP Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 1], [6, 6, 1], [20, 30, 12, 1], [70, 140, 90, 20, 1], [252, 630, 560, 210, 30, 1], [924, 2772, 3150, 1680, 420, 42, 1], [3432, 12012, 16632, 11550, 4200, 756, 56, 1], [12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1]] | +| RevTabl | [[1], [1, 2], [1, 6, 6], [1, 12, 30, 20], [1, 20, 90, 140, 70], [1, 30, 210, 560, 630, 252], [1, 42, 420, 1680, 3150, 2772, 924], [1, 56, 756, 4200, 11550, 16632, 12012, 3432], [1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870]] | +| AntiDiag | [[1], [2], [6, 1], [20, 6], [70, 30, 1], [252, 140, 12], [924, 630, 90, 1], [3432, 2772, 560, 20], [12870, 12012, 3150, 210, 1]] | +| AccTabl | [[1], [2, 3], [6, 12, 13], [20, 50, 62, 63], [70, 210, 300, 320, 321], [252, 882, 1442, 1652, 1682, 1683], [924, 3696, 6846, 8526, 8946, 8988, 8989], [3432, 15444, 32076, 43626, 47826, 48582, 48638, 48639], [12870, 64350, 148434, 220506, 255156, 264396, 265656, 265728, 265729]] | +| RevAccTabl | [[1], [3, 2], [13, 12, 6], [63, 62, 50, 20], [321, 320, 300, 210, 70], [1683, 1682, 1652, 1442, 882, 252], [8989, 8988, 8946, 8526, 6846, 3696, 924], [48639, 48638, 48582, 47826, 43626, 32076, 15444, 3432], [265729, 265728, 265656, 264396, 255156, 220506, 148434, 64350, 12870]] | +| AccRevTabl | [[1], [1, 3], [1, 7, 13], [1, 13, 43, 63], [1, 21, 111, 251, 321], [1, 31, 241, 801, 1431, 1683], [1, 43, 463, 2143, 5293, 8065, 8989], [1, 57, 813, 5013, 16563, 33195, 45207, 48639], [1, 73, 1333, 10573, 45223, 117295, 201379, 252859, 265729]] | SchroederP Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729] | -| EvenSum | [1, 2, 7, 32, 161, 842, 4495, 24320, 132865] | -| OddSum | [0, 1, 6, 31, 160, 841, 4494, 24319, 132864] | -| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| AccSum | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | -| AccRevSum | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | -| RevAccSum | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | -| DiagSum | [1, 2, 7, 26, 101, 404, 1645, 6784] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729] | +| EvenSum | [1, 2, 7, 32, 161, 842, 4495, 24320, 132865] | +| OddSum | [0, 1, 6, 31, 160, 841, 4494, 24319, 132864] | +| AltSum | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| AccSum | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | +| AccRevSum | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | +| AntiDiagSum | [1, 2, 7, 26, 101, 404, 1645, 6784, 28243] | SchroederP Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 6, 60, 1260, 5040, 277200, 5405400, 12612600] | -| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | -| RowMax | [1, 2, 6, 30, 140, 630, 3150, 16632, 84084] | -| Middle | [1, 2, 6, 30, 90, 560, 1680, 11550, 34650] | -| Central | [1, 6, 90, 1680, 34650] | -| LeftSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 5, 37, 305, 2641, 23525, 213445, 1961825, 18205345] | -| NegHalf | [1, -3, 13, -63, 321, -1683, 8989, -48639, 265729] | -| Bin | [1, 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891] | -| InvBin | [1, -1, -5, 35, -29, -751, 3991, 4115, -137885] | -| TransSqrs | [0, 1, 10, 87, 696, 5265, 38298, 270655, 1870816] | -| TransNat0 | [0, 1, 8, 57, 384, 2505, 16008, 100849, 628736] | -| TransNat1 | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 60, 1260, 5040, 277200, 5405400, 12612600] | +| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | +| RowMax | [1, 2, 6, 30, 140, 630, 3150, 16632, 84084] | +| ColMiddle | [1, 2, 6, 30, 90, 560, 1680, 11550, 34650] | +| ColECenter | [1, 6, 90, 1680, 34650] | +| ColOCenter | [2, 30, 560, 11550] | +| ColLeft | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 10, 87, 696, 5265, 38298, 270655, 1870816] | +| TransNat0 | [0, 1, 8, 57, 384, 2505, 16008, 100849, 628736] | +| TransNat1 | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | SchroederP Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ SchroederP Polynomial values as columns | PolyCol7 | [1, 9, 97, 1161, 14721, 192969, 2582881, 35066313, 481003009] | | PolyCol8 | [1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270] | -# SchroederPInv -True +# SchroederP:Inv +[] -SchroederPInv Triangle view +SchroederP:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SchroederPInv Triangle view | Row7 | [-3802752, 5497632, -1986432, 318570, -28560, 1596, -56, 1] | | Row8 | [163825974, -236856384, 85606584, -13748112, 1240470, -71232, 2772, -72, 1] | -SchroederPInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -2, 1, 6, -6, 1, -32, 42, -12, 1, 310, -440, 150, -20, 1, -4932, 7110, -2540, 390, -30, 1, 116424, -168252, 60690, -9660, 840, -42, 1, -3802752, 5497632, -1986432, 318570, -28560, 1596, -56, 1, 163825974, -236856384, 85606584, -13748112, 1240470, -71232, 2772, -72, 1] | -| TriRev | [1, 1, -2, 1, -6, 6, 1, -12, 42, -32, 1, -20, 150, -440, 310, 1, -30, 390, -2540, 7110, -4932, 1, -42, 840, -9660, 60690, -168252, 116424, 1, -56, 1596, -28560, 318570, -1986432, 5497632, -3802752, 1, -72, 2772, -71232, 1240470, -13748112, 85606584, -236856384, 163825974] | -| TriAcc | [1, -2, -1, 6, 0, 1, -32, 10, -2, -1, 310, -130, 20, 0, 1, -4932, 2178, -362, 28, -2, -1, 116424, -51828, 8862, -798, 42, 0, 1, -3802752, 1694880, -291552, 27018, -1542, 54, -2, -1, 163825974, -73030410, 12576174, -1171938, 68532, -2700, 72, 0, 1] | -| TriRevAcc | [1, -1, -2, 1, 0, 6, -1, -2, 10, -32, 1, 0, 20, -130, 310, -1, -2, 28, -362, 2178, -4932, 1, 0, 42, -798, 8862, -51828, 116424, -1, -2, 54, -1542, 27018, -291552, 1694880, -3802752, 1, 0, 72, -2700, 68532, -1171938, 12576174, -73030410, 163825974] | -| TriAccRev | [1, 1, -1, 1, -5, 1, 1, -11, 31, -1, 1, -19, 131, -309, 1, 1, -29, 361, -2179, 4931, -1, 1, -41, 799, -8861, 51829, -116423, 1, 1, -55, 1541, -27019, 291551, -1694881, 3802751, -1, 1, -71, 2701, -68531, 1171939, -12576173, 73030411, -163825973, 1] | -| TriDiag | [1, -2, 6, 1, -32, -6, 310, 42, 1, -4932, -440, -12, 116424, 7110, 150, 1, -3802752, -168252, -2540, -20] | - -SchroederPInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| EvenSum | [1, -2, 7, -44, 461, -7502, 177955, -5817800, 250675801] | -| OddSum | [0, 1, -6, 43, -460, 7501, -177954, 5817799, -250675800] | -| AltSum | [1, -3, 13, -87, 921, -15003, 355909, -11635599, 501351601] | -| AccSum | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | -| AccRevSum | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | -| RevAccSum | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | -| DiagSum | [1, -2, 7, -38, 353, -5384, 123685, -3973564] | - -SchroederPInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, -2, 6, 672, 204600, 3216379140, 7750311334920, 131426737497966879360, 54722432423401919925487384726080] | -| RowGcd | [1, -2, 6, 2, 10, 2, 42, 2, 6] | -| RowMax | [1, 2, 6, 42, 440, 7110, 168252, 5497632, 236856384] | -| Middle | [1, -2, -6, 42, 150, -2540, -9660, 318570, 1240470] | -| Central | [1, -6, 150, -9660, 1240470] | -| LeftSide | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -3, 13, -111, 2001, -62883, 2964109, -193594719, 16680002593] | -| NegHalf | [1, 5, 37, 449, 9121, 293525, 13886965, 907498625, 78196256065] | -| Bin | [1, -1, -5, 59, -629, 8969, -163589, 3149075, -21014069] | -| InvBin | [1, 3, 19, 195, 3051, 69933, 2242339, 96184707, 5316455755] | -| TransSqrs | [0, 1, -2, 3, -4, 5, -6, 7, -8] | -| TransNat0 | [0, 1, -4, 21, -196, 3085, -72696, 2373889, -102265696] | -| TransNat1 | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | - -SchroederPInv Diagonals as rows +SchroederP:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-2, 1], [6, -6, 1], [-32, 42, -12, 1], [310, -440, 150, -20, 1], [-4932, 7110, -2540, 390, -30, 1], [116424, -168252, 60690, -9660, 840, -42, 1], [-3802752, 5497632, -1986432, 318570, -28560, 1596, -56, 1], [163825974, -236856384, 85606584, -13748112, 1240470, -71232, 2772, -72, 1]] | +| RevTabl | [[1], [1, -2], [1, -6, 6], [1, -12, 42, -32], [1, -20, 150, -440, 310], [1, -30, 390, -2540, 7110, -4932], [1, -42, 840, -9660, 60690, -168252, 116424], [1, -56, 1596, -28560, 318570, -1986432, 5497632, -3802752], [1, -72, 2772, -71232, 1240470, -13748112, 85606584, -236856384, 163825974]] | +| AntiDiag | [[1], [-2], [6, 1], [-32, -6], [310, 42, 1], [-4932, -440, -12], [116424, 7110, 150, 1], [-3802752, -168252, -2540, -20], [163825974, 5497632, 60690, 390, 1]] | +| AccTabl | [[1], [-2, -1], [6, 0, 1], [-32, 10, -2, -1], [310, -130, 20, 0, 1], [-4932, 2178, -362, 28, -2, -1], [116424, -51828, 8862, -798, 42, 0, 1], [-3802752, 1694880, -291552, 27018, -1542, 54, -2, -1], [163825974, -73030410, 12576174, -1171938, 68532, -2700, 72, 0, 1]] | +| RevAccTabl | [[1], [-1, -2], [1, 0, 6], [-1, -2, 10, -32], [1, 0, 20, -130, 310], [-1, -2, 28, -362, 2178, -4932], [1, 0, 42, -798, 8862, -51828, 116424], [-1, -2, 54, -1542, 27018, -291552, 1694880, -3802752], [1, 0, 72, -2700, 68532, -1171938, 12576174, -73030410, 163825974]] | +| AccRevTabl | [[1], [1, -1], [1, -5, 1], [1, -11, 31, -1], [1, -19, 131, -309, 1], [1, -29, 361, -2179, 4931, -1], [1, -41, 799, -8861, 51829, -116423, 1], [1, -55, 1541, -27019, 291551, -1694881, 3802751, -1], [1, -71, 2701, -68531, 1171939, -12576173, 73030411, -163825973, 1]] | + +SchroederP:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| EvenSum | [1, -2, 7, -44, 461, -7502, 177955, -5817800, 250675801] | +| OddSum | [0, 1, -6, 43, -460, 7501, -177954, 5817799, -250675800] | +| AltSum | [1, -3, 13, -87, 921, -15003, 355909, -11635599, 501351601] | +| AccSum | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | +| AccRevSum | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | +| AntiDiagSum | [1, -2, 7, -38, 353, -5384, 123685, -3973564, 169384687] | + +SchroederP:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 672, 204600, 3216379140, 7750311334920, 131426737497966879360, 54722432423401919925487384726080] | +| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | +| RowMax | [1, 2, 6, 42, 440, 7110, 168252, 5497632, 236856384] | +| ColMiddle | [1, -2, -6, 42, 150, -2540, -9660, 318570, 1240470] | +| ColECenter | [1, -6, 150, -9660, 1240470] | +| ColOCenter | [-2, 42, -2540, 318570] | +| ColLeft | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, -2, 3, -4, 5, -6, 7, -8] | +| TransNat0 | [0, 1, -4, 21, -196, 3085, -72696, 2373889, -102265696] | +| TransNat1 | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | + +SchroederP:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ SchroederPInv Diagonals as rows | DiagRow7 | [-3802752, -236856384, -4697384400, -51337094160, -383049120960, -2181870518256, -10146680307192, -40261266559800, -140561186800320]| | DiagRow8 | [163825974, 12996105750, 319573375110, 4239074011590, 37728135077490, 252651966609042, 1364519028362490, 6222162826878570, 24737671557915030]| -SchroederPInv Diagonals as columns +SchroederP:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | @@ -188,7 +180,7 @@ SchroederPInv Diagonals as columns | DiagCol7 | [1, -72, 4500, -315480, 26212230, -2584821096, 299376289212, -40261266559800, 6222162826878570] | | DiagCol8 | [1, -90, 6930, -588720, 58326840, -6761650896, 909489300720, -140561186800320, 24737671557915030] | -SchroederPInv Polynomial values as rows +SchroederP:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SchroederPInv Polynomial values as rows | PolyRow7 | [-3802752, -1, 1386000, 1449477, 903296, 186483, -453456, -898375, -1112448] | | PolyRow8 | [163825974, 1, -59708810, -62444367, -38914634, -8033471, 19536246, 38716465, 48043318] | -SchroederPInv Polynomial values as columns +SchroederP:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | @@ -214,10 +206,10 @@ SchroederPInv Polynomial values as columns | PolyCol7 | [1, 5, 13, 17, 121, -1075, 27685, -898375, 38716465] | | PolyCol8 | [1, 6, 22, 48, 246, -1044, 35176, -1112448, 48043318] | -# SchroederPRev -True +# SchroederP:Rev +[] -SchroederPRev Triangle view +SchroederP:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ SchroederPRev Triangle view | Row7 | [1, 56, 756, 4200, 11550, 16632, 12012, 3432] | | Row8 | [1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870] | -SchroederPRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, 70, 1, 30, 210, 560, 630, 252, 1, 42, 420, 1680, 3150, 2772, 924, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870] | -| TriRev | [1, 2, 1, 6, 6, 1, 20, 30, 12, 1, 70, 140, 90, 20, 1, 252, 630, 560, 210, 30, 1, 924, 2772, 3150, 1680, 420, 42, 1, 3432, 12012, 16632, 11550, 4200, 756, 56, 1, 12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1] | -| TriAcc | [1, 1, 3, 1, 7, 13, 1, 13, 43, 63, 1, 21, 111, 251, 321, 1, 31, 241, 801, 1431, 1683, 1, 43, 463, 2143, 5293, 8065, 8989, 1, 57, 813, 5013, 16563, 33195, 45207, 48639, 1, 73, 1333, 10573, 45223, 117295, 201379, 252859, 265729] | -| TriRevAcc | [1, 3, 1, 13, 7, 1, 63, 43, 13, 1, 321, 251, 111, 21, 1, 1683, 1431, 801, 241, 31, 1, 8989, 8065, 5293, 2143, 463, 43, 1, 48639, 45207, 33195, 16563, 5013, 813, 57, 1, 265729, 252859, 201379, 117295, 45223, 10573, 1333, 73, 1] | -| TriAccRev | [1, 2, 3, 6, 12, 13, 20, 50, 62, 63, 70, 210, 300, 320, 321, 252, 882, 1442, 1652, 1682, 1683, 924, 3696, 6846, 8526, 8946, 8988, 8989, 3432, 15444, 32076, 43626, 47826, 48582, 48638, 48639, 12870, 64350, 148434, 220506, 255156, 264396, 265656, 265728, 265729] | -| TriDiag | [1, 1, 1, 2, 1, 6, 1, 12, 6, 1, 20, 30, 1, 30, 90, 20, 1, 42, 210, 140] | - -SchroederPRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729] | -| EvenSum | [1, 1, 7, 31, 161, 841, 4495, 24319, 132865] | -| OddSum | [0, 2, 6, 32, 160, 842, 4494, 24320, 132864] | -| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| AccSum | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | -| AccRevSum | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | -| RevAccSum | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | -| DiagSum | [1, 1, 3, 7, 19, 51, 141, 393] | - -SchroederPRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 6, 60, 1260, 5040, 277200, 5405400, 12612600] | -| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | -| RowMax | [1, 2, 6, 30, 140, 630, 3150, 16632, 84084] | -| Middle | [1, 1, 6, 12, 90, 210, 1680, 4200, 34650] | -| Central | [1, 6, 90, 1680, 34650] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | -| PosHalf | [1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934] | -| NegHalf | [1, 0, -2, 0, 6, 0, -20, 0, 70] | -| Bin | [1, 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891] | -| InvBin | [1, 1, -5, -35, -29, 751, 3991, -4115, -137885] | -| TransSqrs | [0, 2, 30, 312, 2760, 22290, 169806, 1242080, 8817696] | -| TransNat0 | [0, 2, 18, 132, 900, 5910, 37926, 239624, 1497096] | -| TransNat1 | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | - -SchroederPRev Diagonals as rows +SchroederP:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 2], [1, 6, 6], [1, 12, 30, 20], [1, 20, 90, 140, 70], [1, 30, 210, 560, 630, 252], [1, 42, 420, 1680, 3150, 2772, 924], [1, 56, 756, 4200, 11550, 16632, 12012, 3432], [1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870]] | +| RevTabl | [[1], [2, 1], [6, 6, 1], [20, 30, 12, 1], [70, 140, 90, 20, 1], [252, 630, 560, 210, 30, 1], [924, 2772, 3150, 1680, 420, 42, 1], [3432, 12012, 16632, 11550, 4200, 756, 56, 1], [12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1]] | +| AntiDiag | [[1], [1], [1, 2], [1, 6], [1, 12, 6], [1, 20, 30], [1, 30, 90, 20], [1, 42, 210, 140], [1, 56, 420, 560, 70]] | +| AccTabl | [[1], [1, 3], [1, 7, 13], [1, 13, 43, 63], [1, 21, 111, 251, 321], [1, 31, 241, 801, 1431, 1683], [1, 43, 463, 2143, 5293, 8065, 8989], [1, 57, 813, 5013, 16563, 33195, 45207, 48639], [1, 73, 1333, 10573, 45223, 117295, 201379, 252859, 265729]] | +| RevAccTabl | [[1], [3, 1], [13, 7, 1], [63, 43, 13, 1], [321, 251, 111, 21, 1], [1683, 1431, 801, 241, 31, 1], [8989, 8065, 5293, 2143, 463, 43, 1], [48639, 45207, 33195, 16563, 5013, 813, 57, 1], [265729, 252859, 201379, 117295, 45223, 10573, 1333, 73, 1]] | +| AccRevTabl | [[1], [2, 3], [6, 12, 13], [20, 50, 62, 63], [70, 210, 300, 320, 321], [252, 882, 1442, 1652, 1682, 1683], [924, 3696, 6846, 8526, 8946, 8988, 8989], [3432, 15444, 32076, 43626, 47826, 48582, 48638, 48639], [12870, 64350, 148434, 220506, 255156, 264396, 265656, 265728, 265729]] | + +SchroederP:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729] | +| EvenSum | [1, 1, 7, 31, 161, 841, 4495, 24319, 132865] | +| OddSum | [0, 2, 6, 32, 160, 842, 4494, 24320, 132864] | +| AltSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| AccSum | [1, 4, 21, 120, 705, 4188, 24997, 149488, 894465] | +| AccRevSum | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | +| AntiDiagSum | [1, 1, 3, 7, 19, 51, 141, 393, 1107] | + +SchroederP:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 60, 1260, 5040, 277200, 5405400, 12612600] | +| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | +| RowMax | [1, 2, 6, 30, 140, 630, 3150, 16632, 84084] | +| ColMiddle | [1, 1, 6, 12, 90, 210, 1680, 4200, 34650] | +| ColECenter | [1, 6, 90, 1680, 34650] | +| ColOCenter | [1, 12, 210, 4200] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 2, 6, 20, 70, 252, 924, 3432, 12870] | +| TransSqrs | [0, 2, 30, 312, 2760, 22290, 169806, 1242080, 8817696] | +| TransNat0 | [0, 2, 18, 132, 900, 5910, 37926, 239624, 1497096] | +| TransNat1 | [1, 5, 31, 195, 1221, 7593, 46915, 288263, 1762825] | + +SchroederP:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 2, 6, 20, 70, 252, 924, 3432, 12870]| @@ -283,7 +271,7 @@ SchroederPRev Diagonals as rows | DiagRow7 | [1, 72, 1980, 34320, 450450, 4900896, 46558512, 399072960, 3155170590]| | DiagRow8 | [1, 90, 2970, 60060, 900900, 11027016, 116396280, 1097450640, 9465511770]| -SchroederPRev Diagonals as columns +SchroederP:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ SchroederPRev Diagonals as columns | DiagCol7 | [3432, 51480, 411840, 2333760, 10501920, 39907296, 133024320, 399072960, 1097450640] | | DiagCol8 | [12870, 218790, 1969110, 12471030, 62355150, 261891630, 960269310, 3155170590, 9465511770] | -SchroederPRev Polynomial values as rows +SchroederP:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ SchroederPRev Polynomial values as rows | PolyRow7 | [1, 48639, 1961825, 21360031, 125700129, 515550431, 1666406209, 4548342975, 10940804801] | | PolyRow8 | [1, 265729, 18205345, 278905249, 2114588641, 10611099361, 40558166209, 127779121345, 348434664769] | -SchroederPRev Polynomial values as columns +SchroederP:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ SchroederPRev Polynomial values as columns | PolyCol7 | [1, 15, 337, 8415, 220641, 5950575, 163456945, 4548342975, 127779121345] | | PolyCol8 | [1, 17, 433, 12257, 364321, 11138417, 346843729, 10940804801, 348434664769] | -# SchroederPInvRev -True +# SchroederP:Inv:Rev +[] -SchroederPInvRev Triangle view +SchroederP:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ SchroederPInvRev Triangle view | Row7 | [1, -56, 1596, -28560, 318570, -1986432, 5497632, -3802752] | | Row8 | [1, -72, 2772, -71232, 1240470, -13748112, 85606584, -236856384, 163825974] | -SchroederPInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -2, 1, -6, 6, 1, -12, 42, -32, 1, -20, 150, -440, 310, 1, -30, 390, -2540, 7110, -4932, 1, -42, 840, -9660, 60690, -168252, 116424, 1, -56, 1596, -28560, 318570, -1986432, 5497632, -3802752, 1, -72, 2772, -71232, 1240470, -13748112, 85606584, -236856384, 163825974] | -| TriRev | [1, -2, 1, 6, -6, 1, -32, 42, -12, 1, 310, -440, 150, -20, 1, -4932, 7110, -2540, 390, -30, 1, 116424, -168252, 60690, -9660, 840, -42, 1, -3802752, 5497632, -1986432, 318570, -28560, 1596, -56, 1, 163825974, -236856384, 85606584, -13748112, 1240470, -71232, 2772, -72, 1] | -| TriAcc | [1, 1, -1, 1, -5, 1, 1, -11, 31, -1, 1, -19, 131, -309, 1, 1, -29, 361, -2179, 4931, -1, 1, -41, 799, -8861, 51829, -116423, 1, 1, -55, 1541, -27019, 291551, -1694881, 3802751, -1, 1, -71, 2701, -68531, 1171939, -12576173, 73030411, -163825973, 1] | -| TriRevAcc | [1, -1, 1, 1, -5, 1, -1, 31, -11, 1, 1, -309, 131, -19, 1, -1, 4931, -2179, 361, -29, 1, 1, -116423, 51829, -8861, 799, -41, 1, -1, 3802751, -1694881, 291551, -27019, 1541, -55, 1, 1, -163825973, 73030411, -12576173, 1171939, -68531, 2701, -71, 1] | -| TriAccRev | [1, -2, -1, 6, 0, 1, -32, 10, -2, -1, 310, -130, 20, 0, 1, -4932, 2178, -362, 28, -2, -1, 116424, -51828, 8862, -798, 42, 0, 1, -3802752, 1694880, -291552, 27018, -1542, 54, -2, -1, 163825974, -73030410, 12576174, -1171938, 68532, -2700, 72, 0, 1] | -| TriDiag | [1, 1, 1, -2, 1, -6, 1, -12, 6, 1, -20, 42, 1, -30, 150, -32, 1, -42, 390, -440] | - -SchroederPInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| EvenSum | [1, 1, 7, 43, 461, 7501, 177955, 5817799, 250675801] | -| OddSum | [0, -2, -6, -44, -460, -7502, -177954, -5817800, -250675800] | -| AltSum | [1, 3, 13, 87, 921, 15003, 355909, 11635599, 501351601] | -| AccSum | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | -| AccRevSum | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | -| RevAccSum | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | -| DiagSum | [1, 1, -1, -5, -5, 23, 89, -91] | - -SchroederPInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, -2, 6, 672, 204600, 3216379140, 7750311334920, 131426737497966879360, 54722432423401919925487384726080] | -| RowGcd | [1, -2, 6, 2, 10, 2, 42, 2, 6] | -| RowMax | [1, 2, 6, 42, 440, 7110, 168252, 5497632, 236856384] | -| Middle | [1, 1, -6, -12, 150, 390, -9660, -28560, 1240470] | -| Central | [1, -6, 150, -9660, 1240470] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | -| PosHalf | [1, 0, -2, 12, -114, 1800, -42440, 1386000, -59708810] | -| NegHalf | [1, -4, 22, -172, 1966, -32944, 787816, -25804048, 1112263798] | -| Bin | [1, -1, -5, 59, -629, 8969, -163589, 3149075, -21014069] | -| InvBin | [1, -3, 19, -195, 3051, -69933, 2242339, -96184707, 5316455755] | -| TransSqrs | [0, -2, 18, -132, 1580, -30870, 872382, -33234488, 1636251192] | -| TransNat0 | [0, -2, 6, -24, 200, -3090, 72702, -2373896, 102265704] | -| TransNat1 | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | - -SchroederPInvRev Diagonals as rows +SchroederP:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -2], [1, -6, 6], [1, -12, 42, -32], [1, -20, 150, -440, 310], [1, -30, 390, -2540, 7110, -4932], [1, -42, 840, -9660, 60690, -168252, 116424], [1, -56, 1596, -28560, 318570, -1986432, 5497632, -3802752], [1, -72, 2772, -71232, 1240470, -13748112, 85606584, -236856384, 163825974]] | +| RevTabl | [[1], [-2, 1], [6, -6, 1], [-32, 42, -12, 1], [310, -440, 150, -20, 1], [-4932, 7110, -2540, 390, -30, 1], [116424, -168252, 60690, -9660, 840, -42, 1], [-3802752, 5497632, -1986432, 318570, -28560, 1596, -56, 1], [163825974, -236856384, 85606584, -13748112, 1240470, -71232, 2772, -72, 1]] | +| AntiDiag | [[1], [1], [1, -2], [1, -6], [1, -12, 6], [1, -20, 42], [1, -30, 150, -32], [1, -42, 390, -440], [1, -56, 840, -2540, 310]] | +| AccTabl | [[1], [1, -1], [1, -5, 1], [1, -11, 31, -1], [1, -19, 131, -309, 1], [1, -29, 361, -2179, 4931, -1], [1, -41, 799, -8861, 51829, -116423, 1], [1, -55, 1541, -27019, 291551, -1694881, 3802751, -1], [1, -71, 2701, -68531, 1171939, -12576173, 73030411, -163825973, 1]] | +| RevAccTabl | [[1], [-1, 1], [1, -5, 1], [-1, 31, -11, 1], [1, -309, 131, -19, 1], [-1, 4931, -2179, 361, -29, 1], [1, -116423, 51829, -8861, 799, -41, 1], [-1, 3802751, -1694881, 291551, -27019, 1541, -55, 1], [1, -163825973, 73030411, -12576173, 1171939, -68531, 2701, -71, 1]] | +| AccRevTabl | [[1], [-2, -1], [6, 0, 1], [-32, 10, -2, -1], [310, -130, 20, 0, 1], [-4932, 2178, -362, 28, -2, -1], [116424, -51828, 8862, -798, 42, 0, 1], [-3802752, 1694880, -291552, 27018, -1542, 54, -2, -1], [163825974, -73030410, 12576174, -1171938, 68532, -2700, 72, 0, 1]] | + +SchroederP:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| EvenSum | [1, 1, 7, 43, 461, 7501, 177955, 5817799, 250675801] | +| OddSum | [0, -2, -6, -44, -460, -7502, -177954, -5817800, -250675800] | +| AltSum | [1, 3, 13, 87, 921, 15003, 355909, 11635599, 501351601] | +| AccSum | [1, 0, -3, 20, -195, 3084, -72695, 2373888, -102265695] | +| AccRevSum | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | +| AntiDiagSum | [1, 1, -1, -5, -5, 23, 89, -91, -1445] | + +SchroederP:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 6, 672, 204600, 3216379140, 7750311334920, 131426737497966879360, 54722432423401919925487384726080] | +| RowGcd | [1, 2, 6, 2, 10, 2, 42, 2, 6] | +| RowMax | [1, 2, 6, 42, 440, 7110, 168252, 5497632, 236856384] | +| ColMiddle | [1, 1, -6, -12, 150, 390, -9660, -28560, 1240470] | +| ColECenter | [1, -6, 150, -9660, 1240470] | +| ColOCenter | [1, -12, 390, -28560] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974] | +| TransSqrs | [0, -2, 18, -132, 1580, -30870, 872382, -33234488, 1636251192] | +| TransNat0 | [0, -2, 6, -24, 200, -3090, 72702, -2373896, 102265704] | +| TransNat1 | [1, -3, 7, -25, 201, -3091, 72703, -2373897, 102265705] | + +SchroederP:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -2, 6, -32, 310, -4932, 116424, -3802752, 163825974]| @@ -391,7 +375,7 @@ SchroederPInvRev Diagonals as rows | DiagRow7 | [1, -72, 4500, -315480, 26212230, -2584821096, 299376289212, -40261266559800, 6222162826878570]| | DiagRow8 | [1, -90, 6930, -588720, 58326840, -6761650896, 909489300720, -140561186800320, 24737671557915030]| -SchroederPInvRev Diagonals as columns +SchroederP:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ SchroederPInvRev Diagonals as columns | DiagCol7 | [-3802752, -236856384, -4697384400, -51337094160, -383049120960, -2181870518256, -10146680307192, -40261266559800, -140561186800320] | | DiagCol8 | [163825974, 12996105750, 319573375110, 4239074011590, 37728135077490, 252651966609042, 1364519028362490, 6222162826878570, 24737671557915030] | -SchroederPInvRev Polynomial values as rows +SchroederP:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ SchroederPInvRev Polynomial values as rows | PolyRow7 | [1, -1, -193594719, -4766500625, -41740343071, -217201524129, -823069405631, -2517569677489, -6597578781375] | | PolyRow8 | [1, 1, 16680002593, 616022291665, 7192723583521, 46785522577441, 212748846261121, 759206140246513, 2273815711683265] | -SchroederPInvRev Polynomial values as columns +SchroederP:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Seidel.md b/data/md/Seidel.md index 94fb5e8..b81e44d 100644 --- a/data/md/Seidel.md +++ b/data/md/Seidel.md @@ -14,45 +14,41 @@ Seidel Triangle view | Row7 | [0, 61, 122, 178, 224, 256, 272, 272] | | Row8 | [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -Seidel Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 4, 5, 5, 0, 5, 10, 14, 16, 16, 0, 16, 32, 46, 56, 61, 61, 0, 61, 122, 178, 224, 256, 272, 272, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -| TriRev | [1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 4, 2, 0, 16, 16, 14, 10, 5, 0, 61, 61, 56, 46, 32, 16, 0, 272, 272, 256, 224, 178, 122, 61, 0, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 2, 6, 11, 16, 0, 5, 15, 29, 45, 61, 0, 16, 48, 94, 150, 211, 272, 0, 61, 183, 361, 585, 841, 1113, 1385, 0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 5, 3, 1, 0, 16, 11, 6, 2, 0, 61, 45, 29, 15, 5, 0, 272, 211, 150, 94, 48, 16, 0, 1385, 1113, 841, 585, 361, 183, 61, 0, 7936, 6551, 5166, 3842, 2640, 1616, 816, 272, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 10, 14, 16, 16, 16, 32, 46, 56, 61, 61, 61, 122, 178, 224, 256, 272, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 2, 0, 5, 4, 2, 0, 16, 10, 5] | +Seidel Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 2, 2], [0, 2, 4, 5, 5], [0, 5, 10, 14, 16, 16], [0, 16, 32, 46, 56, 61, 61], [0, 61, 122, 178, 224, 256, 272, 272], [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [2, 2, 1, 0], [5, 5, 4, 2, 0], [16, 16, 14, 10, 5, 0], [61, 61, 56, 46, 32, 16, 0], [272, 272, 256, 224, 178, 122, 61, 0], [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 2, 2], [0, 5, 4, 2], [0, 16, 10, 5], [0, 61, 32, 14, 5]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 5], [0, 2, 6, 11, 16], [0, 5, 15, 29, 45, 61], [0, 16, 48, 94, 150, 211, 272], [0, 61, 183, 361, 585, 841, 1113, 1385], [0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [5, 3, 1, 0], [16, 11, 6, 2, 0], [61, 45, 29, 15, 5, 0], [272, 211, 150, 94, 48, 16, 0], [1385, 1113, 841, 585, 361, 183, 61, 0], [7936, 6551, 5166, 3842, 2640, 1616, 816, 272, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [2, 4, 5, 5], [5, 10, 14, 16, 16], [16, 32, 46, 56, 61, 61], [61, 122, 178, 224, 256, 272, 272], [272, 544, 800, 1024, 1202, 1324, 1385, 1385], [1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]] | Seidel Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | -| EvenSum | [1, 0, 1, 2, 9, 26, 149, 618, 4277] | -| OddSum | [0, 1, 1, 3, 7, 35, 123, 767, 3659] | -| AltSum | [1, -1, 0, -1, 2, -9, 26, -149, 618] | -| AccSum | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | -| AccRevSum | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | -| RevAccSum | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | -| DiagSum | [1, 0, 1, 1, 2, 4, 11, 31] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | +| EvenSum | [1, 0, 1, 2, 9, 26, 149, 618, 4277] | +| OddSum | [0, 1, 1, 3, 7, 35, 123, 767, 3659] | +| AltSum | [1, -1, 0, -1, 2, -9, 26, -149, 618] | +| AccSum | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | +| AccRevSum | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | +| AntiDiagSum | [1, 0, 1, 1, 2, 4, 11, 31, 112] | Seidel Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| Middle | [1, 0, 1, 1, 4, 10, 46, 178, 1024] | -| Central | [1, 1, 4, 46, 1024] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| PosHalf | [1, 1, 3, 10, 47, 264, 1799, 14288, 130683] | -| NegHalf | [1, 1, -1, 2, -5, 40, -205, 1808, -14921] | -| Bin | [1, 1, 3, 11, 57, 361, 2763, 24611, 250737] | -| InvBin | [1, 1, -1, -1, 1, 1, -1, -1, 1] | -| TransSqrs | [0, 1, 5, 27, 143, 827, 5175, 35255, 260251] | -| TransNat0 | [0, 1, 3, 11, 45, 211, 1113, 6551, 42585] | -| TransNat1 | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColMiddle | [1, 0, 1, 1, 4, 10, 46, 178, 1024] | +| ColECenter | [1, 1, 4, 46, 1024] | +| ColOCenter | [0, 1, 10, 178] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| TransSqrs | [0, 1, 5, 27, 143, 827, 5175, 35255, 260251] | +| TransNat0 | [0, 1, 3, 11, 45, 211, 1113, 6551, 42585] | +| TransNat1 | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | Seidel Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Seidel Polynomial values as columns | PolyCol7 | [1, 7, 56, 791, 13930, 312655, 8353730, 260912099, 9303587314] | | PolyCol8 | [1, 8, 72, 1160, 23312, 597672, 18244736, 651134056, 26532106368] | -# SeidelRev -True +# Seidel:Rev +[] -SeidelRev Triangle view +Seidel:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SeidelRev Triangle view | Row7 | [272, 272, 256, 224, 178, 122, 61, 0] | | Row8 | [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -SeidelRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 4, 2, 0, 16, 16, 14, 10, 5, 0, 61, 61, 56, 46, 32, 16, 0, 272, 272, 256, 224, 178, 122, 61, 0, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 4, 5, 5, 0, 5, 10, 14, 16, 16, 0, 16, 32, 46, 56, 61, 61, 0, 61, 122, 178, 224, 256, 272, 272, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -| TriAcc | [1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 10, 14, 16, 16, 16, 32, 46, 56, 61, 61, 61, 122, 178, 224, 256, 272, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 5, 5, 4, 2, 16, 16, 14, 10, 5, 61, 61, 56, 46, 32, 16, 272, 272, 256, 224, 178, 122, 61, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 2, 6, 11, 16, 0, 5, 15, 29, 45, 61, 0, 16, 48, 94, 150, 211, 272, 0, 61, 183, 361, 585, 841, 1113, 1385, 0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936] | -| TriDiag | [1, 1, 1, 0, 2, 1, 5, 2, 0, 16, 5, 1, 61, 16, 4, 0, 272, 61, 14, 2] | - -SeidelRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | -| EvenSum | [1, 1, 1, 3, 9, 35, 149, 767, 4277] | -| OddSum | [0, 0, 1, 2, 7, 26, 123, 618, 3659] | -| AltSum | [1, 1, 0, 1, 2, 9, 26, 149, 618] | -| AccSum | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | -| AccRevSum | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | -| RevAccSum | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | -| DiagSum | [1, 1, 1, 3, 7, 22, 81, 349] | - -SeidelRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| Middle | [1, 1, 1, 2, 4, 14, 46, 224, 1024] | -| Central | [1, 1, 4, 46, 1024] | -| LeftSide | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 26, 140, 930, 7280, 66034, 680544] | -| NegHalf | [1, -2, 2, -10, 52, -338, 2576, -23074, 235168] | -| Bin | [1, 1, 3, 11, 57, 361, 2763, 24611, 250737] | -| InvBin | [1, -1, -1, 1, 1, -1, -1, 1, 1] | -| TransSqrs | [0, 0, 1, 6, 39, 242, 1611, 11406, 86795] | -| TransNat0 | [0, 0, 1, 4, 19, 94, 519, 3144, 20903] | -| TransNat1 | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | - -SeidelRev Diagonals as rows +Seidel:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [2, 2, 1, 0], [5, 5, 4, 2, 0], [16, 16, 14, 10, 5, 0], [61, 61, 56, 46, 32, 16, 0], [272, 272, 256, 224, 178, 122, 61, 0], [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 2, 2], [0, 2, 4, 5, 5], [0, 5, 10, 14, 16, 16], [0, 16, 32, 46, 56, 61, 61], [0, 61, 122, 178, 224, 256, 272, 272], [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385]] | +| AntiDiag | [[1], [1], [1, 0], [2, 1], [5, 2, 0], [16, 5, 1], [61, 16, 4, 0], [272, 61, 14, 2], [1385, 272, 56, 10, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [2, 4, 5, 5], [5, 10, 14, 16, 16], [16, 32, 46, 56, 61, 61], [61, 122, 178, 224, 256, 272, 272], [272, 544, 800, 1024, 1202, 1324, 1385, 1385], [1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [5, 5, 4, 2], [16, 16, 14, 10, 5], [61, 61, 56, 46, 32, 16], [272, 272, 256, 224, 178, 122, 61], [1385, 1385, 1324, 1202, 1024, 800, 544, 272], [7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 3, 5], [0, 2, 6, 11, 16], [0, 5, 15, 29, 45, 61], [0, 16, 48, 94, 150, 211, 272], [0, 61, 183, 361, 585, 841, 1113, 1385], [0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936]] | + +Seidel:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | +| EvenSum | [1, 1, 1, 3, 9, 35, 149, 767, 4277] | +| OddSum | [0, 0, 1, 2, 7, 26, 123, 618, 3659] | +| AltSum | [1, 1, 0, 1, 2, 9, 26, 149, 618] | +| AccSum | [1, 2, 5, 16, 61, 272, 1385, 7936, 50521] | +| AccRevSum | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | +| AntiDiagSum | [1, 1, 1, 3, 7, 22, 81, 349, 1723] | + +Seidel:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColMiddle | [1, 1, 1, 2, 4, 14, 46, 224, 1024] | +| ColECenter | [1, 1, 4, 46, 1024] | +| ColOCenter | [1, 2, 14, 224] | +| ColLeft | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 6, 39, 242, 1611, 11406, 86795] | +| TransNat0 | [0, 0, 1, 4, 19, 94, 519, 3144, 20903] | +| TransNat1 | [1, 1, 3, 9, 35, 155, 791, 4529, 28839] | + +Seidel:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ SeidelRev Diagonals as rows | DiagRow7 | [272, 1385, 7664, 46366, 306448, 2204480, 17180144, 144361456, 1301989648]| | DiagRow8 | [1385, 7936, 49136, 329984, 2401024, 18870016, 159575936, 1446351104, 13997185024]| -SeidelRev Diagonals as columns +Seidel:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | @@ -188,7 +180,7 @@ SeidelRev Diagonals as columns | DiagCol7 | [0, 272, 2770, 23536, 196544, 1689872, 15214480, 144361456, 1446351104] | | DiagCol8 | [0, 1385, 15872, 150178, 1383424, 13010000, 127181312, 1301989648, 13997185024] | -SeidelRev Polynomial values as rows +Seidel:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SeidelRev Polynomial values as rows | PolyRow7 | [272, 1385, 14288, 97973, 440144, 1481657, 4084880, 9745973, 20851088] | | PolyRow8 | [1385, 7936, 130683, 1318694, 7871053, 33081660, 109388351, 304397218, 744153009] | -SeidelRev Polynomial values as columns +Seidel:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | diff --git a/data/md/SeidelBoust.md b/data/md/SeidelBoust.md index f92ef96..fce6848 100644 --- a/data/md/SeidelBoust.md +++ b/data/md/SeidelBoust.md @@ -14,45 +14,41 @@ SeidelBoust Triangle view | Row7 | [0, 61, 122, 178, 224, 256, 272, 272] | | Row8 | [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -SeidelBoust Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 1, 1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, 61, 61, 56, 46, 32, 16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -| TriRev | [1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 0, 2, 4, 5, 5, 16, 16, 14, 10, 5, 0, 0, 16, 32, 46, 56, 61, 61, 272, 272, 256, 224, 178, 122, 61, 0, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -| TriAcc | [1, 0, 1, 1, 2, 2, 0, 1, 3, 5, 5, 10, 14, 16, 16, 0, 5, 15, 29, 45, 61, 61, 122, 178, 224, 256, 272, 272, 0, 61, 183, 361, 585, 841, 1113, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936] | -| TriRevAcc | [1, 1, 0, 2, 2, 1, 5, 3, 1, 0, 16, 16, 14, 10, 5, 61, 45, 29, 15, 5, 0, 272, 272, 256, 224, 178, 122, 61, 1385, 1113, 841, 585, 361, 183, 61, 0, 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385] | -| TriAccRev | [1, 1, 1, 0, 1, 2, 2, 4, 5, 5, 0, 2, 6, 11, 16, 16, 32, 46, 56, 61, 61, 0, 16, 48, 94, 150, 211, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936] | -| TriDiag | [1, 0, 1, 1, 0, 1, 5, 1, 0, 0, 5, 2, 61, 5, 4, 2, 0, 61, 10, 2] | +SeidelBoust Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [1, 1, 0], [0, 1, 2, 2], [5, 5, 4, 2, 0], [0, 5, 10, 14, 16, 16], [61, 61, 56, 46, 32, 16, 0], [0, 61, 122, 178, 224, 256, 272, 272], [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0]] | +| RevTabl | [[1], [1, 0], [0, 1, 1], [2, 2, 1, 0], [0, 2, 4, 5, 5], [16, 16, 14, 10, 5, 0], [0, 16, 32, 46, 56, 61, 61], [272, 272, 256, 224, 178, 122, 61, 0], [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385]] | +| AntiDiag | [[1], [0], [1, 1], [0, 1], [5, 1, 0], [0, 5, 2], [61, 5, 4, 2], [0, 61, 10, 2], [1385, 61, 56, 14, 0]] | +| AccTabl | [[1], [0, 1], [1, 2, 2], [0, 1, 3, 5], [5, 10, 14, 16, 16], [0, 5, 15, 29, 45, 61], [61, 122, 178, 224, 256, 272, 272], [0, 61, 183, 361, 585, 841, 1113, 1385], [1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]] | +| RevAccTabl | [[1], [1, 0], [2, 2, 1], [5, 3, 1, 0], [16, 16, 14, 10, 5], [61, 45, 29, 15, 5, 0], [272, 272, 256, 224, 178, 122, 61], [1385, 1113, 841, 585, 361, 183, 61, 0], [7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385]] | +| AccRevTabl | [[1], [1, 1], [0, 1, 2], [2, 4, 5, 5], [0, 2, 6, 11, 16], [16, 32, 46, 56, 61, 61], [0, 16, 48, 94, 150, 211, 272], [272, 544, 800, 1024, 1202, 1324, 1385, 1385], [0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936]] | SeidelBoust Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | -| EvenSum | [1, 0, 1, 2, 9, 26, 149, 618, 4277] | -| OddSum | [0, 1, 1, 3, 7, 35, 123, 767, 3659] | -| AltSum | [1, -1, 0, -1, 2, -9, 26, -149, 618] | -| AccSum | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | -| AccRevSum | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | -| RevAccSum | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | -| DiagSum | [1, 0, 2, 1, 6, 7, 72, 73] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | +| EvenSum | [1, 0, 1, 2, 9, 26, 149, 618, 4277] | +| OddSum | [0, 1, 1, 3, 7, 35, 123, 767, 3659] | +| AltSum | [1, -1, 0, -1, 2, -9, 26, -149, 618] | +| AccSum | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | +| AccRevSum | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | +| AntiDiagSum | [1, 0, 2, 1, 6, 7, 72, 73, 1516] | SeidelBoust Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| Middle | [1, 0, 1, 1, 4, 10, 46, 178, 1024] | -| Central | [1, 1, 4, 46, 1024] | -| LeftSide | [1, 0, 1, 0, 5, 0, 61, 0, 1385] | -| RightSide | [1, 1, 0, 2, 0, 16, 0, 272, 0] | -| PosHalf | [1, 1, 6, 10, 140, 264, 7280, 14288, 680544] | -| NegHalf | [1, 1, 2, 2, 52, 40, 2576, 1808, 235168] | -| Bin | [1, 1, 3, 11, 57, 361, 2763, 24611, 250737] | -| InvBin | [1, 1, -1, -1, 1, 1, -1, -1, 1] | -| TransSqrs | [0, 1, 1, 27, 39, 827, 1611, 35255, 86795] | -| TransNat0 | [0, 1, 1, 11, 19, 211, 519, 6551, 20903] | -| TransNat1 | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColMiddle | [1, 0, 1, 1, 4, 10, 46, 178, 1024] | +| ColECenter | [1, 1, 4, 46, 1024] | +| ColOCenter | [0, 1, 10, 178] | +| ColLeft | [1, 0, 1, 0, 5, 0, 61, 0, 1385] | +| ColRight | [1, 1, 0, 2, 0, 16, 0, 272, 0] | +| TransSqrs | [0, 1, 1, 27, 39, 827, 1611, 35255, 86795] | +| TransNat0 | [0, 1, 1, 11, 19, 211, 519, 6551, 20903] | +| TransNat1 | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | SeidelBoust Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ SeidelBoust Polynomial values as columns | PolyCol7 | [1, 7, 8, 791, 922, 312655, 364754, 260912099, 304397218] | | PolyCol8 | [1, 8, 9, 1160, 1325, 597672, 683045, 651134056, 744153009] | -# SeidelBoustRev -True +# SeidelBoust:Rev +[] -SeidelBoustRev Triangle view +SeidelBoust:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SeidelBoustRev Triangle view | Row7 | [272, 272, 256, 224, 178, 122, 61, 0] | | Row8 | [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -SeidelBoustRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 0, 2, 4, 5, 5, 16, 16, 14, 10, 5, 0, 0, 16, 32, 46, 56, 61, 61, 272, 272, 256, 224, 178, 122, 61, 0, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385] | -| TriRev | [1, 0, 1, 1, 1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, 61, 61, 56, 46, 32, 16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0] | -| TriAcc | [1, 1, 1, 0, 1, 2, 2, 4, 5, 5, 0, 2, 6, 11, 16, 16, 32, 46, 56, 61, 61, 0, 16, 48, 94, 150, 211, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936] | -| TriRevAcc | [1, 1, 1, 2, 1, 0, 5, 5, 4, 2, 16, 11, 6, 2, 0, 61, 61, 56, 46, 32, 16, 272, 211, 150, 94, 48, 16, 0, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 7936, 6551, 5166, 3842, 2640, 1616, 816, 272, 0] | -| TriAccRev | [1, 0, 1, 1, 2, 2, 0, 1, 3, 5, 5, 10, 14, 16, 16, 0, 5, 15, 29, 45, 61, 61, 122, 178, 224, 256, 272, 272, 0, 61, 183, 361, 585, 841, 1113, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936] | -| TriDiag | [1, 1, 0, 0, 2, 1, 0, 2, 1, 16, 2, 1, 0, 16, 4, 0, 272, 16, 14, 5] | - -SeidelBoustRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | -| EvenSum | [1, 1, 1, 3, 9, 35, 149, 767, 4277] | -| OddSum | [0, 0, 1, 2, 7, 26, 123, 618, 3659] | -| AltSum | [1, 1, 0, 1, 2, 9, 26, 149, 618] | -| AccSum | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | -| AccRevSum | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | -| RevAccSum | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | -| DiagSum | [1, 1, 0, 3, 3, 19, 20, 307] | - -SeidelBoustRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | -| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | -| Middle | [1, 1, 1, 2, 4, 14, 46, 224, 1024] | -| Central | [1, 1, 4, 46, 1024] | -| LeftSide | [1, 1, 0, 2, 0, 16, 0, 272, 0] | -| RightSide | [1, 0, 1, 0, 5, 0, 61, 0, 1385] | -| PosHalf | [1, 2, 3, 26, 47, 930, 1799, 66034, 130683] | -| NegHalf | [1, -2, -1, -10, -5, -338, -205, -23074, -14921] | -| Bin | [1, 1, 3, 11, 57, 361, 2763, 24611, 250737] | -| InvBin | [1, -1, -1, 1, 1, -1, -1, 1, 1] | -| TransSqrs | [0, 0, 5, 6, 143, 242, 5175, 11406, 260251] | -| TransNat0 | [0, 0, 3, 4, 45, 94, 1113, 3144, 42585] | -| TransNat1 | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | - -SeidelBoustRev Diagonals as rows +SeidelBoust:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [0, 1, 1], [2, 2, 1, 0], [0, 2, 4, 5, 5], [16, 16, 14, 10, 5, 0], [0, 16, 32, 46, 56, 61, 61], [272, 272, 256, 224, 178, 122, 61, 0], [0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385]] | +| RevTabl | [[1], [0, 1], [1, 1, 0], [0, 1, 2, 2], [5, 5, 4, 2, 0], [0, 5, 10, 14, 16, 16], [61, 61, 56, 46, 32, 16, 0], [0, 61, 122, 178, 224, 256, 272, 272], [1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0]] | +| AntiDiag | [[1], [1], [0, 0], [2, 1], [0, 2, 1], [16, 2, 1], [0, 16, 4, 0], [272, 16, 14, 5], [0, 272, 32, 10, 5]] | +| AccTabl | [[1], [1, 1], [0, 1, 2], [2, 4, 5, 5], [0, 2, 6, 11, 16], [16, 32, 46, 56, 61, 61], [0, 16, 48, 94, 150, 211, 272], [272, 544, 800, 1024, 1202, 1324, 1385, 1385], [0, 272, 816, 1616, 2640, 3842, 5166, 6551, 7936]] | +| RevAccTabl | [[1], [1, 1], [2, 1, 0], [5, 5, 4, 2], [16, 11, 6, 2, 0], [61, 61, 56, 46, 32, 16], [272, 211, 150, 94, 48, 16, 0], [1385, 1385, 1324, 1202, 1024, 800, 544, 272], [7936, 6551, 5166, 3842, 2640, 1616, 816, 272, 0]] | +| AccRevTabl | [[1], [0, 1], [1, 2, 2], [0, 1, 3, 5], [5, 10, 14, 16, 16], [0, 5, 15, 29, 45, 61], [61, 122, 178, 224, 256, 272, 272], [0, 61, 183, 361, 585, 841, 1113, 1385], [1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936]] | + +SeidelBoust:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 16, 61, 272, 1385, 7936] | +| EvenSum | [1, 1, 1, 3, 9, 35, 149, 767, 4277] | +| OddSum | [0, 0, 1, 2, 7, 26, 123, 618, 3659] | +| AltSum | [1, 1, 0, 1, 2, 9, 26, 149, 618] | +| AccSum | [1, 2, 3, 16, 35, 272, 791, 7936, 28839] | +| AccRevSum | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | +| AntiDiagSum | [1, 1, 0, 3, 3, 19, 20, 307, 319] | + +SeidelBoust:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 20, 560, 314272, 165389056, 23981211622400] | +| RowGcd | [1, 1, 1, 2, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 2, 5, 16, 61, 272, 1385] | +| ColMiddle | [1, 1, 1, 2, 4, 14, 46, 224, 1024] | +| ColECenter | [1, 1, 4, 46, 1024] | +| ColOCenter | [1, 2, 14, 224] | +| ColLeft | [1, 1, 0, 2, 0, 16, 0, 272, 0] | +| ColRight | [1, 0, 1, 0, 5, 0, 61, 0, 1385] | +| TransSqrs | [0, 0, 5, 6, 143, 242, 5175, 11406, 260251] | +| TransNat0 | [0, 0, 3, 4, 45, 94, 1113, 3144, 42585] | +| TransNat1 | [1, 1, 5, 9, 61, 155, 1385, 4529, 50521] | + +SeidelBoust:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 1, 0, 5, 0, 61, 0, 1385]| @@ -175,7 +167,7 @@ SeidelBoustRev Diagonals as rows | DiagRow7 | [272, 272, 7664, 23536, 306448, 1689872, 17180144, 144361456, 1301989648]| | DiagRow8 | [0, 7936, 15872, 329984, 1383424, 18870016, 127181312, 1446351104, 13997185024]| -SeidelBoustRev Diagonals as columns +SeidelBoust:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 0, 2, 0, 16, 0, 272, 0] | @@ -188,7 +180,7 @@ SeidelBoustRev Diagonals as columns | DiagCol7 | [0, 1385, 2770, 46366, 196544, 2204480, 15214480, 144361456, 1446351104] | | DiagCol8 | [1385, 1385, 49136, 150178, 2401024, 13010000, 159575936, 1301989648, 13997185024] | -SeidelBoustRev Polynomial values as rows +SeidelBoust:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SeidelBoustRev Polynomial values as rows | PolyRow7 | [272, 1385, 14288, 97973, 440144, 1481657, 4084880, 9745973, 20851088] | | PolyRow8 | [0, 7936, 680544, 13483518, 120436288, 674417460, 2786619936, 9303587314, 26532106368] | -SeidelBoustRev Polynomial values as columns +SeidelBoust:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 0, 2, 0, 16, 0, 272, 0] | diff --git a/data/md/Sierpinski.md b/data/md/Sierpinski.md index ea75cd8..2171c7c 100644 --- a/data/md/Sierpinski.md +++ b/data/md/Sierpinski.md @@ -14,45 +14,41 @@ Sierpinski Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 0, 0, 0, 0, 0, 0, 0, 1] | -Sierpinski Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriRev | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriAcc | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2] | -| TriRevAcc | [1, 2, 1, 2, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 5, 4, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriAccRev | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2] | -| TriDiag | [1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0] | +Sierpinski Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 0, 0, 1], [1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| RevTabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 0, 0, 1], [1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 0], [1, 1, 1], [1, 0, 1], [1, 1, 0, 1], [1, 0, 0, 0], [1, 1, 1, 0, 1]] | +| AccTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 1, 1, 2], [1, 2, 2, 2, 3, 4], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 1, 1, 1, 1, 1, 1, 2]] | +| RevAccTabl | [[1], [2, 1], [2, 1, 1], [4, 3, 2, 1], [2, 1, 1, 1, 1], [4, 3, 2, 2, 2, 1], [4, 3, 3, 2, 2, 1, 1], [8, 7, 6, 5, 4, 3, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 1, 1, 2], [1, 2, 2, 2, 3, 4], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 1, 1, 1, 1, 1, 1, 2]] | Sierpinski Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 2, 4, 2, 4, 4, 8, 2] | -| EvenSum | [1, 1, 2, 2, 2, 2, 4, 4, 2] | -| OddSum | [0, 1, 0, 2, 0, 2, 0, 4, 0] | -| AltSum | [1, 0, 2, 0, 2, 0, 4, 0, 2] | -| AccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| AccRevSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| RevAccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| DiagSum | [1, 1, 2, 1, 3, 2, 3, 1] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 2, 4, 2, 4, 4, 8, 2] | +| EvenSum | [1, 1, 2, 2, 2, 2, 4, 4, 2] | +| OddSum | [0, 1, 0, 2, 0, 2, 0, 4, 0] | +| AltSum | [1, 0, 2, 0, 2, 0, 4, 0, 2] | +| AccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | +| AccRevSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | +| AntiDiagSum | [1, 1, 2, 1, 3, 2, 3, 1, 4] | Sierpinski Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, 1, 0, 0, 0, 1, 0] | -| Central | [1, 0, 0, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 5, 15, 17, 51, 85, 255, 257] | -| NegHalf | [1, -1, 5, -5, 17, -17, 85, -85, 257] | -| Bin | [1, 2, 2, 8, 2, 12, 32, 128, 2] | -| InvBin | [1, 0, 2, 0, 2, 0, 32, 0, 2] | -| TransSqrs | [0, 1, 4, 14, 16, 42, 56, 140, 64] | -| TransNat0 | [0, 1, 2, 6, 4, 10, 12, 28, 8] | -| TransNat1 | [1, 3, 4, 10, 6, 14, 16, 36, 10] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, 1, 0, 0, 0, 1, 0] | +| ColECenter | [1, 0, 0, 0, 0] | +| ColOCenter | [1, 1, 0, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 14, 16, 42, 56, 140, 64] | +| TransNat0 | [0, 1, 2, 6, 4, 10, 12, 28, 8] | +| TransNat1 | [1, 3, 4, 10, 6, 14, 16, 36, 10] | Sierpinski Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Sierpinski Polynomial values as columns | PolyCol7 | [1, 8, 50, 400, 2402, 19216, 120100, 960800, 5764802] | | PolyCol8 | [1, 9, 65, 585, 4097, 36873, 266305, 2396745, 16777217] | -# SierpinskiInv -True +# Sierpinski:Inv +[] -SierpinskiInv Triangle view +Sierpinski:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SierpinskiInv Triangle view | Row7 | [-1, 1, 1, -1, 1, -1, -1, 1] | | Row8 | [-1, 0, 0, 0, 0, 0, 0, 0, 1] | -SierpinskiInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 0, 0, 0, 1, 1, -1, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriRev | [1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1] | -| TriAcc | [1, -1, 0, -1, -1, 0, 1, 0, -1, 0, -1, -1, -1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, -1, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -1, -1, -1, -1, -1, -1, -1, 0] | -| TriRevAcc | [1, 0, -1, 0, -1, -1, 0, -1, 0, 1, 0, -1, -1, -1, -1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 1, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -1, -1, -1, -1, -1, -1, -1] | -| TriAccRev | [1, 1, 0, 1, 1, 0, 1, 0, -1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] | -| TriDiag | [1, -1, -1, 1, 1, 0, -1, -1, 1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 0, 0] | - -SierpinskiInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | -| AccRevSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | -| RevAccSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | -| DiagSum | [1, -1, 0, 1, -1, 0, 1, -1] | - -SierpinskiInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, -1, 0, -1, 0, 0, 0, -1, 0] | -| Central | [1, 0, 0, 0, 0] | -| LeftSide | [1, -1, -1, 1, -1, 1, 1, -1, -1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, 3, -15, 15, 45, -45, -255] | -| NegHalf | [1, 3, -3, -9, -15, -45, 45, 135, -255] | -| Bin | [1, 0, 0, -4, 0, -8, -28, 0, 0] | -| InvBin | [1, 2, 0, 0, 0, 0, -28, -96, 0] | -| TransSqrs | [0, 1, 4, 4, 16, 8, 16, 0, 64] | -| TransNat0 | [0, 1, 2, 0, 4, 0, 0, 0, 8] | -| TransNat1 | [1, 1, 2, 0, 4, 0, 0, 0, 8] | - -SierpinskiInv Diagonals as rows +Sierpinski:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [-1, 0, 0, 0, 1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [-1, 1, 1, -1, 1, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| RevTabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, 0, 0, -1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [1, -1, -1, 1, -1, 1, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0, -1]] | +| AntiDiag | [[1], [-1], [-1, 1], [1, 0], [-1, -1, 1], [1, 0, -1], [1, -1, 0, 1], [-1, 0, 0, 0], [-1, 1, -1, 0, 1]] | +| AccTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [-1, -1, -1, -1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [-1, 0, 1, 0, 1, 0, -1, 0], [-1, -1, -1, -1, -1, -1, -1, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, -1], [0, -1, 0, 1], [0, -1, -1, -1, -1], [0, -1, 0, 0, 0, 1], [0, -1, -1, 0, 0, 1, 1], [0, -1, 0, 1, 0, 1, 0, -1], [0, -1, -1, -1, -1, -1, -1, -1, -1]] | +| AccRevTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 1, 1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [1, 0, -1, 0, -1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0]] | + +Sierpinski:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | +| AccRevSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | +| AntiDiagSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | + +Sierpinski:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, 0, -1, 0, 0, 0, -1, 0] | +| ColECenter | [1, 0, 0, 0, 0] | +| ColOCenter | [-1, -1, 0, -1] | +| ColLeft | [1, -1, -1, 1, -1, 1, 1, -1, -1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 4, 16, 8, 16, 0, 64] | +| TransNat0 | [0, 1, 2, 0, 4, 0, 0, 0, 8] | +| TransNat1 | [1, 1, 2, 0, 4, 0, 0, 0, 8] | + +Sierpinski:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ SierpinskiInv Diagonals as rows | DiagRow7 | [-1, 0, 0, 0, 0, 0, 0, 0, -1]| | DiagRow8 | [-1, -1, -1, -1, -1, -1, -1, -1, 0]| -SierpinskiInv Diagonals as columns +Sierpinski:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, -1, 1, -1, 1, 1, -1, -1] | @@ -188,7 +180,7 @@ SierpinskiInv Diagonals as columns | DiagCol7 | [1, 0, 0, 0, 0, 0, 0, 0, -1] | | DiagCol8 | [1, -1, -1, 1, -1, 1, 1, -1, 0] | -SierpinskiInv Polynomial values as rows +Sierpinski:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SierpinskiInv Polynomial values as rows | PolyRow7 | [-1, 0, 45, 1280, 11475, 59904, 226625, 691200, 1805895] | | PolyRow8 | [-1, 0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215] | -SierpinskiInv Polynomial values as columns +Sierpinski:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, -1, 1, -1, 1, 1, -1, -1] | @@ -214,10 +206,10 @@ SierpinskiInv Polynomial values as columns | PolyCol7 | [1, 6, 48, 288, 2400, 14400, 115200, 691200, 5764800] | | PolyCol8 | [1, 7, 63, 441, 4095, 28665, 257985, 1805895, 16777215] | -# SierpinskiRev -True +# Sierpinski:Rev +[] -SierpinskiRev Triangle view +Sierpinski:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ SierpinskiRev Triangle view | Row7 | [1, 1, 1, 1, 1, 1, 1, 1] | | Row8 | [1, 0, 0, 0, 0, 0, 0, 0, 1] | -SierpinskiRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriRev | [1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriAcc | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2] | -| TriRevAcc | [1, 2, 1, 2, 1, 1, 4, 3, 2, 1, 2, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 5, 4, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriAccRev | [1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2] | -| TriDiag | [1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0] | - -SierpinskiRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 2, 4, 2, 4, 4, 8, 2] | -| EvenSum | [1, 1, 2, 2, 2, 2, 4, 4, 2] | -| OddSum | [0, 1, 0, 2, 0, 2, 0, 4, 0] | -| AltSum | [1, 0, 2, 0, 2, 0, 4, 0, 2] | -| AccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| AccRevSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| RevAccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | -| DiagSum | [1, 1, 2, 1, 3, 2, 3, 1] | - -SierpinskiRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, 1, 0, 0, 0, 1, 0] | -| Central | [1, 0, 0, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 5, 15, 17, 51, 85, 255, 257] | -| NegHalf | [1, -1, 5, -5, 17, -17, 85, -85, 257] | -| Bin | [1, 2, 2, 8, 2, 12, 32, 128, 2] | -| InvBin | [1, 0, 2, 0, 2, 0, 32, 0, 2] | -| TransSqrs | [0, 1, 4, 14, 16, 42, 56, 140, 64] | -| TransNat0 | [0, 1, 2, 6, 4, 10, 12, 28, 8] | -| TransNat1 | [1, 3, 4, 10, 6, 14, 16, 36, 10] | - -SierpinskiRev Diagonals as rows +Sierpinski:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 0, 0, 1], [1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| RevTabl | [[1], [1, 1], [1, 0, 1], [1, 1, 1, 1], [1, 0, 0, 0, 1], [1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 0], [1, 1, 1], [1, 0, 1], [1, 1, 0, 1], [1, 0, 0, 0], [1, 1, 1, 0, 1]] | +| AccTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 1, 1, 2], [1, 2, 2, 2, 3, 4], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 1, 1, 1, 1, 1, 1, 2]] | +| RevAccTabl | [[1], [2, 1], [2, 1, 1], [4, 3, 2, 1], [2, 1, 1, 1, 1], [4, 3, 2, 2, 2, 1], [4, 3, 3, 2, 2, 1, 1], [8, 7, 6, 5, 4, 3, 2, 1], [2, 1, 1, 1, 1, 1, 1, 1, 1]] | +| AccRevTabl | [[1], [1, 2], [1, 1, 2], [1, 2, 3, 4], [1, 1, 1, 1, 2], [1, 2, 2, 2, 3, 4], [1, 1, 2, 2, 3, 3, 4], [1, 2, 3, 4, 5, 6, 7, 8], [1, 1, 1, 1, 1, 1, 1, 1, 2]] | + +Sierpinski:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 2, 4, 2, 4, 4, 8, 2] | +| EvenSum | [1, 1, 2, 2, 2, 2, 4, 4, 2] | +| OddSum | [0, 1, 0, 2, 0, 2, 0, 4, 0] | +| AltSum | [1, 0, 2, 0, 2, 0, 4, 0, 2] | +| AccSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | +| AccRevSum | [1, 3, 4, 10, 6, 14, 16, 36, 10] | +| AntiDiagSum | [1, 1, 2, 1, 3, 2, 3, 1, 4] | + +Sierpinski:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, 1, 0, 0, 0, 1, 0] | +| ColECenter | [1, 0, 0, 0, 0] | +| ColOCenter | [1, 1, 0, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 14, 16, 42, 56, 140, 64] | +| TransNat0 | [0, 1, 2, 6, 4, 10, 12, 28, 8] | +| TransNat1 | [1, 3, 4, 10, 6, 14, 16, 36, 10] | + +Sierpinski:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -283,7 +271,7 @@ SierpinskiRev Diagonals as rows | DiagRow7 | [1, 0, 0, 0, 0, 0, 0, 0, 1]| | DiagRow8 | [1, 1, 1, 1, 1, 1, 1, 1, 0]| -SierpinskiRev Diagonals as columns +Sierpinski:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ SierpinskiRev Diagonals as columns | DiagCol7 | [1, 0, 0, 0, 0, 0, 0, 0, 1] | | DiagCol8 | [1, 1, 1, 1, 1, 1, 1, 1, 0] | -SierpinskiRev Polynomial values as rows +Sierpinski:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ SierpinskiRev Polynomial values as rows | PolyRow7 | [1, 8, 255, 3280, 21845, 97656, 335923, 960800, 2396745] | | PolyRow8 | [1, 2, 257, 6562, 65537, 390626, 1679617, 5764802, 16777217] | -SierpinskiRev Polynomial values as columns +Sierpinski:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ SierpinskiRev Polynomial values as columns | PolyCol7 | [1, 8, 50, 400, 2402, 19216, 120100, 960800, 5764802] | | PolyCol8 | [1, 9, 65, 585, 4097, 36873, 266305, 2396745, 16777217] | -# SierpinskiInvRev -True +# Sierpinski:Inv:Rev +[] -SierpinskiInvRev Triangle view +Sierpinski:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ SierpinskiInvRev Triangle view | Row7 | [1, -1, -1, 1, -1, 1, 1, -1] | | Row8 | [1, 0, 0, 0, 0, 0, 0, 0, -1] | -SierpinskiInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1] | -| TriRev | [1, -1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 0, 0, 0, 1, 1, -1, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1] | -| TriAcc | [1, 1, 0, 1, 1, 0, 1, 0, -1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] | -| TriRevAcc | [1, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 1, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] | -| TriAccRev | [1, -1, 0, -1, -1, 0, 1, 0, -1, 0, -1, -1, -1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, -1, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, -1, -1, -1, -1, -1, -1, -1, 0] | -| TriDiag | [1, 1, 1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, -1, 0, 1, 1, 0, 0, 0] | - -SierpinskiInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| OddSum | [0, -1, 0, 0, 0, 0, 0, 0, 0] | -| AltSum | [1, 2, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | -| AccRevSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | -| RevAccSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | -| DiagSum | [1, 1, 0, 1, -1, 0, 1, 1] | - -SierpinskiInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| Middle | [1, 1, 0, -1, 0, 0, 0, 1, 0] | -| Central | [1, 0, 0, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, -1, 1, -1, 1, 1, -1, -1] | -| PosHalf | [1, 1, 3, 3, 15, 15, 45, 45, 255] | -| NegHalf | [1, -3, 3, -9, 15, -45, 45, -135, 255] | -| Bin | [1, 0, 0, -4, 0, -8, -28, 0, 0] | -| InvBin | [1, -2, 0, 0, 0, 0, -28, 96, 0] | -| TransSqrs | [0, -1, -4, 4, -16, 8, 16, 0, -64] | -| TransNat0 | [0, -1, -2, 0, -4, 0, 0, 0, -8] | -| TransNat1 | [1, -1, -2, 0, -4, 0, 0, 0, -8] | - -SierpinskiInvRev Diagonals as rows +Sierpinski:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, 0, 0, -1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [1, -1, -1, 1, -1, 1, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0, -1]] | +| RevTabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [-1, 0, 0, 0, 1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [-1, 1, 1, -1, 1, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, 0], [1, -1, -1], [1, 0, -1], [1, -1, 0, 1], [1, 0, 0, 0], [1, -1, -1, 0, -1]] | +| AccTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 1, 1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [1, 0, -1, 0, -1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0]] | +| RevAccTabl | [[1], [0, 1], [0, 1, 1], [0, -1, 0, 1], [0, 1, 1, 1, 1], [0, -1, 0, 0, 0, 1], [0, -1, -1, 0, 0, 1, 1], [0, 1, 0, -1, 0, -1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1, 1]] | +| AccRevTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [-1, -1, -1, -1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [-1, 0, 1, 0, 1, 0, -1, 0], [-1, -1, -1, -1, -1, -1, -1, -1, 0]] | + +Sierpinski:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, -1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, 2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | +| AccRevSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | +| AntiDiagSum | [1, 1, 0, 1, -1, 0, 1, 1, -2] | + +Sierpinski:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, 1, 0, -1, 0, 0, 0, 1, 0] | +| ColECenter | [1, 0, 0, 0, 0] | +| ColOCenter | [1, -1, 0, 1] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, -1, 1, -1, 1, 1, -1, -1] | +| TransSqrs | [0, -1, -4, 4, -16, 8, 16, 0, -64] | +| TransNat0 | [0, -1, -2, 0, -4, 0, 0, 0, -8] | +| TransNat1 | [1, -1, -2, 0, -4, 0, 0, 0, -8] | + +Sierpinski:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, -1, 1, -1, 1, 1, -1, -1]| @@ -391,7 +375,7 @@ SierpinskiInvRev Diagonals as rows | DiagRow7 | [1, 0, 0, 0, 0, 0, 0, 0, -1]| | DiagRow8 | [1, -1, -1, 1, -1, 1, 1, -1, 0]| -SierpinskiInvRev Diagonals as columns +Sierpinski:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ SierpinskiInvRev Diagonals as columns | DiagCol7 | [-1, 0, 0, 0, 0, 0, 0, 0, -1] | | DiagCol8 | [-1, -1, -1, -1, -1, -1, -1, -1, 0] | -SierpinskiInvRev Polynomial values as rows +Sierpinski:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ SierpinskiInvRev Polynomial values as rows | PolyRow7 | [1, 0, -45, -1280, -11475, -59904, -226625, -691200, -1805895] | | PolyRow8 | [1, 0, -255, -6560, -65535, -390624, -1679615, -5764800, -16777215] | -SierpinskiInvRev Polynomial values as columns +Sierpinski:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -430,3 +414,107 @@ SierpinskiInvRev Polynomial values as columns | PolyCol7 | [1, -6, -48, 288, -2400, 14400, 115200, -691200, -5764800] | | PolyCol8 | [1, -7, -63, 441, -4095, 28665, 257985, -1805895, -16777215] | +# Sierpinski:Rev:Inv +[] + +Sierpinski:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [-1, 0, 1] | +| Row3 | [1, -1, -1, 1] | +| Row4 | [-1, 0, 0, 0, 1] | +| Row5 | [1, -1, 0, 0, -1, 1] | +| Row6 | [1, 0, -1, 0, -1, 0, 1] | +| Row7 | [-1, 1, 1, -1, 1, -1, -1, 1] | +| Row8 | [-1, 0, 0, 0, 0, 0, 0, 0, 1] | + +Sierpinski:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [-1, 0, 1], [1, -1, -1, 1], [-1, 0, 0, 0, 1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [-1, 1, 1, -1, 1, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 1]] | +| RevTabl | [[1], [1, -1], [1, 0, -1], [1, -1, -1, 1], [1, 0, 0, 0, -1], [1, -1, 0, 0, -1, 1], [1, 0, -1, 0, -1, 0, 1], [1, -1, -1, 1, -1, 1, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0, -1]] | +| AntiDiag | [[1], [-1], [-1, 1], [1, 0], [-1, -1, 1], [1, 0, -1], [1, -1, 0, 1], [-1, 0, 0, 0], [-1, 1, -1, 0, 1]] | +| AccTabl | [[1], [-1, 0], [-1, -1, 0], [1, 0, -1, 0], [-1, -1, -1, -1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [-1, 0, 1, 0, 1, 0, -1, 0], [-1, -1, -1, -1, -1, -1, -1, -1, 0]] | +| RevAccTabl | [[1], [0, -1], [0, -1, -1], [0, -1, 0, 1], [0, -1, -1, -1, -1], [0, -1, 0, 0, 0, 1], [0, -1, -1, 0, 0, 1, 1], [0, -1, 0, 1, 0, 1, 0, -1], [0, -1, -1, -1, -1, -1, -1, -1, -1]] | +| AccRevTabl | [[1], [1, 0], [1, 1, 0], [1, 0, -1, 0], [1, 1, 1, 1, 0], [1, 0, 0, 0, -1, 0], [1, 1, 0, 0, -1, -1, 0], [1, 0, -1, 0, -1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 1, 1, 0]] | + +Sierpinski:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| OddSum | [0, 1, 0, 0, 0, 0, 0, 0, 0] | +| AltSum | [1, -2, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, -1, -2, 0, -4, 0, 0, 0, -8] | +| AccRevSum | [1, 1, 2, 0, 4, 0, 0, 0, 8] | +| AntiDiagSum | [1, -1, 0, 1, -1, 0, 1, -1, 0] | + +Sierpinski:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColMiddle | [1, -1, 0, -1, 0, 0, 0, -1, 0] | +| ColECenter | [1, 0, 0, 0, 0] | +| ColOCenter | [-1, -1, 0, -1] | +| ColLeft | [1, -1, -1, 1, -1, 1, 1, -1, -1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 4, 4, 16, 8, 16, 0, 64] | +| TransNat0 | [0, 1, 2, 0, 4, 0, 0, 0, 8] | +| TransNat1 | [1, 1, 2, 0, 4, 0, 0, 0, 8] | + +Sierpinski:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, 0, -1, 0, -1, 0, -1, 0, -1]| +| DiagRow2 | [-1, -1, 0, 0, -1, -1, 0, 0, -1]| +| DiagRow3 | [1, 0, 0, 0, 1, 0, 0, 0, 1]| +| DiagRow4 | [-1, -1, -1, -1, 0, 0, 0, 0, -1]| +| DiagRow5 | [1, 0, 1, 0, 0, 0, 0, 0, 1]| +| DiagRow6 | [1, 1, 0, 0, 0, 0, 0, 0, 1]| +| DiagRow7 | [-1, 0, 0, 0, 0, 0, 0, 0, -1]| +| DiagRow8 | [-1, -1, -1, -1, -1, -1, -1, -1, 0]| + +Sierpinski:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, -1, 1, -1, 1, 1, -1, -1] | +| DiagCol1 | [1, 0, -1, 0, -1, 0, 1, 0, -1] | +| DiagCol2 | [1, -1, 0, 0, -1, 1, 0, 0, -1] | +| DiagCol3 | [1, 0, 0, 0, -1, 0, 0, 0, -1] | +| DiagCol4 | [1, -1, -1, 1, 0, 0, 0, 0, -1] | +| DiagCol5 | [1, 0, -1, 0, 0, 0, 0, 0, -1] | +| DiagCol6 | [1, -1, 0, 0, 0, 0, 0, 0, -1] | +| DiagCol7 | [1, 0, 0, 0, 0, 0, 0, 0, -1] | +| DiagCol8 | [1, -1, -1, 1, -1, 1, 1, -1, 0] | + +Sierpinski:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [-1, 0, 3, 8, 15, 24, 35, 48, 63] | +| PolyRow3 | [1, 0, 3, 16, 45, 96, 175, 288, 441] | +| PolyRow4 | [-1, 0, 15, 80, 255, 624, 1295, 2400, 4095] | +| PolyRow5 | [1, 0, 15, 160, 765, 2496, 6475, 14400, 28665] | +| PolyRow6 | [1, 0, 45, 640, 3825, 14976, 45325, 115200, 257985] | +| PolyRow7 | [-1, 0, 45, 1280, 11475, 59904, 226625, 691200, 1805895] | +| PolyRow8 | [-1, 0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215] | + +Sierpinski:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, -1, 1, -1, 1, 1, -1, -1] | +| PolyCol1 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| PolyCol2 | [1, 1, 3, 3, 15, 15, 45, 45, 255] | +| PolyCol3 | [1, 2, 8, 16, 80, 160, 640, 1280, 6560] | +| PolyCol4 | [1, 3, 15, 45, 255, 765, 3825, 11475, 65535] | +| PolyCol5 | [1, 4, 24, 96, 624, 2496, 14976, 59904, 390624] | +| PolyCol6 | [1, 5, 35, 175, 1295, 6475, 45325, 226625, 1679615] | +| PolyCol7 | [1, 6, 48, 288, 2400, 14400, 115200, 691200, 5764800] | +| PolyCol8 | [1, 7, 63, 441, 4095, 28665, 257985, 1805895, 16777215] | + diff --git a/data/md/StirlingCyc.md b/data/md/StirlingCyc.md index f1021df..962ba76 100644 --- a/data/md/StirlingCyc.md +++ b/data/md/StirlingCyc.md @@ -1,5 +1,5 @@ # StirlingCyc -['A008275', 'A008276', 'A048994', 'A054654', 'A094638', 'A130534', 'A132393'] +['A132393', 'A008275', 'A008276', 'A048994', 'A054654', 'A094638', 'A130534'] StirlingCyc Triangle view | Row | Seq | @@ -14,45 +14,41 @@ StirlingCyc Triangle view | Row7 | [0, 720, 1764, 1624, 735, 175, 21, 1] | | Row8 | [0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1] | -StirlingCyc Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 6, 11, 6, 0, 1, 10, 35, 50, 24, 0, 1, 15, 85, 225, 274, 120, 0, 1, 21, 175, 735, 1624, 1764, 720, 0, 1, 28, 322, 1960, 6769, 13132, 13068, 5040, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 6, 17, 23, 24, 0, 24, 74, 109, 119, 120, 0, 120, 394, 619, 704, 719, 720, 0, 720, 2484, 4108, 4843, 5018, 5039, 5040, 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 6, 5, 2, 0, 24, 23, 17, 6, 0, 120, 119, 109, 74, 24, 0, 720, 719, 704, 619, 394, 120, 0, 5040, 5039, 5018, 4843, 4108, 2484, 720, 0, 40320, 40319, 40291, 39969, 38009, 31240, 18108, 5040, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 7, 18, 24, 24, 1, 11, 46, 96, 120, 120, 1, 16, 101, 326, 600, 720, 720, 1, 22, 197, 932, 2556, 4320, 5040, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 6, 3, 0, 24, 11, 1, 0, 120, 50, 6] | +StirlingCyc Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 2, 3, 1], [0, 6, 11, 6, 1], [0, 24, 50, 35, 10, 1], [0, 120, 274, 225, 85, 15, 1], [0, 720, 1764, 1624, 735, 175, 21, 1], [0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 3, 2, 0], [1, 6, 11, 6, 0], [1, 10, 35, 50, 24, 0], [1, 15, 85, 225, 274, 120, 0], [1, 21, 175, 735, 1624, 1764, 720, 0], [1, 28, 322, 1960, 6769, 13132, 13068, 5040, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 2, 1], [0, 6, 3], [0, 24, 11, 1], [0, 120, 50, 6], [0, 720, 274, 35, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 2, 5, 6], [0, 6, 17, 23, 24], [0, 24, 74, 109, 119, 120], [0, 120, 394, 619, 704, 719, 720], [0, 720, 2484, 4108, 4843, 5018, 5039, 5040], [0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [6, 5, 2, 0], [24, 23, 17, 6, 0], [120, 119, 109, 74, 24, 0], [720, 719, 704, 619, 394, 120, 0], [5040, 5039, 5018, 4843, 4108, 2484, 720, 0], [40320, 40319, 40291, 39969, 38009, 31240, 18108, 5040, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 6, 6], [1, 7, 18, 24, 24], [1, 11, 46, 96, 120, 120], [1, 16, 101, 326, 600, 720, 720], [1, 22, 197, 932, 2556, 4320, 5040, 5040], [1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320]] | StirlingCyc Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 0, 1, 3, 12, 60, 360, 2520, 20160] | -| OddSum | [0, 1, 1, 3, 12, 60, 360, 2520, 20160] | -| AltSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | -| AccRevSum | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | -| RevAccSum | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | -| DiagSum | [1, 0, 1, 1, 3, 9, 36, 176] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 0, 1, 3, 12, 60, 360, 2520, 20160] | +| OddSum | [0, 1, 1, 3, 12, 60, 360, 2520, 20160] | +| AltSum | [1, -1, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | +| AccRevSum | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | +| AntiDiagSum | [1, 0, 1, 1, 3, 9, 36, 176, 1030] | StirlingCyc Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | -| Middle | [1, 0, 1, 2, 11, 50, 225, 1624, 6769] | -| Central | [1, 1, 11, 225, 6769] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| NegHalf | [1, 1, -1, 3, -15, 105, -945, 10395, -135135] | -| Bin | [1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447] | -| InvBin | [1, 1, -1, -2, 19, -79, 76, 2640, -36945] | -| TransSqrs | [0, 1, 5, 23, 120, 724, 5012, 39332, 345832] | -| TransNat0 | [0, 1, 3, 11, 50, 274, 1764, 13068, 109584] | -| TransNat1 | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | +| ColMiddle | [1, 0, 1, 2, 11, 50, 225, 1624, 6769] | +| ColECenter | [1, 1, 11, 225, 6769] | +| ColOCenter | [0, 2, 50, 1624] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 23, 120, 724, 5012, 39332, 345832] | +| TransNat0 | [0, 1, 3, 11, 50, 274, 1764, 13068, 109584] | +| TransNat1 | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | StirlingCyc Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingCyc Polynomial values as columns | PolyCol7 | [1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960] | | PolyCol8 | [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200] | -# StirlingCycInv -True +# StirlingCyc:Inv +[] -StirlingCycInv Triangle view +StirlingCyc:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingCycInv Triangle view | Row7 | [0, 1, -63, 301, -350, 140, -21, 1] | | Row8 | [0, -1, 127, -966, 1701, -1050, 266, -28, 1] | -StirlingCycInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, -1, 7, -6, 1, 0, 1, -15, 25, -10, 1, 0, -1, 31, -90, 65, -15, 1, 0, 1, -63, 301, -350, 140, -21, 1, 0, -1, 127, -966, 1701, -1050, 266, -28, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -3, 1, 0, 1, -6, 7, -1, 0, 1, -10, 25, -15, 1, 0, 1, -15, 65, -90, 31, -1, 0, 1, -21, 140, -350, 301, -63, 1, 0, 1, -28, 266, -1050, 1701, -966, 127, -1, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 1, -2, -1, 0, -1, 6, 0, 1, 0, 1, -14, 11, 1, 2, 0, -1, 30, -60, 5, -10, -9, 0, 1, -62, 239, -111, 29, 8, 9, 0, -1, 126, -840, 861, -189, 77, 49, 50] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, -1, -2, 1, 0, 1, 0, 6, -1, 0, 2, 1, 11, -14, 1, 0, -9, -10, 5, -60, 30, -1, 0, 9, 8, 29, -111, 239, -62, 1, 0, 50, 49, 77, -189, 861, -840, 126, -1, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -2, -1, -1, 1, -5, 2, 1, 1, 1, -9, 16, 1, 2, 2, 1, -14, 51, -39, -8, -9, -9, 1, -20, 120, -230, 71, 8, 9, 9, 1, -27, 239, -811, 890, -76, 51, 50, 50] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 1, 1, 0, -1, -3, 0, 1, 7, 1, 0, -1, -15, -6] | - -StirlingCycInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | -| EvenSum | [1, 0, 1, -3, 8, -25, 97, -434, 2095] | -| OddSum | [0, 1, -1, 2, -7, 27, -106, 443, -2045] | -| AltSum | [1, -1, 2, -5, 15, -52, 203, -877, 4140] | -| AccSum | [1, 1, -1, -2, 6, 1, -45, 113, 133] | -| AccRevSum | [1, 1, -1, -2, 6, 1, -45, 113, 133] | -| RevAccSum | [1, 2, 1, -3, 0, 13, -27, -32, 367] | -| DiagSum | [1, 0, 1, -1, 2, -4, 9, -22] | - -StirlingCycInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 42, 150, 36270, 270900, 9440379900] | -| RowGcd | [1, 1, 1, -3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | -| Middle | [1, 0, -1, 1, 7, -15, -90, 301, 1701] | -| Central | [1, -1, 7, -90, 1701] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, -1, 9, -23, -25, 583, -3087] | -| NegHalf | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | -| Bin | [1, 1, -1, -5, 15, 56, -455, -237, 16947] | -| InvBin | [1, 1, 3, 13, 71, 456, 3337, 27203, 243203] | -| TransSqrs | [0, 1, 3, -2, -11, 31, 14, -349, 1047] | -| TransNat0 | [0, 1, 1, -2, -1, 11, -18, -41, 317] | -| TransNat1 | [1, 2, 1, -3, 0, 13, -27, -32, 367] | - -StirlingCycInv Diagonals as rows +StirlingCyc:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 1, -3, 1], [0, -1, 7, -6, 1], [0, 1, -15, 25, -10, 1], [0, -1, 31, -90, 65, -15, 1], [0, 1, -63, 301, -350, 140, -21, 1], [0, -1, 127, -966, 1701, -1050, 266, -28, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -3, 1, 0], [1, -6, 7, -1, 0], [1, -10, 25, -15, 1, 0], [1, -15, 65, -90, 31, -1, 0], [1, -21, 140, -350, 301, -63, 1, 0], [1, -28, 266, -1050, 1701, -966, 127, -1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 1, 1], [0, -1, -3], [0, 1, 7, 1], [0, -1, -15, -6], [0, 1, 31, 25, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 1, -2, -1], [0, -1, 6, 0, 1], [0, 1, -14, 11, 1, 2], [0, -1, 30, -60, 5, -10, -9], [0, 1, -62, 239, -111, 29, 8, 9], [0, -1, 126, -840, 861, -189, 77, 49, 50]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [-1, -2, 1, 0], [1, 0, 6, -1, 0], [2, 1, 11, -14, 1, 0], [-9, -10, 5, -60, 30, -1, 0], [9, 8, 29, -111, 239, -62, 1, 0], [50, 49, 77, -189, 861, -840, 126, -1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -1, -1], [1, -5, 2, 1, 1], [1, -9, 16, 1, 2, 2], [1, -14, 51, -39, -8, -9, -9], [1, -20, 120, -230, 71, 8, 9, 9], [1, -27, 239, -811, 890, -76, 51, 50, 50]] | + +StirlingCyc:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | +| EvenSum | [1, 0, 1, -3, 8, -25, 97, -434, 2095] | +| OddSum | [0, 1, -1, 2, -7, 27, -106, 443, -2045] | +| AltSum | [1, -1, 2, -5, 15, -52, 203, -877, 4140] | +| AccSum | [1, 1, -1, -2, 6, 1, -45, 113, 133] | +| AccRevSum | [1, 2, 1, -3, 0, 13, -27, -32, 367] | +| AntiDiagSum | [1, 0, 1, -1, 2, -4, 9, -22, 58] | + +StirlingCyc:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | +| ColMiddle | [1, 0, -1, 1, 7, -15, -90, 301, 1701] | +| ColECenter | [1, -1, 7, -90, 1701] | +| ColOCenter | [0, 1, -15, 301] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -2, -11, 31, 14, -349, 1047] | +| TransNat0 | [0, 1, 1, -2, -1, 11, -18, -41, 317] | +| TransNat1 | [1, 2, 1, -3, 0, 13, -27, -32, 367] | + +StirlingCyc:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ StirlingCycInv Diagonals as rows | DiagRow7 | [0, -1, -255, -9330, -145750, -1379400, -9321312, -49329280, -216627840]| | DiagRow8 | [0, 1, 511, 28501, 611501, 7508501, 63436373, 408741333, 2141764053]| -StirlingCycInv Diagonals as columns +StirlingCyc:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ StirlingCycInv Diagonals as columns | DiagCol7 | [1, -28, 462, -5880, 63987, -627396, 5715424, -49329280, 408741333] | | DiagCol8 | [1, -36, 750, -11880, 159027, -1899612, 20912320, -216627840, 2141764053] | -StirlingCycInv Polynomial values as rows +StirlingCyc:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingCycInv Polynomial values as rows | PolyRow7 | [0, 9, -178, 111, 2388, 4805, -2046, -34293, -103864] | | PolyRow8 | [0, 50, 90, -3072, -3220, 23670, 87510, 111860, -138312] | -StirlingCycInv Polynomial values as columns +StirlingCyc:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ StirlingCycInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 203, 679, 644, -7749, -34293, 111860] | | PolyCol8 | [1, 8, 56, 328, 1464, 3656, -7240, -103864, -138312] | -# StirlingCycRev -True +# StirlingCyc:Rev +[] -StirlingCycRev Triangle view +StirlingCyc:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ StirlingCycRev Triangle view | Row7 | [1, 21, 175, 735, 1624, 1764, 720, 0] | | Row8 | [1, 28, 322, 1960, 6769, 13132, 13068, 5040, 0] | -StirlingCycRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 6, 11, 6, 0, 1, 10, 35, 50, 24, 0, 1, 15, 85, 225, 274, 120, 0, 1, 21, 175, 735, 1624, 1764, 720, 0, 1, 28, 322, 1960, 6769, 13132, 13068, 5040, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 7, 18, 24, 24, 1, 11, 46, 96, 120, 120, 1, 16, 101, 326, 600, 720, 720, 1, 22, 197, 932, 2556, 4320, 5040, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 24, 24, 18, 7, 1, 120, 120, 96, 46, 11, 1, 720, 720, 600, 326, 101, 16, 1, 5040, 5040, 4320, 2556, 932, 197, 22, 1, 40320, 40320, 35280, 22212, 9080, 2311, 351, 29, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 6, 17, 23, 24, 0, 24, 74, 109, 119, 120, 0, 120, 394, 619, 704, 719, 720, 0, 720, 2484, 4108, 4843, 5018, 5039, 5040, 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 6, 2, 1, 10, 11, 0, 1, 15, 35, 6] | - -StirlingCycRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| EvenSum | [1, 1, 1, 3, 12, 60, 360, 2520, 20160] | -| OddSum | [0, 0, 1, 3, 12, 60, 360, 2520, 20160] | -| AltSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | -| AccRevSum | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | -| RevAccSum | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | -| DiagSum | [1, 1, 1, 2, 4, 9, 22, 57] | - -StirlingCycRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | -| Middle | [1, 1, 1, 3, 11, 35, 225, 735, 6769] | -| Central | [1, 1, 11, 225, 6769] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 24, 120, 720, 5040, 40320, 362880] | -| NegHalf | [1, -2, 2, 0, 0, 0, 0, 0, 0] | -| Bin | [1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447] | -| InvBin | [1, -1, -1, 2, 19, 79, 76, -2640, -36945] | -| TransSqrs | [0, 0, 1, 11, 104, 984, 9764, 103340, 1172968] | -| TransNat0 | [0, 0, 1, 7, 46, 326, 2556, 22212, 212976] | -| TransNat1 | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | - -StirlingCycRev Diagonals as rows +StirlingCyc:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 3, 2, 0], [1, 6, 11, 6, 0], [1, 10, 35, 50, 24, 0], [1, 15, 85, 225, 274, 120, 0], [1, 21, 175, 735, 1624, 1764, 720, 0], [1, 28, 322, 1960, 6769, 13132, 13068, 5040, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 2, 3, 1], [0, 6, 11, 6, 1], [0, 24, 50, 35, 10, 1], [0, 120, 274, 225, 85, 15, 1], [0, 720, 1764, 1624, 735, 175, 21, 1], [0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 3, 0], [1, 6, 2], [1, 10, 11, 0], [1, 15, 35, 6], [1, 21, 85, 50, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 6, 6], [1, 7, 18, 24, 24], [1, 11, 46, 96, 120, 120], [1, 16, 101, 326, 600, 720, 720], [1, 22, 197, 932, 2556, 4320, 5040, 5040], [1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [6, 6, 4, 1], [24, 24, 18, 7, 1], [120, 120, 96, 46, 11, 1], [720, 720, 600, 326, 101, 16, 1], [5040, 5040, 4320, 2556, 932, 197, 22, 1], [40320, 40320, 35280, 22212, 9080, 2311, 351, 29, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 2, 5, 6], [0, 6, 17, 23, 24], [0, 24, 74, 109, 119, 120], [0, 120, 394, 619, 704, 719, 720], [0, 720, 2484, 4108, 4843, 5018, 5039, 5040], [0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320]] | + +StirlingCyc:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| EvenSum | [1, 1, 1, 3, 12, 60, 360, 2520, 20160] | +| OddSum | [0, 0, 1, 3, 12, 60, 360, 2520, 20160] | +| AltSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 2, 5, 17, 74, 394, 2484, 18108, 149904] | +| AccRevSum | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | +| AntiDiagSum | [1, 1, 1, 2, 4, 9, 22, 57, 157] | + +StirlingCyc:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | +| ColMiddle | [1, 1, 1, 3, 11, 35, 225, 735, 6769] | +| ColECenter | [1, 1, 11, 225, 6769] | +| ColOCenter | [1, 3, 35, 735] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 11, 104, 984, 9764, 103340, 1172968] | +| TransNat0 | [0, 0, 1, 7, 46, 326, 2556, 22212, 212976] | +| TransNat1 | [1, 1, 3, 13, 70, 446, 3276, 27252, 253296] | + +StirlingCyc:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ StirlingCycRev Diagonals as rows | DiagRow7 | [1, 28, 546, 9450, 157773, 2637558, 44990231, 790943153, 14409322928]| | DiagRow8 | [1, 36, 870, 18150, 357423, 6926634, 135036473, 2681453775, 54631129553]| -StirlingCycRev Diagonals as columns +StirlingCyc:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ StirlingCycRev Diagonals as columns | DiagCol7 | [0, 5040, 109584, 1172700, 8409500, 45995730, 206070150, 790943153, 2681453775] | | DiagCol8 | [0, 40320, 1026576, 12753576, 105258076, 657206836, 3336118786, 14409322928, 54631129553] | -StirlingCycRev Polynomial values as rows +StirlingCyc:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ StirlingCycRev Polynomial values as rows | PolyRow7 | [1, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 118514880, 253586025] | | PolyRow8 | [1, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 14454403425] | -StirlingCycRev Polynomial values as columns +StirlingCyc:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ StirlingCycRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000] | | PolyCol8 | [1, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425] | -# StirlingCycInvRev -True +# StirlingCyc:Inv:Rev +[] -StirlingCycInvRev Triangle view +StirlingCyc:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ StirlingCycInvRev Triangle view | Row7 | [1, -21, 140, -350, 301, -63, 1, 0] | | Row8 | [1, -28, 266, -1050, 1701, -966, 127, -1, 0] | -StirlingCycInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -3, 1, 0, 1, -6, 7, -1, 0, 1, -10, 25, -15, 1, 0, 1, -15, 65, -90, 31, -1, 0, 1, -21, 140, -350, 301, -63, 1, 0, 1, -28, 266, -1050, 1701, -966, 127, -1, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, -1, 7, -6, 1, 0, 1, -15, 25, -10, 1, 0, -1, 31, -90, 65, -15, 1, 0, 1, -63, 301, -350, 140, -21, 1, 0, -1, 127, -966, 1701, -1050, 266, -28, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -2, -1, -1, 1, -5, 2, 1, 1, 1, -9, 16, 1, 2, 2, 1, -14, 51, -39, -8, -9, -9, 1, -20, 120, -230, 71, 8, 9, 9, 1, -27, 239, -811, 890, -76, 51, 50, 50] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, -1, -1, -2, 1, 1, 1, 2, -5, 1, 2, 2, 1, 16, -9, 1, -9, -9, -8, -39, 51, -14, 1, 9, 9, 8, 71, -230, 120, -20, 1, 50, 50, 51, -76, 890, -811, 239, -27, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 1, -2, -1, 0, -1, 6, 0, 1, 0, 1, -14, 11, 1, 2, 0, -1, 30, -60, 5, -10, -9, 0, 1, -62, 239, -111, 29, 8, 9, 0, -1, 126, -840, 861, -189, 77, 49, 50] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -3, 0, 1, -6, 1, 1, -10, 7, 0, 1, -15, 25, -1] | - -StirlingCycInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | -| EvenSum | [1, 1, 1, 2, 8, 27, 97, 443, 2095] | -| OddSum | [0, 0, -1, -3, -7, -25, -106, -434, -2045] | -| AltSum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | -| AccSum | [1, 2, 1, -3, 0, 13, -27, -32, 367] | -| AccRevSum | [1, 2, 1, -3, 0, 13, -27, -32, 367] | -| RevAccSum | [1, 1, -1, -2, 6, 1, -45, 113, 133] | -| DiagSum | [1, 1, 1, 0, -2, -4, -2, 10] | - -StirlingCycInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, -3, 42, 150, 36270, 270900, 9440379900] | -| RowGcd | [1, 1, 1, -3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | -| Middle | [1, 1, -1, -3, 7, 25, -90, -350, 1701] | -| Central | [1, -1, 7, -90, 1701] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, -2, -6, 14, 26, -178, 90] | -| NegHalf | [1, -2, 6, -22, 94, -454, 2430, -14214, 89918] | -| Bin | [1, 1, -1, -5, 15, 56, -455, -237, 16947] | -| InvBin | [1, -1, 3, -13, 71, -456, 3337, -27203, 243203] | -| TransSqrs | [0, 0, -1, 1, 13, -29, -94, 666, -825] | -| TransNat0 | [0, 0, -1, -1, 5, -1, -36, 104, 83] | -| TransNat1 | [1, 1, -1, -2, 6, 1, -45, 113, 133] | - -StirlingCycInvRev Diagonals as rows +StirlingCyc:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -3, 1, 0], [1, -6, 7, -1, 0], [1, -10, 25, -15, 1, 0], [1, -15, 65, -90, 31, -1, 0], [1, -21, 140, -350, 301, -63, 1, 0], [1, -28, 266, -1050, 1701, -966, 127, -1, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 1, -3, 1], [0, -1, 7, -6, 1], [0, 1, -15, 25, -10, 1], [0, -1, 31, -90, 65, -15, 1], [0, 1, -63, 301, -350, 140, -21, 1], [0, -1, 127, -966, 1701, -1050, 266, -28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -3, 0], [1, -6, 1], [1, -10, 7, 0], [1, -15, 25, -1], [1, -21, 65, -15, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -2, -1, -1], [1, -5, 2, 1, 1], [1, -9, 16, 1, 2, 2], [1, -14, 51, -39, -8, -9, -9], [1, -20, 120, -230, 71, 8, 9, 9], [1, -27, 239, -811, 890, -76, 51, 50, 50]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [-1, -1, -2, 1], [1, 1, 2, -5, 1], [2, 2, 1, 16, -9, 1], [-9, -9, -8, -39, 51, -14, 1], [9, 9, 8, 71, -230, 120, -20, 1], [50, 50, 51, -76, 890, -811, 239, -27, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 1, -2, -1], [0, -1, 6, 0, 1], [0, 1, -14, 11, 1, 2], [0, -1, 30, -60, 5, -10, -9], [0, 1, -62, 239, -111, 29, 8, 9], [0, -1, 126, -840, 861, -189, 77, 49, 50]] | + +StirlingCyc:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | +| EvenSum | [1, 1, 1, 2, 8, 27, 97, 443, 2095] | +| OddSum | [0, 0, -1, -3, -7, -25, -106, -434, -2045] | +| AltSum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | +| AccSum | [1, 2, 1, -3, 0, 13, -27, -32, 367] | +| AccRevSum | [1, 1, -1, -2, 6, 1, -45, 113, 133] | +| AntiDiagSum | [1, 1, 1, 0, -2, -4, -2, 10, 30] | + +StirlingCyc:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | +| ColMiddle | [1, 1, -1, -3, 7, 25, -90, -350, 1701] | +| ColECenter | [1, -1, 7, -90, 1701] | +| ColOCenter | [1, -3, 25, -350] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, 1, 13, -29, -94, 666, -825] | +| TransNat0 | [0, 0, -1, -1, 5, -1, -36, 104, 83] | +| TransNat1 | [1, 1, -1, -2, 6, 1, -45, 113, 133] | + +StirlingCyc:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ StirlingCycInvRev Diagonals as rows | DiagRow7 | [1, -28, 462, -5880, 63987, -627396, 5715424, -49329280, 408741333]| | DiagRow8 | [1, -36, 750, -11880, 159027, -1899612, 20912320, -216627840, 2141764053]| -StirlingCycInvRev Diagonals as columns +StirlingCyc:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ StirlingCycInvRev Diagonals as columns | DiagCol7 | [0, -1, -255, -9330, -145750, -1379400, -9321312, -49329280, -216627840] | | DiagCol8 | [0, 1, 511, 28501, 611501, 7508501, 63436373, 408741333, 2141764053] | -StirlingCycInvRev Polynomial values as rows +StirlingCyc:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ StirlingCycInvRev Polynomial values as rows | PolyRow7 | [1, 9, 583, 1549, -3603, -33479, -123821, -331827, -739751] | | PolyRow8 | [1, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745] | -StirlingCycInvRev Polynomial values as columns +StirlingCyc:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/StirlingCyc2.md b/data/md/StirlingCyc2.md index 57cc9a9..6d757a5 100644 --- a/data/md/StirlingCyc2.md +++ b/data/md/StirlingCyc2.md @@ -14,45 +14,41 @@ StirlingCyc2 Triangle view | Row7 | [0, 720, 924, 210, 0, 0, 0, 0] | | Row8 | [0, 5040, 7308, 2380, 105, 0, 0, 0, 0] | -StirlingCyc2 Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 24, 20, 0, 0, 0, 0, 120, 130, 15, 0, 0, 0, 0, 720, 924, 210, 0, 0, 0, 0, 0, 5040, 7308, 2380, 105, 0, 0, 0, 0] | -| TriRev | [1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 6, 0, 0, 0, 0, 20, 24, 0, 0, 0, 0, 15, 130, 120, 0, 0, 0, 0, 0, 210, 924, 720, 0, 0, 0, 0, 0, 105, 2380, 7308, 5040, 0] | -| TriAcc | [1, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 6, 9, 9, 9, 0, 24, 44, 44, 44, 44, 0, 120, 250, 265, 265, 265, 265, 0, 720, 1644, 1854, 1854, 1854, 1854, 1854, 0, 5040, 12348, 14728, 14833, 14833, 14833, 14833, 14833] | -| TriRevAcc | [1, 0, 0, 1, 1, 0, 2, 2, 2, 0, 9, 9, 9, 6, 0, 44, 44, 44, 44, 24, 0, 265, 265, 265, 265, 250, 120, 0, 1854, 1854, 1854, 1854, 1854, 1644, 720, 0, 14833, 14833, 14833, 14833, 14833, 14728, 12348, 5040, 0] | -| TriAccRev | [1, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 3, 9, 9, 0, 0, 0, 20, 44, 44, 0, 0, 0, 15, 145, 265, 265, 0, 0, 0, 0, 210, 1134, 1854, 1854, 0, 0, 0, 0, 105, 2485, 9793, 14833, 14833] | -| TriDiag | [1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 6, 0, 0, 24, 3, 0, 0, 120, 20, 0] | +StirlingCyc2 Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 0], [0, 1, 0], [0, 2, 0, 0], [0, 6, 3, 0, 0], [0, 24, 20, 0, 0, 0], [0, 120, 130, 15, 0, 0, 0], [0, 720, 924, 210, 0, 0, 0, 0], [0, 5040, 7308, 2380, 105, 0, 0, 0, 0]] | +| RevTabl | [[1], [0, 0], [0, 1, 0], [0, 0, 2, 0], [0, 0, 3, 6, 0], [0, 0, 0, 20, 24, 0], [0, 0, 0, 15, 130, 120, 0], [0, 0, 0, 0, 210, 924, 720, 0], [0, 0, 0, 0, 105, 2380, 7308, 5040, 0]] | +| AntiDiag | [[1], [0], [0, 0], [0, 1], [0, 2, 0], [0, 6, 0], [0, 24, 3, 0], [0, 120, 20, 0], [0, 720, 130, 0, 0]] | +| AccTabl | [[1], [0, 0], [0, 1, 1], [0, 2, 2, 2], [0, 6, 9, 9, 9], [0, 24, 44, 44, 44, 44], [0, 120, 250, 265, 265, 265, 265], [0, 720, 1644, 1854, 1854, 1854, 1854, 1854], [0, 5040, 12348, 14728, 14833, 14833, 14833, 14833, 14833]] | +| RevAccTabl | [[1], [0, 0], [1, 1, 0], [2, 2, 2, 0], [9, 9, 9, 6, 0], [44, 44, 44, 44, 24, 0], [265, 265, 265, 265, 250, 120, 0], [1854, 1854, 1854, 1854, 1854, 1644, 720, 0], [14833, 14833, 14833, 14833, 14833, 14728, 12348, 5040, 0]] | +| AccRevTabl | [[1], [0, 0], [0, 1, 1], [0, 0, 2, 2], [0, 0, 3, 9, 9], [0, 0, 0, 20, 44, 44], [0, 0, 0, 15, 145, 265, 265], [0, 0, 0, 0, 210, 1134, 1854, 1854], [0, 0, 0, 0, 105, 2485, 9793, 14833, 14833]] | StirlingCyc2 Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| EvenSum | [1, 0, 0, 0, 3, 20, 130, 924, 7413] | -| OddSum | [0, 0, 1, 2, 6, 24, 135, 930, 7420] | -| AltSum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | -| AccSum | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | -| AccRevSum | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | -| RevAccSum | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | -| DiagSum | [1, 0, 0, 1, 2, 6, 27, 140] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| EvenSum | [1, 0, 0, 0, 3, 20, 130, 924, 7413] | +| OddSum | [0, 0, 1, 2, 6, 24, 135, 930, 7420] | +| AltSum | [1, 0, -1, -2, -3, -4, -5, -6, -7] | +| AccSum | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | +| AccRevSum | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | +| AntiDiagSum | [1, 0, 0, 1, 2, 6, 27, 140, 850] | StirlingCyc2 Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 120, 1560, 55440, 2484720] | -| RowGcd | [1, 1, 1, 2, 3, 4, 5, 6, 7] | -| RowMax | [1, 0, 1, 2, 6, 24, 130, 924, 7308] | -| Middle | [1, 0, 1, 2, 3, 20, 15, 210, 105] | -| Central | [1, 1, 3, 15, 105] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 0, 2, 8, 60, 544, 6040, 79008, 1190672] | -| NegHalf | [1, 0, -2, 8, -36, 224, -1880, 19872, -251888] | -| Bin | [1, 0, 2, 6, 42, 320, 2970, 31794, 385574] | -| InvBin | [1, 0, -2, 6, -6, -80, 930, -7014, 38374] | -| TransSqrs | [0, 0, 1, 2, 18, 104, 775, 6306, 57372] | -| TransNat0 | [0, 0, 1, 2, 12, 64, 425, 3198, 27216] | -| TransNat1 | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 120, 1560, 55440, 2484720] | +| RowGcd | [1, 1, 1, 2, 3, 4, 5, 6, 7] | +| RowMax | [1, 0, 1, 2, 6, 24, 130, 924, 7308] | +| ColMiddle | [1, 0, 1, 2, 3, 20, 15, 210, 105] | +| ColECenter | [1, 1, 3, 15, 105] | +| ColOCenter | [0, 2, 20, 210] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 2, 18, 104, 775, 6306, 57372] | +| TransNat0 | [0, 0, 1, 2, 12, 64, 425, 3198, 27216] | +| TransNat1 | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | StirlingCyc2 Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingCyc2 Polynomial values as columns | PolyCol7 | [1, 0, 7, 14, 189, 1148, 12355, 122346, 1461817] | | PolyCol8 | [1, 0, 8, 16, 240, 1472, 16960, 172416, 2156672] | -# StirlingCyc2Rev -True +# StirlingCyc2:Rev +[] -StirlingCyc2Rev Triangle view +StirlingCyc2:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingCyc2Rev Triangle view | Row7 | [0, 0, 0, 0, 210, 924, 720, 0] | | Row8 | [0, 0, 0, 0, 105, 2380, 7308, 5040, 0] | -StirlingCyc2Rev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 6, 0, 0, 0, 0, 20, 24, 0, 0, 0, 0, 15, 130, 120, 0, 0, 0, 0, 0, 210, 924, 720, 0, 0, 0, 0, 0, 105, 2380, 7308, 5040, 0] | -| TriRev | [1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 24, 20, 0, 0, 0, 0, 120, 130, 15, 0, 0, 0, 0, 720, 924, 210, 0, 0, 0, 0, 0, 5040, 7308, 2380, 105, 0, 0, 0, 0] | -| TriAcc | [1, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 3, 9, 9, 0, 0, 0, 20, 44, 44, 0, 0, 0, 15, 145, 265, 265, 0, 0, 0, 0, 210, 1134, 1854, 1854, 0, 0, 0, 0, 105, 2485, 9793, 14833, 14833] | -| TriRevAcc | [1, 0, 0, 1, 1, 0, 2, 2, 0, 0, 9, 9, 3, 0, 0, 44, 44, 20, 0, 0, 0, 265, 265, 145, 15, 0, 0, 0, 1854, 1854, 1134, 210, 0, 0, 0, 0, 14833, 14833, 9793, 2485, 105, 0, 0, 0, 0] | -| TriAccRev | [1, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 6, 9, 9, 9, 0, 24, 44, 44, 44, 44, 0, 120, 250, 265, 265, 265, 265, 0, 720, 1644, 1854, 1854, 1854, 1854, 1854, 0, 5040, 12348, 14728, 14833, 14833, 14833, 14833, 14833] | -| TriDiag | [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 6] | - -StirlingCyc2Rev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | -| EvenSum | [1, 0, 0, 2, 3, 24, 130, 930, 7413] | -| OddSum | [0, 0, 1, 0, 6, 20, 135, 924, 7420] | -| AltSum | [1, 0, -1, 2, -3, 4, -5, 6, -7] | -| AccSum | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | -| AccRevSum | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | -| RevAccSum | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | -| DiagSum | [1, 0, 0, 1, 0, 2, 3, 6] | - -StirlingCyc2Rev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 2, 6, 120, 1560, 55440, 2484720] | -| RowGcd | [1, 1, 1, 2, 3, 4, 5, 6, 7] | -| RowMax | [1, 0, 1, 2, 6, 24, 130, 924, 7308] | -| Middle | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| Central | [1, 1, 3, 15, 105] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 0, 2, 4, 24, 128, 880, 6816, 60032] | -| NegHalf | [1, 0, -2, -4, 0, 32, 160, 576, 1792] | -| Bin | [1, 0, 2, 6, 42, 320, 2970, 31794, 385574] | -| InvBin | [1, 0, -2, -6, -6, 80, 930, 7014, 38374] | -| TransSqrs | [0, 0, 1, 8, 66, 564, 5215, 52380, 571228] | -| TransNat0 | [0, 0, 1, 4, 24, 156, 1165, 9780, 91448] | -| TransNat1 | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | - -StirlingCyc2Rev Diagonals as rows +StirlingCyc2:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 0], [0, 1, 0], [0, 0, 2, 0], [0, 0, 3, 6, 0], [0, 0, 0, 20, 24, 0], [0, 0, 0, 15, 130, 120, 0], [0, 0, 0, 0, 210, 924, 720, 0], [0, 0, 0, 0, 105, 2380, 7308, 5040, 0]] | +| RevTabl | [[1], [0, 0], [0, 1, 0], [0, 2, 0, 0], [0, 6, 3, 0, 0], [0, 24, 20, 0, 0, 0], [0, 120, 130, 15, 0, 0, 0], [0, 720, 924, 210, 0, 0, 0, 0], [0, 5040, 7308, 2380, 105, 0, 0, 0, 0]] | +| AntiDiag | [[1], [0], [0, 0], [0, 1], [0, 0, 0], [0, 0, 2], [0, 0, 3, 0], [0, 0, 0, 6], [0, 0, 0, 20, 0]] | +| AccTabl | [[1], [0, 0], [0, 1, 1], [0, 0, 2, 2], [0, 0, 3, 9, 9], [0, 0, 0, 20, 44, 44], [0, 0, 0, 15, 145, 265, 265], [0, 0, 0, 0, 210, 1134, 1854, 1854], [0, 0, 0, 0, 105, 2485, 9793, 14833, 14833]] | +| RevAccTabl | [[1], [0, 0], [1, 1, 0], [2, 2, 0, 0], [9, 9, 3, 0, 0], [44, 44, 20, 0, 0, 0], [265, 265, 145, 15, 0, 0, 0], [1854, 1854, 1134, 210, 0, 0, 0, 0], [14833, 14833, 9793, 2485, 105, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [0, 0], [0, 1, 1], [0, 2, 2, 2], [0, 6, 9, 9, 9], [0, 24, 44, 44, 44, 44], [0, 120, 250, 265, 265, 265, 265], [0, 720, 1644, 1854, 1854, 1854, 1854, 1854], [0, 5040, 12348, 14728, 14833, 14833, 14833, 14833, 14833]] | + +StirlingCyc2:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 1, 2, 9, 44, 265, 1854, 14833] | +| EvenSum | [1, 0, 0, 2, 3, 24, 130, 930, 7413] | +| OddSum | [0, 0, 1, 0, 6, 20, 135, 924, 7420] | +| AltSum | [1, 0, -1, 2, -3, 4, -5, 6, -7] | +| AccSum | [1, 0, 2, 4, 21, 108, 690, 5052, 42049] | +| AccRevSum | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | +| AntiDiagSum | [1, 0, 0, 1, 0, 2, 3, 6, 20] | + +StirlingCyc2:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 2, 6, 120, 1560, 55440, 2484720] | +| RowGcd | [1, 1, 1, 2, 3, 4, 5, 6, 7] | +| RowMax | [1, 0, 1, 2, 6, 24, 130, 924, 7308] | +| ColMiddle | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| ColECenter | [1, 1, 3, 15, 105] | +| ColOCenter | [0, 0, 0, 0] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 8, 66, 564, 5215, 52380, 571228] | +| TransNat0 | [0, 0, 1, 4, 24, 156, 1165, 9780, 91448] | +| TransNat1 | [1, 0, 2, 6, 33, 200, 1430, 11634, 106281] | + +StirlingCyc2:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ StirlingCyc2Rev Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 135135, 9459450]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 2027025]| -StirlingCyc2Rev Diagonals as columns +StirlingCyc2:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ StirlingCyc2Rev Diagonals as columns | DiagCol7 | [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 0] | | DiagCol8 | [0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025] | -StirlingCyc2Rev Polynomial values as rows +StirlingCyc2:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingCyc2Rev Polynomial values as rows | PolyRow7 | [0, 1854, 79008, 766422, 3949056, 14268750, 41049504, 100741158, 219881472] | | PolyRow8 | [0, 14833, 1190672, 16936857, 114972928, 515440625, 1770482448, 5050688377, 12563812352] | -StirlingCyc2Rev Polynomial values as columns +StirlingCyc2:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | diff --git a/data/md/StirlingCycB.md b/data/md/StirlingCycB.md index 31d6057..cf346f7 100644 --- a/data/md/StirlingCycB.md +++ b/data/md/StirlingCycB.md @@ -14,45 +14,41 @@ StirlingCycB Triangle view | Row7 | [135135, 264207, 177331, 57379, 10045, 973, 49, 1] | | Row8 | [2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1] | -StirlingCycB Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 3, 4, 1, 15, 23, 9, 1, 105, 176, 86, 16, 1, 945, 1689, 950, 230, 25, 1, 10395, 19524, 12139, 3480, 505, 36, 1, 135135, 264207, 177331, 57379, 10045, 973, 49, 1, 2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1] | -| TriRev | [1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025] | -| TriAcc | [1, 1, 2, 3, 7, 8, 15, 38, 47, 48, 105, 281, 367, 383, 384, 945, 2634, 3584, 3814, 3839, 3840, 10395, 29919, 42058, 45538, 46043, 46079, 46080, 135135, 399342, 576673, 634052, 644097, 645070, 645119, 645120, 2027025, 6125265, 9049437, 10087453, 10295507, 10320147, 10321855, 10321919, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 7, 3, 48, 47, 38, 15, 384, 383, 367, 281, 105, 3840, 3839, 3814, 3584, 2634, 945, 46080, 46079, 46043, 45538, 42058, 29919, 10395, 645120, 645119, 645070, 644097, 634052, 576673, 399342, 135135, 10321920, 10321919, 10321855, 10320147, 10295507, 10087453, 9049437, 6125265, 2027025] | -| TriAccRev | [1, 1, 2, 1, 5, 8, 1, 10, 33, 48, 1, 17, 103, 279, 384, 1, 26, 256, 1206, 2895, 3840, 1, 37, 542, 4022, 16161, 35685, 46080, 1, 50, 1023, 11068, 68447, 245778, 509985, 645120, 1, 65, 1773, 26413, 234467, 1272483, 4196655, 8294895, 10321920] | -| TriDiag | [1, 1, 3, 1, 15, 4, 105, 23, 1, 945, 176, 9, 10395, 1689, 86, 1, 135135, 19524, 950, 16] | +StirlingCycB Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [3, 4, 1], [15, 23, 9, 1], [105, 176, 86, 16, 1], [945, 1689, 950, 230, 25, 1], [10395, 19524, 12139, 3480, 505, 36, 1], [135135, 264207, 177331, 57379, 10045, 973, 49, 1], [2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1]] | +| RevTabl | [[1], [1, 1], [1, 4, 3], [1, 9, 23, 15], [1, 16, 86, 176, 105], [1, 25, 230, 950, 1689, 945], [1, 36, 505, 3480, 12139, 19524, 10395], [1, 49, 973, 10045, 57379, 177331, 264207, 135135], [1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025]] | +| AntiDiag | [[1], [1], [3, 1], [15, 4], [105, 23, 1], [945, 176, 9], [10395, 1689, 86, 1], [135135, 19524, 950, 16], [2027025, 264207, 12139, 230, 1]] | +| AccTabl | [[1], [1, 2], [3, 7, 8], [15, 38, 47, 48], [105, 281, 367, 383, 384], [945, 2634, 3584, 3814, 3839, 3840], [10395, 29919, 42058, 45538, 46043, 46079, 46080], [135135, 399342, 576673, 634052, 644097, 645070, 645119, 645120], [2027025, 6125265, 9049437, 10087453, 10295507, 10320147, 10321855, 10321919, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 7, 3], [48, 47, 38, 15], [384, 383, 367, 281, 105], [3840, 3839, 3814, 3584, 2634, 945], [46080, 46079, 46043, 45538, 42058, 29919, 10395], [645120, 645119, 645070, 644097, 634052, 576673, 399342, 135135], [10321920, 10321919, 10321855, 10320147, 10295507, 10087453, 9049437, 6125265, 2027025]] | +| AccRevTabl | [[1], [1, 2], [1, 5, 8], [1, 10, 33, 48], [1, 17, 103, 279, 384], [1, 26, 256, 1206, 2895, 3840], [1, 37, 542, 4022, 16161, 35685, 46080], [1, 50, 1023, 11068, 68447, 245778, 509985, 645120], [1, 65, 1773, 26413, 234467, 1272483, 4196655, 8294895, 10321920]] | StirlingCycB Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | -| AccRevSum | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | -| RevAccSum | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | -| DiagSum | [1, 1, 4, 19, 129, 1130, 12171, 155625] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | +| AccRevSum | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | +| AntiDiagSum | [1, 1, 4, 19, 129, 1130, 12171, 155625, 2303602] | StirlingCycB Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 12, 1035, 794640, 2324992950, 4810656157072920, 182911242997871793405, 409032268138980471585600] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 4, 23, 176, 1689, 19524, 264207, 4098240] | -| Middle | [1, 1, 4, 23, 86, 950, 3480, 57379, 208054] | -| Central | [1, 4, 86, 3480, 208054] | -| LeftSide | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 21, 231, 3465, 65835, 1514205, 40883535, 1267389585] | -| NegHalf | [1, -1, 5, -45, 585, -9945, 208845, -5221125, 151412625] | -| Bin | [1, 2, 12, 112, 1390, 21316, 387016, 8089152, 190810614] | -| InvBin | [1, 0, -4, 28, -146, 176, 13096, -332856, 6220278] | -| TransSqrs | [0, 1, 8, 68, 680, 7984, 108416, 1676800, 29146624] | -| TransNat0 | [0, 1, 6, 44, 400, 4384, 56448, 836352, 14026752] | -| TransNat1 | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1035, 794640, 2324992950, 4810656157072920, 182911242997871793405, 409032268138980471585600] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 23, 176, 1689, 19524, 264207, 4098240] | +| ColMiddle | [1, 1, 4, 23, 86, 950, 3480, 57379, 208054] | +| ColECenter | [1, 4, 86, 3480, 208054] | +| ColOCenter | [1, 23, 950, 57379] | +| ColLeft | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 8, 68, 680, 7984, 108416, 1676800, 29146624] | +| TransNat0 | [0, 1, 6, 44, 400, 4384, 56448, 836352, 14026752] | +| TransNat1 | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | StirlingCycB Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingCycB Polynomial values as columns | PolyCol7 | [1, 8, 80, 960, 13440, 215040, 3870720, 77414400, 1703116800] | | PolyCol8 | [1, 9, 99, 1287, 19305, 328185, 6235515, 130945815, 3011753745] | -# StirlingCycBInv -True +# StirlingCycB:Inv +[] -StirlingCycBInv Triangle view +StirlingCycB:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingCycBInv Triangle view | Row7 | [-1, 1093, -9219, 12411, -5075, 791, -49, 1] | | Row8 | [1, -3280, 47188, -96096, 58086, -13776, 1428, -64, 1] | -StirlingCycBInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 1, -4, 1, -1, 13, -9, 1, 1, -40, 58, -16, 1, -1, 121, -330, 170, -25, 1, 1, -364, 1771, -1520, 395, -36, 1, -1, 1093, -9219, 12411, -5075, 791, -49, 1, 1, -3280, 47188, -96096, 58086, -13776, 1428, -64, 1] | -| TriRev | [1, 1, -1, 1, -4, 1, 1, -9, 13, -1, 1, -16, 58, -40, 1, 1, -25, 170, -330, 121, -1, 1, -36, 395, -1520, 1771, -364, 1, 1, -49, 791, -5075, 12411, -9219, 1093, -1, 1, -64, 1428, -13776, 58086, -96096, 47188, -3280, 1] | -| TriAcc | [1, -1, 0, 1, -3, -2, -1, 12, 3, 4, 1, -39, 19, 3, 4, -1, 120, -210, -40, -65, -64, 1, -363, 1408, -112, 283, 247, 248, -1, 1092, -8127, 4284, -791, 0, -49, -48, 1, -3279, 43909, -52187, 5899, -7877, -6449, -6513, -6512] | -| TriRevAcc | [1, 0, -1, -2, -3, 1, 4, 3, 12, -1, 4, 3, 19, -39, 1, -64, -65, -40, -210, 120, -1, 248, 247, 283, -112, 1408, -363, 1, -48, -49, 0, -791, 4284, -8127, 1092, -1, -6512, -6513, -6449, -7877, 5899, -52187, 43909, -3279, 1] | -| TriAccRev | [1, 1, 0, 1, -3, -2, 1, -8, 5, 4, 1, -15, 43, 3, 4, 1, -24, 146, -184, -63, -64, 1, -35, 360, -1160, 611, 247, 248, 1, -48, 743, -4332, 8079, -1140, -47, -48, 1, -63, 1365, -12411, 45675, -50421, -3233, -6513, -6512] | -| TriDiag | [1, -1, 1, 1, -1, -4, 1, 13, 1, -1, -40, -9, 1, 121, 58, 1, -1, -364, -330, -16] | - -StirlingCycBInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -2, 4, 4, -64, 248, -48, -6512] | -| EvenSum | [1, -1, 2, -10, 60, -356, 2168, -14344, 106704] | -| OddSum | [0, 1, -4, 14, -56, 292, -1920, 14296, -113216] | -| AltSum | [1, -2, 6, -24, 116, -648, 4088, -28640, 219920] | -| AccSum | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | -| AccRevSum | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | -| RevAccSum | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | -| DiagSum | [1, -1, 2, -5, 15, -50, 181, -711] | - -StirlingCycBInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -4, 117, 2320, 308550, 24881416560, 3415119838107075, 10927777609869120] | -| RowGcd | [1, 1, -4, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 4, 13, 58, 330, 1771, 12411, 96096] | -| Middle | [1, -1, -4, 13, 58, -330, -1520, 12411, 58086] | -| Central | [1, -4, 58, -1520, 58086] | -| LeftSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, 27, -103, -105, 6101, -64141, 350129] | -| NegHalf | [1, 3, 13, 79, 601, 5339, 53861, 607527, 7560625] | -| Bin | [1, 0, -6, 12, 126, -1120, -308, 87080, -752314] | -| InvBin | [1, 2, 10, 68, 574, 5732, 65292, 830216, 11606854] | -| TransSqrs | [0, 1, 0, -14, 64, -44, -1504, 12776, -46080] | -| TransNat0 | [0, 1, -2, -2, 32, -124, 24, 3256, -25600] | -| TransNat1 | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | - -StirlingCycBInv Diagonals as rows +StirlingCycB:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -4, 1], [-1, 13, -9, 1], [1, -40, 58, -16, 1], [-1, 121, -330, 170, -25, 1], [1, -364, 1771, -1520, 395, -36, 1], [-1, 1093, -9219, 12411, -5075, 791, -49, 1], [1, -3280, 47188, -96096, 58086, -13776, 1428, -64, 1]] | +| RevTabl | [[1], [1, -1], [1, -4, 1], [1, -9, 13, -1], [1, -16, 58, -40, 1], [1, -25, 170, -330, 121, -1], [1, -36, 395, -1520, 1771, -364, 1], [1, -49, 791, -5075, 12411, -9219, 1093, -1], [1, -64, 1428, -13776, 58086, -96096, 47188, -3280, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -4], [1, 13, 1], [-1, -40, -9], [1, 121, 58, 1], [-1, -364, -330, -16], [1, 1093, 1771, 170, 1]] | +| AccTabl | [[1], [-1, 0], [1, -3, -2], [-1, 12, 3, 4], [1, -39, 19, 3, 4], [-1, 120, -210, -40, -65, -64], [1, -363, 1408, -112, 283, 247, 248], [-1, 1092, -8127, 4284, -791, 0, -49, -48], [1, -3279, 43909, -52187, 5899, -7877, -6449, -6513, -6512]] | +| RevAccTabl | [[1], [0, -1], [-2, -3, 1], [4, 3, 12, -1], [4, 3, 19, -39, 1], [-64, -65, -40, -210, 120, -1], [248, 247, 283, -112, 1408, -363, 1], [-48, -49, 0, -791, 4284, -8127, 1092, -1], [-6512, -6513, -6449, -7877, 5899, -52187, 43909, -3279, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -3, -2], [1, -8, 5, 4], [1, -15, 43, 3, 4], [1, -24, 146, -184, -63, -64], [1, -35, 360, -1160, 611, 247, 248], [1, -48, 743, -4332, 8079, -1140, -47, -48], [1, -63, 1365, -12411, 45675, -50421, -3233, -6513, -6512]] | + +StirlingCycB:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 4, 4, -64, 248, -48, -6512] | +| EvenSum | [1, -1, 2, -10, 60, -356, 2168, -14344, 106704] | +| OddSum | [0, 1, -4, 14, -56, 292, -1920, 14296, -113216] | +| AltSum | [1, -2, 6, -24, 116, -648, 4088, -28640, 219920] | +| AccSum | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | +| AccRevSum | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | +| AntiDiagSum | [1, -1, 2, -5, 15, -50, 181, -711, 3036] | + +StirlingCycB:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 117, 2320, 308550, 24881416560, 3415119838107075, 10927777609869120] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 13, 58, 330, 1771, 12411, 96096] | +| ColMiddle | [1, -1, -4, 13, 58, -330, -1520, 12411, 58086] | +| ColECenter | [1, -4, 58, -1520, 58086] | +| ColOCenter | [-1, 13, -330, 12411] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 0, -14, 64, -44, -1504, 12776, -46080] | +| TransNat0 | [0, 1, -2, -2, 32, -124, 24, 3256, -25600] | +| TransNat1 | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | + +StirlingCycB:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ StirlingCycBInv Diagonals as rows | DiagRow7 | [-1, -3280, -239220, -5278240, -61885450, -484962192, -2863440580, -13685763520, -55483708995]| | DiagRow8 | [1, 9841, 1205941, 38153621, 595122671, 5929706783, 43154434323, 248440887123, 1191663940038]| -StirlingCycBInv Diagonals as columns +StirlingCycB:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -188,7 +180,7 @@ StirlingCycBInv Diagonals as columns | DiagCol7 | [1, -64, 2388, -68160, 1652442, -35901888, 721488196, -13685763520, 248440887123] | | DiagCol8 | [1, -81, 3765, -132165, 3899247, -102189087, 2458702675, -55483708995, 1191663940038] | -StirlingCycBInv Polynomial values as rows +StirlingCycB:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingCycBInv Polynomial values as rows | PolyRow7 | [-1, -48, 5701, 3008, -22365, -61136, -77143, -19104, 157543] | | PolyRow8 | [1, -6512, -14575, 85264, 221185, 108176, -531887, -1679600, -2797951] | -StirlingCycBInv Polynomial values as columns +StirlingCycB:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -214,10 +206,10 @@ StirlingCycBInv Polynomial values as columns | PolyCol7 | [1, 6, 22, -8, -524, -232, 23864, -19104, -1679600] | | PolyCol8 | [1, 7, 33, 39, -703, -2745, 32609, 157543, -2797951] | -# StirlingCycBRev -True +# StirlingCycB:Rev +[] -StirlingCycBRev Triangle view +StirlingCycB:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ StirlingCycBRev Triangle view | Row7 | [1, 49, 973, 10045, 57379, 177331, 264207, 135135] | | Row8 | [1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025] | -StirlingCycBRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025] | -| TriRev | [1, 1, 1, 3, 4, 1, 15, 23, 9, 1, 105, 176, 86, 16, 1, 945, 1689, 950, 230, 25, 1, 10395, 19524, 12139, 3480, 505, 36, 1, 135135, 264207, 177331, 57379, 10045, 973, 49, 1, 2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1] | -| TriAcc | [1, 1, 2, 1, 5, 8, 1, 10, 33, 48, 1, 17, 103, 279, 384, 1, 26, 256, 1206, 2895, 3840, 1, 37, 542, 4022, 16161, 35685, 46080, 1, 50, 1023, 11068, 68447, 245778, 509985, 645120, 1, 65, 1773, 26413, 234467, 1272483, 4196655, 8294895, 10321920] | -| TriRevAcc | [1, 2, 1, 8, 5, 1, 48, 33, 10, 1, 384, 279, 103, 17, 1, 3840, 2895, 1206, 256, 26, 1, 46080, 35685, 16161, 4022, 542, 37, 1, 645120, 509985, 245778, 68447, 11068, 1023, 50, 1, 10321920, 8294895, 4196655, 1272483, 234467, 26413, 1773, 65, 1] | -| TriAccRev | [1, 1, 2, 3, 7, 8, 15, 38, 47, 48, 105, 281, 367, 383, 384, 945, 2634, 3584, 3814, 3839, 3840, 10395, 29919, 42058, 45538, 46043, 46079, 46080, 135135, 399342, 576673, 634052, 644097, 645070, 645119, 645120, 2027025, 6125265, 9049437, 10087453, 10295507, 10320147, 10321855, 10321919, 10321920] | -| TriDiag | [1, 1, 1, 1, 1, 4, 1, 9, 3, 1, 16, 23, 1, 25, 86, 15, 1, 36, 230, 176] | - -StirlingCycBRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | -| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | -| AccRevSum | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | -| RevAccSum | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | -| DiagSum | [1, 1, 2, 5, 13, 40, 127, 443] | - -StirlingCycBRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 12, 1035, 794640, 2324992950, 4810656157072920, 182911242997871793405, 409032268138980471585600] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 4, 23, 176, 1689, 19524, 264207, 4098240] | -| Middle | [1, 1, 4, 9, 86, 230, 3480, 10045, 208054] | -| Central | [1, 4, 86, 3480, 208054] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| PosHalf | [1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425] | -| NegHalf | [1, -1, -1, -3, -15, -105, -945, -10395, -135135] | -| Bin | [1, 2, 12, 112, 1390, 21316, 387016, 8089152, 190810614] | -| InvBin | [1, 0, -4, -28, -146, -176, 13096, 332856, 6220278] | -| TransSqrs | [0, 1, 16, 236, 3624, 60144, 1089920, 21578752, 465321472] | -| TransNat0 | [0, 1, 10, 100, 1136, 14816, 220032, 3679488, 68548608] | -| TransNat1 | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | - -StirlingCycBRev Diagonals as rows +StirlingCycB:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 4, 3], [1, 9, 23, 15], [1, 16, 86, 176, 105], [1, 25, 230, 950, 1689, 945], [1, 36, 505, 3480, 12139, 19524, 10395], [1, 49, 973, 10045, 57379, 177331, 264207, 135135], [1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025]] | +| RevTabl | [[1], [1, 1], [3, 4, 1], [15, 23, 9, 1], [105, 176, 86, 16, 1], [945, 1689, 950, 230, 25, 1], [10395, 19524, 12139, 3480, 505, 36, 1], [135135, 264207, 177331, 57379, 10045, 973, 49, 1], [2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 4], [1, 9, 3], [1, 16, 23], [1, 25, 86, 15], [1, 36, 230, 176], [1, 49, 505, 950, 105]] | +| AccTabl | [[1], [1, 2], [1, 5, 8], [1, 10, 33, 48], [1, 17, 103, 279, 384], [1, 26, 256, 1206, 2895, 3840], [1, 37, 542, 4022, 16161, 35685, 46080], [1, 50, 1023, 11068, 68447, 245778, 509985, 645120], [1, 65, 1773, 26413, 234467, 1272483, 4196655, 8294895, 10321920]] | +| RevAccTabl | [[1], [2, 1], [8, 5, 1], [48, 33, 10, 1], [384, 279, 103, 17, 1], [3840, 2895, 1206, 256, 26, 1], [46080, 35685, 16161, 4022, 542, 37, 1], [645120, 509985, 245778, 68447, 11068, 1023, 50, 1], [10321920, 8294895, 4196655, 1272483, 234467, 26413, 1773, 65, 1]] | +| AccRevTabl | [[1], [1, 2], [3, 7, 8], [15, 38, 47, 48], [105, 281, 367, 383, 384], [945, 2634, 3584, 3814, 3839, 3840], [10395, 29919, 42058, 45538, 46043, 46079, 46080], [135135, 399342, 576673, 634052, 644097, 645070, 645119, 645120], [2027025, 6125265, 9049437, 10087453, 10295507, 10320147, 10321855, 10321919, 10321920]] | + +StirlingCycB:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920] | +| EvenSum | [1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| OddSum | [0, 1, 4, 24, 192, 1920, 23040, 322560, 5160960] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 14, 92, 784, 8224, 102528, 1481472, 24348672] | +| AccRevSum | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | +| AntiDiagSum | [1, 1, 2, 5, 13, 40, 127, 443, 1610] | + +StirlingCycB:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1035, 794640, 2324992950, 4810656157072920, 182911242997871793405, 409032268138980471585600] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 23, 176, 1689, 19524, 264207, 4098240] | +| ColMiddle | [1, 1, 4, 9, 86, 230, 3480, 10045, 208054] | +| ColECenter | [1, 4, 86, 3480, 208054] | +| ColOCenter | [1, 9, 230, 10045] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| TransSqrs | [0, 1, 16, 236, 3624, 60144, 1089920, 21578752, 465321472] | +| TransNat0 | [0, 1, 10, 100, 1136, 14816, 220032, 3679488, 68548608] | +| TransNat1 | [1, 3, 18, 148, 1520, 18656, 266112, 4324608, 78870528] | + +StirlingCycB:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]| @@ -283,7 +271,7 @@ StirlingCycBRev Diagonals as rows | DiagRow7 | [1, 64, 2796, 106800, 3889578, 140529312, 5141534684, 192666441968, 7442156684963]| | DiagRow8 | [1, 81, 4335, 197835, 8439783, 351523887, 14632679633, 617014151325, 26569595376038]| -StirlingCycBRev Diagonals as columns +StirlingCycB:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ StirlingCycBRev Diagonals as columns | DiagCol7 | [135135, 4098240, 53809164, 444647600, 2702025590, 13137458400, 53845005500, 192666441968, 617014151325] | | DiagCol8 | [2027025, 71697105, 1094071221, 10431670821, 72578259391, 401014719391, 1854829867891, 7442156684963, 26569595376038] | -StirlingCycBRev Polynomial values as rows +StirlingCycB:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ StirlingCycBRev Polynomial values as rows | PolyRow7 | [1, 645120, 40883535, 536166400, 3493178325, 15276957696, 51611487235, 145494835200, 358709927625] | | PolyRow8 | [1, 10321920, 1267389585, 24663654400, 213083877825, 1161048784896, 4696645338385, 15422452531200, 43403901242625] | -StirlingCycBRev Polynomial values as columns +StirlingCycB:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ StirlingCycBRev Polynomial values as columns | PolyCol7 | [1, 8, 176, 6336, 316800, 20275200, 1581465600, 145494835200, 15422452531200] | | PolyCol8 | [1, 9, 225, 9225, 525825, 38385225, 3416285025, 358709927625, 43403901242625] | -# StirlingCycBInvRev -True +# StirlingCycB:Inv:Rev +[] -StirlingCycBInvRev Triangle view +StirlingCycB:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ StirlingCycBInvRev Triangle view | Row7 | [1, -49, 791, -5075, 12411, -9219, 1093, -1] | | Row8 | [1, -64, 1428, -13776, 58086, -96096, 47188, -3280, 1] | -StirlingCycBInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -4, 1, 1, -9, 13, -1, 1, -16, 58, -40, 1, 1, -25, 170, -330, 121, -1, 1, -36, 395, -1520, 1771, -364, 1, 1, -49, 791, -5075, 12411, -9219, 1093, -1, 1, -64, 1428, -13776, 58086, -96096, 47188, -3280, 1] | -| TriRev | [1, -1, 1, 1, -4, 1, -1, 13, -9, 1, 1, -40, 58, -16, 1, -1, 121, -330, 170, -25, 1, 1, -364, 1771, -1520, 395, -36, 1, -1, 1093, -9219, 12411, -5075, 791, -49, 1, 1, -3280, 47188, -96096, 58086, -13776, 1428, -64, 1] | -| TriAcc | [1, 1, 0, 1, -3, -2, 1, -8, 5, 4, 1, -15, 43, 3, 4, 1, -24, 146, -184, -63, -64, 1, -35, 360, -1160, 611, 247, 248, 1, -48, 743, -4332, 8079, -1140, -47, -48, 1, -63, 1365, -12411, 45675, -50421, -3233, -6513, -6512] | -| TriRevAcc | [1, 0, 1, -2, -3, 1, 4, 5, -8, 1, 4, 3, 43, -15, 1, -64, -63, -184, 146, -24, 1, 248, 247, 611, -1160, 360, -35, 1, -48, -47, -1140, 8079, -4332, 743, -48, 1, -6512, -6513, -3233, -50421, 45675, -12411, 1365, -63, 1] | -| TriAccRev | [1, -1, 0, 1, -3, -2, -1, 12, 3, 4, 1, -39, 19, 3, 4, -1, 120, -210, -40, -65, -64, 1, -363, 1408, -112, 283, 247, 248, -1, 1092, -8127, 4284, -791, 0, -49, -48, 1, -3279, 43909, -52187, 5899, -7877, -6449, -6513, -6512] | -| TriDiag | [1, 1, 1, -1, 1, -4, 1, -9, 1, 1, -16, 13, 1, -25, 58, -1, 1, -36, 170, -40] | - -StirlingCycBInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -2, 4, 4, -64, 248, -48, -6512] | -| EvenSum | [1, 1, 2, 14, 60, 292, 2168, 14296, 106704] | -| OddSum | [0, -1, -4, -10, -56, -356, -1920, -14344, -113216] | -| AltSum | [1, 2, 6, 24, 116, 648, 4088, 28640, 219920] | -| AccSum | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | -| AccRevSum | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | -| RevAccSum | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | -| DiagSum | [1, 1, 0, -3, -7, -2, 33, 95] | - -StirlingCycBInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -4, 117, 2320, 308550, 24881416560, 3415119838107075, 10927777609869120] | -| RowGcd | [1, 1, -4, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 4, 13, 58, 330, 1771, 12411, 96096] | -| Middle | [1, 1, -4, -9, 58, 170, -1520, -5075, 58086] | -| Central | [1, -4, 58, -1520, 58086] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| PosHalf | [1, 1, -3, -3, 41, -87, -571, 5701, -14575] | -| NegHalf | [1, -3, 13, -71, 457, -3355, 27509, -248127, 2434129] | -| Bin | [1, 0, -6, 12, 126, -1120, -308, 87080, -752314] | -| InvBin | [1, -2, 10, -68, 574, -5732, 65292, -830216, 11606854] | -| TransSqrs | [0, -1, 0, 34, -128, -404, 7136, -35160, -53248] | -| TransNat0 | [0, -1, -2, 14, -16, -196, 1464, -3592, -26496] | -| TransNat1 | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | - -StirlingCycBInvRev Diagonals as rows +StirlingCycB:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -4, 1], [1, -9, 13, -1], [1, -16, 58, -40, 1], [1, -25, 170, -330, 121, -1], [1, -36, 395, -1520, 1771, -364, 1], [1, -49, 791, -5075, 12411, -9219, 1093, -1], [1, -64, 1428, -13776, 58086, -96096, 47188, -3280, 1]] | +| RevTabl | [[1], [-1, 1], [1, -4, 1], [-1, 13, -9, 1], [1, -40, 58, -16, 1], [-1, 121, -330, 170, -25, 1], [1, -364, 1771, -1520, 395, -36, 1], [-1, 1093, -9219, 12411, -5075, 791, -49, 1], [1, -3280, 47188, -96096, 58086, -13776, 1428, -64, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -4], [1, -9, 1], [1, -16, 13], [1, -25, 58, -1], [1, -36, 170, -40], [1, -49, 395, -330, 1]] | +| AccTabl | [[1], [1, 0], [1, -3, -2], [1, -8, 5, 4], [1, -15, 43, 3, 4], [1, -24, 146, -184, -63, -64], [1, -35, 360, -1160, 611, 247, 248], [1, -48, 743, -4332, 8079, -1140, -47, -48], [1, -63, 1365, -12411, 45675, -50421, -3233, -6513, -6512]] | +| RevAccTabl | [[1], [0, 1], [-2, -3, 1], [4, 5, -8, 1], [4, 3, 43, -15, 1], [-64, -63, -184, 146, -24, 1], [248, 247, 611, -1160, 360, -35, 1], [-48, -47, -1140, 8079, -4332, 743, -48, 1], [-6512, -6513, -3233, -50421, 45675, -12411, 1365, -63, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -3, -2], [-1, 12, 3, 4], [1, -39, 19, 3, 4], [-1, 120, -210, -40, -65, -64], [1, -363, 1408, -112, 283, 247, 248], [-1, 1092, -8127, 4284, -791, 0, -49, -48], [1, -3279, 43909, -52187, 5899, -7877, -6449, -6513, -6512]] | + +StirlingCycB:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 4, 4, -64, 248, -48, -6512] | +| EvenSum | [1, 1, 2, 14, 60, 292, 2168, 14296, 106704] | +| OddSum | [0, -1, -4, -10, -56, -356, -1920, -14344, -113216] | +| AltSum | [1, 2, 6, 24, 116, 648, 4088, 28640, 219920] | +| AccSum | [1, 1, -4, 2, 36, -188, 272, 3208, -32112] | +| AccRevSum | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | +| AntiDiagSum | [1, 1, 0, -3, -7, -2, 33, 95, 18] | + +StirlingCycB:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 117, 2320, 308550, 24881416560, 3415119838107075, 10927777609869120] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 13, 58, 330, 1771, 12411, 96096] | +| ColMiddle | [1, 1, -4, -9, 58, 170, -1520, -5075, 58086] | +| ColECenter | [1, -4, 58, -1520, 58086] | +| ColOCenter | [1, -9, 170, -5075] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| TransSqrs | [0, -1, 0, 34, -128, -404, 7136, -35160, -53248] | +| TransNat0 | [0, -1, -2, 14, -16, -196, 1464, -3592, -26496] | +| TransNat1 | [1, -1, -4, 18, -12, -260, 1712, -3640, -33008] | + +StirlingCycB:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 1, -1, 1, -1, 1, -1, 1]| @@ -391,7 +375,7 @@ StirlingCycBInvRev Diagonals as rows | DiagRow7 | [1, -64, 2388, -68160, 1652442, -35901888, 721488196, -13685763520, 248440887123]| | DiagRow8 | [1, -81, 3765, -132165, 3899247, -102189087, 2458702675, -55483708995, 1191663940038]| -StirlingCycBInvRev Diagonals as columns +StirlingCycB:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ StirlingCycBInvRev Diagonals as columns | DiagCol7 | [-1, -3280, -239220, -5278240, -61885450, -484962192, -2863440580, -13685763520, -55483708995] | | DiagCol8 | [1, 9841, 1205941, 38153621, 595122671, 5929706783, 43154434323, 248440887123, 1191663940038] | -StirlingCycBInvRev Polynomial values as rows +StirlingCycB:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ StirlingCycBInvRev Polynomial values as rows | PolyRow7 | [1, -48, -64141, -570368, -2114835, -4667344, -5955233, 919584, 30625337] | | PolyRow8 | [1, -6512, 350129, 8227600, 55216705, 215770256, 610205233, 1375888144, 2590253825] | -StirlingCycBInvRev Polynomial values as columns +StirlingCycB:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/StirlingSet.md b/data/md/StirlingSet.md index 0d59287..0c9ed47 100644 --- a/data/md/StirlingSet.md +++ b/data/md/StirlingSet.md @@ -1,5 +1,5 @@ # StirlingSet -['A008277', 'A008278', 'A048993', 'A080417', 'A106800', 'A151511', 'A151512', 'A154959', 'A213735'] +['A048993', 'A008277', 'A008278', 'A080417', 'A106800', 'A151511', 'A151512', 'A154959', 'A213735'] StirlingSet Triangle view | Row | Seq | @@ -14,45 +14,41 @@ StirlingSet Triangle view | Row7 | [0, 1, 63, 301, 350, 140, 21, 1] | | Row8 | [0, 1, 127, 966, 1701, 1050, 266, 28, 1] | -StirlingSet Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1] | -| TriRev | [1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 8, 14, 15, 0, 1, 16, 41, 51, 52, 0, 1, 32, 122, 187, 202, 203, 0, 1, 64, 365, 715, 855, 876, 877, 0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140] | -| TriRevAcc | [1, 1, 0, 2, 1, 0, 5, 4, 1, 0, 15, 14, 8, 1, 0, 52, 51, 41, 16, 1, 0, 203, 202, 187, 122, 32, 1, 0, 877, 876, 855, 715, 365, 64, 1, 0, 4140, 4139, 4111, 3845, 2795, 1094, 128, 1, 0] | -| TriAccRev | [1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 7, 14, 15, 15, 1, 11, 36, 51, 52, 52, 1, 16, 81, 171, 202, 203, 203, 1, 22, 162, 512, 813, 876, 877, 877, 1, 29, 295, 1345, 3046, 4012, 4139, 4140, 4140] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 3, 0, 1, 7, 1, 0, 1, 15, 6] | +StirlingSet Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 1], [0, 1, 3, 1], [0, 1, 7, 6, 1], [0, 1, 15, 25, 10, 1], [0, 1, 31, 90, 65, 15, 1], [0, 1, 63, 301, 350, 140, 21, 1], [0, 1, 127, 966, 1701, 1050, 266, 28, 1]] | +| RevTabl | [[1], [1, 0], [1, 1, 0], [1, 3, 1, 0], [1, 6, 7, 1, 0], [1, 10, 25, 15, 1, 0], [1, 15, 65, 90, 31, 1, 0], [1, 21, 140, 350, 301, 63, 1, 0], [1, 28, 266, 1050, 1701, 966, 127, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 1], [0, 1, 3], [0, 1, 7, 1], [0, 1, 15, 6], [0, 1, 31, 25, 1]] | +| AccTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 4, 5], [0, 1, 8, 14, 15], [0, 1, 16, 41, 51, 52], [0, 1, 32, 122, 187, 202, 203], [0, 1, 64, 365, 715, 855, 876, 877], [0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140]] | +| RevAccTabl | [[1], [1, 0], [2, 1, 0], [5, 4, 1, 0], [15, 14, 8, 1, 0], [52, 51, 41, 16, 1, 0], [203, 202, 187, 122, 32, 1, 0], [877, 876, 855, 715, 365, 64, 1, 0], [4140, 4139, 4111, 3845, 2795, 1094, 128, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 5, 5], [1, 7, 14, 15, 15], [1, 11, 36, 51, 52, 52], [1, 16, 81, 171, 202, 203, 203], [1, 22, 162, 512, 813, 876, 877, 877], [1, 29, 295, 1345, 3046, 4012, 4139, 4140, 4140]] | StirlingSet Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | -| EvenSum | [1, 0, 1, 3, 8, 25, 97, 434, 2095] | -| OddSum | [0, 1, 1, 2, 7, 27, 106, 443, 2045] | -| AltSum | [1, -1, 0, 1, 1, -2, -9, -9, 50] | -| AccSum | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | -| AccRevSum | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | -| RevAccSum | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| DiagSum | [1, 0, 1, 1, 2, 4, 9, 22] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | +| EvenSum | [1, 0, 1, 3, 8, 25, 97, 434, 2095] | +| OddSum | [0, 1, 1, 2, 7, 27, 106, 443, 2045] | +| AltSum | [1, -1, 0, 1, 1, -2, -9, -9, 50] | +| AccSum | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | +| AccRevSum | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| AntiDiagSum | [1, 0, 1, 1, 2, 4, 9, 22, 58] | StirlingSet Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | -| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | -| Middle | [1, 0, 1, 1, 7, 15, 90, 301, 1701] | -| Central | [1, 1, 7, 90, 1701] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | -| NegHalf | [1, 1, -1, -1, 9, -23, -25, 583, -3087] | -| Bin | [1, 1, 3, 13, 71, 456, 3337, 27203, 243203] | -| InvBin | [1, 1, -1, -5, 15, 56, -455, -237, 16947] | -| TransSqrs | [0, 1, 5, 22, 99, 471, 2386, 12867, 73681] | -| TransNat0 | [0, 1, 3, 10, 37, 151, 674, 3263, 17007] | -| TransNat1 | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | +| ColMiddle | [1, 0, 1, 1, 7, 15, 90, 301, 1701] | +| ColECenter | [1, 1, 7, 90, 1701] | +| ColOCenter | [0, 1, 15, 301] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 5, 22, 99, 471, 2386, 12867, 73681] | +| TransNat0 | [0, 1, 3, 10, 37, 151, 674, 3263, 17007] | +| TransNat1 | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | StirlingSet Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingSet Polynomial values as columns | PolyCol7 | [1, 7, 56, 497, 4809, 50134, 558215, 6593839, 82187658] | | PolyCol8 | [1, 8, 72, 712, 7624, 87496, 1067976, 13781448, 187104200] | -# StirlingSetInv -True +# StirlingSet:Inv +[] -StirlingSetInv Triangle view +StirlingSet:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingSetInv Triangle view | Row7 | [0, 720, -1764, 1624, -735, 175, -21, 1] | | Row8 | [0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1] | -StirlingSetInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1] | -| TriRev | [1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0] | -| TriAcc | [1, 0, 1, 0, -1, 0, 0, 2, -1, 0, 0, -6, 5, -1, 0, 0, 24, -26, 9, -1, 0, 0, -120, 154, -71, 14, -1, 0, 0, 720, -1044, 580, -155, 20, -1, 0, 0, -5040, 8028, -5104, 1665, -295, 27, -1, 0] | -| TriRevAcc | [1, 1, 0, 0, -1, 0, 0, -1, 2, 0, 0, -1, 5, -6, 0, 0, -1, 9, -26, 24, 0, 0, -1, 14, -71, 154, -120, 0, 0, -1, 20, -155, 580, -1044, 720, 0, 0, -1, 27, -295, 1665, -5104, 8028, -5040, 0] | -| TriAccRev | [1, 1, 1, 1, 0, 0, 1, -2, 0, 0, 1, -5, 6, 0, 0, 1, -9, 26, -24, 0, 0, 1, -14, 71, -154, 120, 0, 0, 1, -20, 155, -580, 1044, -720, 0, 0, 1, -27, 295, -1665, 5104, -8028, 5040, 0, 0] | -| TriDiag | [1, 0, 0, 1, 0, -1, 0, 2, 1, 0, -6, -3, 0, 24, 11, 1, 0, -120, -50, -6] | - -StirlingSetInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 0, 1, -3, 12, -60, 360, -2520, 20160] | -| OddSum | [0, 1, -1, 3, -12, 60, -360, 2520, -20160] | -| AltSum | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| AccSum | [1, 1, -1, 1, -2, 6, -24, 120, -720] | -| AccRevSum | [1, 1, -1, 1, -2, 6, -24, 120, -720] | -| RevAccSum | [1, 2, 1, -1, 2, -6, 24, -120, 720] | -| DiagSum | [1, 0, 1, -1, 3, -9, 36, -176] | - -StirlingSetInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | -| Middle | [1, 0, -1, 2, 11, -50, -225, 1624, 6769] | -| Central | [1, -1, 11, -225, 6769] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 1, -1, 3, -15, 105, -945, 10395, -135135] | -| NegHalf | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| Bin | [1, 1, -1, -2, 19, -79, 76, 2640, -36945] | -| InvBin | [1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447] | -| TransSqrs | [0, 1, 3, -1, 0, 4, -28, 188, -1368] | -| TransNat0 | [0, 1, 1, -1, 2, -6, 24, -120, 720] | -| TransNat1 | [1, 2, 1, -1, 2, -6, 24, -120, 720] | - -StirlingSetInv Diagonals as rows +StirlingSet:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, -1, 1], [0, 2, -3, 1], [0, -6, 11, -6, 1], [0, 24, -50, 35, -10, 1], [0, -120, 274, -225, 85, -15, 1], [0, 720, -1764, 1624, -735, 175, -21, 1], [0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1]] | +| RevTabl | [[1], [1, 0], [1, -1, 0], [1, -3, 2, 0], [1, -6, 11, -6, 0], [1, -10, 35, -50, 24, 0], [1, -15, 85, -225, 274, -120, 0], [1, -21, 175, -735, 1624, -1764, 720, 0], [1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, -1], [0, 2, 1], [0, -6, -3], [0, 24, 11, 1], [0, -120, -50, -6], [0, 720, 274, 35, 1]] | +| AccTabl | [[1], [0, 1], [0, -1, 0], [0, 2, -1, 0], [0, -6, 5, -1, 0], [0, 24, -26, 9, -1, 0], [0, -120, 154, -71, 14, -1, 0], [0, 720, -1044, 580, -155, 20, -1, 0], [0, -5040, 8028, -5104, 1665, -295, 27, -1, 0]] | +| RevAccTabl | [[1], [1, 0], [0, -1, 0], [0, -1, 2, 0], [0, -1, 5, -6, 0], [0, -1, 9, -26, 24, 0], [0, -1, 14, -71, 154, -120, 0], [0, -1, 20, -155, 580, -1044, 720, 0], [0, -1, 27, -295, 1665, -5104, 8028, -5040, 0]] | +| AccRevTabl | [[1], [1, 1], [1, 0, 0], [1, -2, 0, 0], [1, -5, 6, 0, 0], [1, -9, 26, -24, 0, 0], [1, -14, 71, -154, 120, 0, 0], [1, -20, 155, -580, 1044, -720, 0, 0], [1, -27, 295, -1665, 5104, -8028, 5040, 0, 0]] | + +StirlingSet:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 0, 1, -3, 12, -60, 360, -2520, 20160] | +| OddSum | [0, 1, -1, 3, -12, 60, -360, 2520, -20160] | +| AltSum | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | +| AccSum | [1, 1, -1, 1, -2, 6, -24, 120, -720] | +| AccRevSum | [1, 2, 1, -1, 2, -6, 24, -120, 720] | +| AntiDiagSum | [1, 0, 1, -1, 3, -9, 36, -176, 1030] | + +StirlingSet:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | +| ColMiddle | [1, 0, -1, 2, 11, -50, -225, 1624, 6769] | +| ColECenter | [1, -1, 11, -225, 6769] | +| ColOCenter | [0, 2, -50, 1624] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 3, -1, 0, 4, -28, 188, -1368] | +| TransNat0 | [0, 1, 1, -1, 2, -6, 24, -120, 720] | +| TransNat1 | [1, 2, 1, -1, 2, -6, 24, -120, 720] | + +StirlingSet:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ StirlingSetInv Diagonals as rows | DiagRow7 | [0, -5040, -109584, -1172700, -8409500, -45995730, -206070150, -790943153, -2681453775]| | DiagRow8 | [0, 40320, 1026576, 12753576, 105258076, 657206836, 3336118786, 14409322928, 54631129553]| -StirlingSetInv Diagonals as columns +StirlingSet:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ StirlingSetInv Diagonals as columns | DiagCol7 | [1, -28, 546, -9450, 157773, -2637558, 44990231, -790943153, 14409322928] | | DiagCol8 | [1, -36, 870, -18150, 357423, -6926634, 135036473, -2681453775, 54631129553] | -StirlingSetInv Polynomial values as rows +StirlingSet:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingSetInv Polynomial values as rows | PolyRow7 | [0, 0, 0, 0, 0, 0, 0, 5040, 40320] | | PolyRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 40320] | -StirlingSetInv Polynomial values as columns +StirlingSet:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ StirlingSetInv Polynomial values as columns | PolyCol7 | [1, 7, 42, 210, 840, 2520, 5040, 5040, 0] | | PolyCol8 | [1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320] | -# StirlingSetRev -True +# StirlingSet:Rev +[] -StirlingSetRev Triangle view +StirlingSet:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ StirlingSetRev Triangle view | Row7 | [1, 21, 140, 350, 301, 63, 1, 0] | | Row8 | [1, 28, 266, 1050, 1701, 966, 127, 1, 0] | -StirlingSetRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1] | -| TriAcc | [1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 7, 14, 15, 15, 1, 11, 36, 51, 52, 52, 1, 16, 81, 171, 202, 203, 203, 1, 22, 162, 512, 813, 876, 877, 877, 1, 29, 295, 1345, 3046, 4012, 4139, 4140, 4140] | -| TriRevAcc | [1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 15, 15, 14, 7, 1, 52, 52, 51, 36, 11, 1, 203, 203, 202, 171, 81, 16, 1, 877, 877, 876, 813, 512, 162, 22, 1, 4140, 4140, 4139, 4012, 3046, 1345, 295, 29, 1] | -| TriAccRev | [1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 8, 14, 15, 0, 1, 16, 41, 51, 52, 0, 1, 32, 122, 187, 202, 203, 0, 1, 64, 365, 715, 855, 876, 877, 0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140] | -| TriDiag | [1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 6, 1, 1, 10, 7, 0, 1, 15, 25, 1] | - -StirlingSetRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | -| EvenSum | [1, 1, 1, 2, 8, 27, 97, 443, 2095] | -| OddSum | [0, 0, 1, 3, 7, 25, 106, 434, 2045] | -| AltSum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | -| AccSum | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| AccRevSum | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | -| RevAccSum | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | -| DiagSum | [1, 1, 1, 2, 4, 8, 18, 42] | - -StirlingSetRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | -| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | -| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | -| Middle | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | -| Central | [1, 1, 7, 90, 1701] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 6, 22, 94, 454, 2430, 14214, 89918] | -| NegHalf | [1, -2, 2, 2, -6, -14, 26, 178, 90] | -| Bin | [1, 1, 3, 13, 71, 456, 3337, 27203, 243203] | -| InvBin | [1, -1, -1, 5, 15, -56, -455, 237, 16947] | -| TransSqrs | [0, 0, 1, 7, 43, 261, 1606, 10158, 66529] | -| TransNat0 | [0, 0, 1, 5, 23, 109, 544, 2876, 16113] | -| TransNat1 | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | - -StirlingSetRev Diagonals as rows +StirlingSet:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, 1, 0], [1, 3, 1, 0], [1, 6, 7, 1, 0], [1, 10, 25, 15, 1, 0], [1, 15, 65, 90, 31, 1, 0], [1, 21, 140, 350, 301, 63, 1, 0], [1, 28, 266, 1050, 1701, 966, 127, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 1], [0, 1, 3, 1], [0, 1, 7, 6, 1], [0, 1, 15, 25, 10, 1], [0, 1, 31, 90, 65, 15, 1], [0, 1, 63, 301, 350, 140, 21, 1], [0, 1, 127, 966, 1701, 1050, 266, 28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, 1], [1, 3, 0], [1, 6, 1], [1, 10, 7, 0], [1, 15, 25, 1], [1, 21, 65, 15, 0]] | +| AccTabl | [[1], [1, 1], [1, 2, 2], [1, 4, 5, 5], [1, 7, 14, 15, 15], [1, 11, 36, 51, 52, 52], [1, 16, 81, 171, 202, 203, 203], [1, 22, 162, 512, 813, 876, 877, 877], [1, 29, 295, 1345, 3046, 4012, 4139, 4140, 4140]] | +| RevAccTabl | [[1], [1, 1], [2, 2, 1], [5, 5, 4, 1], [15, 15, 14, 7, 1], [52, 52, 51, 36, 11, 1], [203, 203, 202, 171, 81, 16, 1], [877, 877, 876, 813, 512, 162, 22, 1], [4140, 4140, 4139, 4012, 3046, 1345, 295, 29, 1]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 2], [0, 1, 4, 5], [0, 1, 8, 14, 15], [0, 1, 16, 41, 51, 52], [0, 1, 32, 122, 187, 202, 203], [0, 1, 64, 365, 715, 855, 876, 877], [0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140]] | + +StirlingSet:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 2, 5, 15, 52, 203, 877, 4140] | +| EvenSum | [1, 1, 1, 2, 8, 27, 97, 443, 2095] | +| OddSum | [0, 0, 1, 3, 7, 25, 106, 434, 2045] | +| AltSum | [1, 1, 0, -1, 1, 2, -9, 9, 50] | +| AccSum | [1, 2, 5, 15, 52, 203, 877, 4140, 21147] | +| AccRevSum | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | +| AntiDiagSum | [1, 1, 1, 2, 4, 8, 18, 42, 102] | + +StirlingSet:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 3, 42, 150, 36270, 270900, 9440379900] | +| RowGcd | [1, 1, 1, 3, 1, 5, 1, 7, 1] | +| RowMax | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | +| ColMiddle | [1, 1, 1, 3, 7, 25, 90, 350, 1701] | +| ColECenter | [1, 1, 7, 90, 1701] | +| ColOCenter | [1, 3, 25, 350] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 7, 43, 261, 1606, 10158, 66529] | +| TransNat0 | [0, 0, 1, 5, 23, 109, 544, 2876, 16113] | +| TransNat1 | [1, 1, 3, 10, 38, 161, 747, 3753, 20253] | + +StirlingSet:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -283,7 +271,7 @@ StirlingSetRev Diagonals as rows | DiagRow7 | [1, 28, 462, 5880, 63987, 627396, 5715424, 49329280, 408741333]| | DiagRow8 | [1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053]| -StirlingSetRev Diagonals as columns +StirlingSet:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ StirlingSetRev Diagonals as columns | DiagCol7 | [0, 1, 255, 9330, 145750, 1379400, 9321312, 49329280, 216627840] | | DiagCol8 | [0, 1, 511, 28501, 611501, 7508501, 63436373, 408741333, 2141764053] | -StirlingSetRev Polynomial values as rows +StirlingSet:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ StirlingSetRev Polynomial values as rows | PolyRow7 | [1, 877, 10299, 51193, 170389, 447981, 1007407, 2026249, 3747753] | | PolyRow8 | [1, 4140, 75905, 498118, 2032785, 6282416, 16157905, 36458010, 74565473] | -StirlingSetRev Polynomial values as columns +StirlingSet:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ StirlingSetRev Polynomial values as columns | PolyCol7 | [1, 1, 8, 71, 729, 8842, 125399, 2026249, 36458010] | | PolyCol8 | [1, 1, 9, 89, 1009, 13457, 210105, 3747753, 74565473] | -# StirlingSetInvRev -True +# StirlingSet:Inv:Rev +[] -StirlingSetInvRev Triangle view +StirlingSet:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ StirlingSetInvRev Triangle view | Row7 | [1, -21, 175, -735, 1624, -1764, 720, 0] | | Row8 | [1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0] | -StirlingSetInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0] | -| TriRev | [1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1] | -| TriAcc | [1, 1, 1, 1, 0, 0, 1, -2, 0, 0, 1, -5, 6, 0, 0, 1, -9, 26, -24, 0, 0, 1, -14, 71, -154, 120, 0, 0, 1, -20, 155, -580, 1044, -720, 0, 0, 1, -27, 295, -1665, 5104, -8028, 5040, 0, 0] | -| TriRevAcc | [1, 1, 1, 0, 0, 1, 0, 0, -2, 1, 0, 0, 6, -5, 1, 0, 0, -24, 26, -9, 1, 0, 0, 120, -154, 71, -14, 1, 0, 0, -720, 1044, -580, 155, -20, 1, 0, 0, 5040, -8028, 5104, -1665, 295, -27, 1] | -| TriAccRev | [1, 0, 1, 0, -1, 0, 0, 2, -1, 0, 0, -6, 5, -1, 0, 0, 24, -26, 9, -1, 0, 0, -120, 154, -71, 14, -1, 0, 0, 720, -1044, 580, -155, 20, -1, 0, 0, -5040, 8028, -5104, 1665, -295, 27, -1, 0] | -| TriDiag | [1, 1, 1, 0, 1, -1, 1, -3, 0, 1, -6, 2, 1, -10, 11, 0, 1, -15, 35, -6] | - -StirlingSetInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | -| EvenSum | [1, 1, 1, 3, 12, 60, 360, 2520, 20160] | -| OddSum | [0, 0, -1, -3, -12, -60, -360, -2520, -20160] | -| AltSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| AccSum | [1, 2, 1, -1, 2, -6, 24, -120, 720] | -| AccRevSum | [1, 2, 1, -1, 2, -6, 24, -120, 720] | -| RevAccSum | [1, 1, -1, 1, -2, 6, -24, 120, -720] | -| DiagSum | [1, 1, 1, 0, -2, -3, 2, 15] | - -StirlingSetInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | -| Middle | [1, 1, -1, -3, 11, 35, -225, -735, 6769] | -| Central | [1, -1, 11, -225, 6769] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 2, 0, 0, 0, 0, 0, 0] | -| NegHalf | [1, -2, 6, -24, 120, -720, 5040, -40320, 362880] | -| Bin | [1, 1, -1, -2, 19, -79, 76, 2640, -36945] | -| InvBin | [1, -1, 3, -16, 115, -1021, 10696, -128472, 1734447] | -| TransSqrs | [0, 0, -1, 5, -16, 64, -316, 1868, -12888] | -| TransNat0 | [0, 0, -1, 1, -2, 6, -24, 120, -720] | -| TransNat1 | [1, 1, -1, 1, -2, 6, -24, 120, -720] | - -StirlingSetInvRev Diagonals as rows +StirlingSet:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [1, -1, 0], [1, -3, 2, 0], [1, -6, 11, -6, 0], [1, -10, 35, -50, 24, 0], [1, -15, 85, -225, 274, -120, 0], [1, -21, 175, -735, 1624, -1764, 720, 0], [1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0]] | +| RevTabl | [[1], [0, 1], [0, -1, 1], [0, 2, -3, 1], [0, -6, 11, -6, 1], [0, 24, -50, 35, -10, 1], [0, -120, 274, -225, 85, -15, 1], [0, 720, -1764, 1624, -735, 175, -21, 1], [0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1]] | +| AntiDiag | [[1], [1], [1, 0], [1, -1], [1, -3, 0], [1, -6, 2], [1, -10, 11, 0], [1, -15, 35, -6], [1, -21, 85, -50, 0]] | +| AccTabl | [[1], [1, 1], [1, 0, 0], [1, -2, 0, 0], [1, -5, 6, 0, 0], [1, -9, 26, -24, 0, 0], [1, -14, 71, -154, 120, 0, 0], [1, -20, 155, -580, 1044, -720, 0, 0], [1, -27, 295, -1665, 5104, -8028, 5040, 0, 0]] | +| RevAccTabl | [[1], [1, 1], [0, 0, 1], [0, 0, -2, 1], [0, 0, 6, -5, 1], [0, 0, -24, 26, -9, 1], [0, 0, 120, -154, 71, -14, 1], [0, 0, -720, 1044, -580, 155, -20, 1], [0, 0, 5040, -8028, 5104, -1665, 295, -27, 1]] | +| AccRevTabl | [[1], [0, 1], [0, -1, 0], [0, 2, -1, 0], [0, -6, 5, -1, 0], [0, 24, -26, 9, -1, 0], [0, -120, 154, -71, 14, -1, 0], [0, 720, -1044, 580, -155, 20, -1, 0], [0, -5040, 8028, -5104, 1665, -295, 27, -1, 0]] | + +StirlingSet:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 0, 0, 0, 0, 0, 0, 0] | +| EvenSum | [1, 1, 1, 3, 12, 60, 360, 2520, 20160] | +| OddSum | [0, 0, -1, -3, -12, -60, -360, -2520, -20160] | +| AltSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| AccSum | [1, 2, 1, -1, 2, -6, 24, -120, 720] | +| AccRevSum | [1, 1, -1, 1, -2, 6, -24, 120, -720] | +| AntiDiagSum | [1, 1, 1, 0, -2, -3, 2, 15, 15] | + +StirlingSet:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 6, 66, 4200, 4192200, 5115600, 19083776176080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 1, 3, 11, 50, 274, 1764, 13132] | +| ColMiddle | [1, 1, -1, -3, 11, 35, -225, -735, 6769] | +| ColECenter | [1, -1, 11, -225, 6769] | +| ColOCenter | [1, -3, 35, -735] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, -1, 5, -16, 64, -316, 1868, -12888] | +| TransNat0 | [0, 0, -1, 1, -2, 6, -24, 120, -720] | +| TransNat1 | [1, 1, -1, 1, -2, 6, -24, 120, -720] | + +StirlingSet:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ StirlingSetInvRev Diagonals as rows | DiagRow7 | [1, -28, 546, -9450, 157773, -2637558, 44990231, -790943153, 14409322928]| | DiagRow8 | [1, -36, 870, -18150, 357423, -6926634, 135036473, -2681453775, 54631129553]| -StirlingSetInvRev Diagonals as columns +StirlingSet:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ StirlingSetInvRev Diagonals as columns | DiagCol7 | [0, -5040, -109584, -1172700, -8409500, -45995730, -206070150, -790943153, -2681453775] | | DiagCol8 | [0, 40320, 1026576, 12753576, 105258076, 657206836, 3336118786, 14409322928, 54631129553] | -StirlingSetInvRev Polynomial values as rows +StirlingSet:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ StirlingSetInvRev Polynomial values as rows | PolyRow7 | [1, 0, 10395, 209440, 1514205, 6664896, 21827575, 58715280, 137227545] | | PolyRow8 | [1, 0, -135135, -4188800, -40883535, -226606464, -894930575, -2818333440, -7547514975] | -StirlingSetInvRev Polynomial values as columns +StirlingSet:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/StirlingSet2.md b/data/md/StirlingSet2.md index 40bce1f..c799fc0 100644 --- a/data/md/StirlingSet2.md +++ b/data/md/StirlingSet2.md @@ -14,45 +14,41 @@ StirlingSet2 Triangle view | Row7 | [0, 1, 56, 105, 0, 0, 0, 0] | | Row8 | [0, 1, 119, 490, 105, 0, 0, 0, 0] | -StirlingSet2 Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0] | -| TriRev | [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 15, 25, 1, 0, 0, 0, 0, 0, 105, 56, 1, 0, 0, 0, 0, 0, 105, 490, 119, 1, 0] | -| TriAcc | [1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 4, 4, 0, 1, 11, 11, 11, 11, 0, 1, 26, 41, 41, 41, 41, 0, 1, 57, 162, 162, 162, 162, 162, 0, 1, 120, 610, 715, 715, 715, 715, 715] | -| TriRevAcc | [1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 4, 4, 4, 1, 0, 11, 11, 11, 11, 1, 0, 41, 41, 41, 41, 26, 1, 0, 162, 162, 162, 162, 162, 57, 1, 0, 715, 715, 715, 715, 715, 610, 120, 1, 0] | -| TriAccRev | [1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 4, 4, 0, 0, 0, 10, 11, 11, 0, 0, 0, 15, 40, 41, 41, 0, 0, 0, 0, 105, 161, 162, 162, 0, 0, 0, 0, 105, 595, 714, 715, 715] | -| TriDiag | [1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 10, 0] | +StirlingSet2 Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 0], [0, 1, 0], [0, 1, 0, 0], [0, 1, 3, 0, 0], [0, 1, 10, 0, 0, 0], [0, 1, 25, 15, 0, 0, 0], [0, 1, 56, 105, 0, 0, 0, 0], [0, 1, 119, 490, 105, 0, 0, 0, 0]] | +| RevTabl | [[1], [0, 0], [0, 1, 0], [0, 0, 1, 0], [0, 0, 3, 1, 0], [0, 0, 0, 10, 1, 0], [0, 0, 0, 15, 25, 1, 0], [0, 0, 0, 0, 105, 56, 1, 0], [0, 0, 0, 0, 105, 490, 119, 1, 0]] | +| AntiDiag | [[1], [0], [0, 0], [0, 1], [0, 1, 0], [0, 1, 0], [0, 1, 3, 0], [0, 1, 10, 0], [0, 1, 25, 0, 0]] | +| AccTabl | [[1], [0, 0], [0, 1, 1], [0, 1, 1, 1], [0, 1, 4, 4, 4], [0, 1, 11, 11, 11, 11], [0, 1, 26, 41, 41, 41, 41], [0, 1, 57, 162, 162, 162, 162, 162], [0, 1, 120, 610, 715, 715, 715, 715, 715]] | +| RevAccTabl | [[1], [0, 0], [1, 1, 0], [1, 1, 1, 0], [4, 4, 4, 1, 0], [11, 11, 11, 11, 1, 0], [41, 41, 41, 41, 26, 1, 0], [162, 162, 162, 162, 162, 57, 1, 0], [715, 715, 715, 715, 715, 610, 120, 1, 0]] | +| AccRevTabl | [[1], [0, 0], [0, 1, 1], [0, 0, 1, 1], [0, 0, 3, 4, 4], [0, 0, 0, 10, 11, 11], [0, 0, 0, 15, 40, 41, 41], [0, 0, 0, 0, 105, 161, 162, 162], [0, 0, 0, 0, 105, 595, 714, 715, 715]] | StirlingSet2 Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 1, 1, 4, 11, 41, 162, 715] | -| EvenSum | [1, 0, 0, 0, 3, 10, 25, 56, 224] | -| OddSum | [0, 0, 1, 1, 1, 1, 16, 106, 491] | -| AltSum | [1, 0, -1, -1, 2, 9, 9, -50, -267] | -| AccSum | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | -| AccRevSum | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | -| RevAccSum | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | -| DiagSum | [1, 0, 0, 1, 1, 1, 4, 11] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 1, 1, 4, 11, 41, 162, 715] | +| EvenSum | [1, 0, 0, 0, 3, 10, 25, 56, 224] | +| OddSum | [0, 0, 1, 1, 1, 1, 16, 106, 491] | +| AltSum | [1, 0, -1, -1, 2, 9, 9, -50, -267] | +| AccSum | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | +| AccRevSum | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | +| AntiDiagSum | [1, 0, 0, 1, 1, 1, 4, 11, 26] | StirlingSet2 Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 3, 10, 75, 840, 24990] | -| RowGcd | [1, 1, 1, 1, 3, 10, 5, 7, 7] | -| RowMax | [1, 0, 1, 1, 3, 10, 25, 105, 490] | -| Middle | [1, 0, 1, 1, 3, 10, 15, 105, 105] | -| Central | [1, 1, 3, 15, 105] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 0, 2, 4, 20, 96, 552, 3536, 25104] | -| NegHalf | [1, 0, -2, 4, 4, -64, 248, -48, -6512] | -| Bin | [1, 0, 2, 3, 22, 105, 681, 4858, 38130] | -| InvBin | [1, 0, -2, 3, 14, -95, 69, 2506, -16766] | -| TransSqrs | [0, 0, 1, 1, 13, 41, 236, 1170, 6567] | -| TransNat0 | [0, 0, 1, 1, 7, 21, 96, 428, 2129] | -| TransNat1 | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 3, 10, 75, 840, 24990] | +| RowGcd | [1, 1, 1, 1, 3, 10, 5, 7, 7] | +| RowMax | [1, 0, 1, 1, 3, 10, 25, 105, 490] | +| ColMiddle | [1, 0, 1, 1, 3, 10, 15, 105, 105] | +| ColECenter | [1, 1, 3, 15, 105] | +| ColOCenter | [0, 1, 10, 105] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 1, 13, 41, 236, 1170, 6567] | +| TransNat0 | [0, 0, 1, 1, 7, 21, 96, 428, 2129] | +| TransNat1 | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | StirlingSet2 Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingSet2 Polynomial values as columns | PolyCol7 | [1, 0, 7, 7, 154, 497, 6377, 38766, 426013] | | PolyCol8 | [1, 0, 8, 8, 200, 648, 9288, 57352, 688584] | -# StirlingSet2Rev -True +# StirlingSet2:Rev +[] -StirlingSet2Rev Triangle view +StirlingSet2:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingSet2Rev Triangle view | Row7 | [0, 0, 0, 0, 105, 56, 1, 0] | | Row8 | [0, 0, 0, 0, 105, 490, 119, 1, 0] | -StirlingSet2Rev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 15, 25, 1, 0, 0, 0, 0, 0, 105, 56, 1, 0, 0, 0, 0, 0, 105, 490, 119, 1, 0] | -| TriRev | [1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0] | -| TriAcc | [1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 4, 4, 0, 0, 0, 10, 11, 11, 0, 0, 0, 15, 40, 41, 41, 0, 0, 0, 0, 105, 161, 162, 162, 0, 0, 0, 0, 105, 595, 714, 715, 715] | -| TriRevAcc | [1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 4, 4, 3, 0, 0, 11, 11, 10, 0, 0, 0, 41, 41, 40, 15, 0, 0, 0, 162, 162, 161, 105, 0, 0, 0, 0, 715, 715, 714, 595, 105, 0, 0, 0, 0] | -| TriAccRev | [1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 4, 4, 0, 1, 11, 11, 11, 11, 0, 1, 26, 41, 41, 41, 41, 0, 1, 57, 162, 162, 162, 162, 162, 0, 1, 120, 610, 715, 715, 715, 715, 715] | -| TriDiag | [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 1] | - -StirlingSet2Rev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, 1, 1, 4, 11, 41, 162, 715] | -| EvenSum | [1, 0, 0, 1, 3, 1, 25, 106, 224] | -| OddSum | [0, 0, 1, 0, 1, 10, 16, 56, 491] | -| AltSum | [1, 0, -1, 1, 2, -9, 9, 50, -267] | -| AccSum | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | -| AccRevSum | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | -| RevAccSum | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | -| DiagSum | [1, 0, 0, 1, 0, 1, 3, 1] | - -StirlingSet2Rev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 1, 1, 3, 10, 75, 840, 24990] | -| RowGcd | [1, 1, 1, 1, 3, 10, 5, 7, 7] | -| RowMax | [1, 0, 1, 1, 3, 10, 25, 105, 490] | -| Middle | [1, 0, 1, 0, 3, 0, 15, 0, 105] | -| Central | [1, 1, 3, 15, 105] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 0, 2, 2, 14, 42, 222, 1066, 6078] | -| NegHalf | [1, 0, -2, -2, 10, 38, -22, -618, -1766] | -| Bin | [1, 0, 2, 3, 22, 105, 681, 4858, 38130] | -| InvBin | [1, 0, -2, -3, 14, 95, 69, -2506, -16766] | -| TransSqrs | [0, 0, 1, 4, 21, 106, 560, 3116, 18263] | -| TransNat0 | [0, 0, 1, 2, 9, 34, 150, 706, 3591] | -| TransNat1 | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | - -StirlingSet2Rev Diagonals as rows +StirlingSet2:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 0], [0, 1, 0], [0, 0, 1, 0], [0, 0, 3, 1, 0], [0, 0, 0, 10, 1, 0], [0, 0, 0, 15, 25, 1, 0], [0, 0, 0, 0, 105, 56, 1, 0], [0, 0, 0, 0, 105, 490, 119, 1, 0]] | +| RevTabl | [[1], [0, 0], [0, 1, 0], [0, 1, 0, 0], [0, 1, 3, 0, 0], [0, 1, 10, 0, 0, 0], [0, 1, 25, 15, 0, 0, 0], [0, 1, 56, 105, 0, 0, 0, 0], [0, 1, 119, 490, 105, 0, 0, 0, 0]] | +| AntiDiag | [[1], [0], [0, 0], [0, 1], [0, 0, 0], [0, 0, 1], [0, 0, 3, 0], [0, 0, 0, 1], [0, 0, 0, 10, 0]] | +| AccTabl | [[1], [0, 0], [0, 1, 1], [0, 0, 1, 1], [0, 0, 3, 4, 4], [0, 0, 0, 10, 11, 11], [0, 0, 0, 15, 40, 41, 41], [0, 0, 0, 0, 105, 161, 162, 162], [0, 0, 0, 0, 105, 595, 714, 715, 715]] | +| RevAccTabl | [[1], [0, 0], [1, 1, 0], [1, 1, 0, 0], [4, 4, 3, 0, 0], [11, 11, 10, 0, 0, 0], [41, 41, 40, 15, 0, 0, 0], [162, 162, 161, 105, 0, 0, 0, 0], [715, 715, 714, 595, 105, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [0, 0], [0, 1, 1], [0, 1, 1, 1], [0, 1, 4, 4, 4], [0, 1, 11, 11, 11, 11], [0, 1, 26, 41, 41, 41, 41], [0, 1, 57, 162, 162, 162, 162, 162], [0, 1, 120, 610, 715, 715, 715, 715, 715]] | + +StirlingSet2:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, 1, 1, 4, 11, 41, 162, 715] | +| EvenSum | [1, 0, 0, 1, 3, 1, 25, 106, 224] | +| OddSum | [0, 0, 1, 0, 1, 10, 16, 56, 491] | +| AltSum | [1, 0, -1, 1, 2, -9, 9, 50, -267] | +| AccSum | [1, 0, 2, 2, 11, 32, 137, 590, 2844] | +| AccRevSum | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | +| AntiDiagSum | [1, 0, 0, 1, 0, 1, 3, 1, 10] | + +StirlingSet2:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 1, 1, 3, 10, 75, 840, 24990] | +| RowGcd | [1, 1, 1, 1, 3, 10, 5, 7, 7] | +| RowMax | [1, 0, 1, 1, 3, 10, 25, 105, 490] | +| ColMiddle | [1, 0, 1, 0, 3, 0, 15, 0, 105] | +| ColECenter | [1, 1, 3, 15, 105] | +| ColOCenter | [0, 0, 0, 0] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 4, 21, 106, 560, 3116, 18263] | +| TransNat0 | [0, 0, 1, 2, 9, 34, 150, 706, 3591] | +| TransNat1 | [1, 0, 2, 3, 13, 45, 191, 868, 4306] | + +StirlingSet2:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ StirlingSet2Rev Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 135135, 4729725]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 2027025]| -StirlingSet2Rev Diagonals as columns +StirlingSet2:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ StirlingSet2Rev Diagonals as columns | DiagCol7 | [0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0] | | DiagCol8 | [0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025] | -StirlingSet2Rev Polynomial values as rows +StirlingSet2:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingSet2Rev Polynomial values as rows | PolyRow7 | [0, 162, 3536, 22842, 88320, 256250, 618192, 1310946, 2527232] | | PolyRow8 | [0, 715, 25104, 216513, 1032448, 3534375, 9778320, 23311309, 49778688] | -StirlingSet2Rev Polynomial values as columns +StirlingSet2:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 0, 0, 0, 0, 0, 0, 0, 0] | diff --git a/data/md/StirlingSetB.md b/data/md/StirlingSetB.md index 7f2f91e..7a3c75a 100644 --- a/data/md/StirlingSetB.md +++ b/data/md/StirlingSetB.md @@ -14,45 +14,41 @@ StirlingSetB Triangle view | Row7 | [10299, 32803, 39515, 22491, 6265, 833, 49, 1] | | Row8 | [75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1] | -StirlingSetB Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1] | -| TriRev | [1, 1, 1, 1, 4, 3, 1, 9, 19, 11, 1, 16, 70, 104, 49, 1, 25, 190, 550, 641, 257, 1, 36, 425, 2080, 4531, 4380, 1539, 1, 49, 833, 6265, 22491, 39515, 32803, 10299, 1, 64, 1484, 16016, 87206, 247072, 365324, 266768, 75905] | -| TriAcc | [1, 1, 2, 3, 7, 8, 11, 30, 39, 40, 49, 153, 223, 239, 240, 257, 898, 1448, 1638, 1663, 1664, 1539, 5919, 10450, 12530, 12955, 12991, 12992, 10299, 43102, 82617, 105108, 111373, 112206, 112255, 112256, 75905, 342673, 707997, 955069, 1042275, 1058291, 1059775, 1059839, 1059840] | -| TriRevAcc | [1, 2, 1, 8, 7, 3, 40, 39, 30, 11, 240, 239, 223, 153, 49, 1664, 1663, 1638, 1448, 898, 257, 12992, 12991, 12955, 12530, 10450, 5919, 1539, 112256, 112255, 112206, 111373, 105108, 82617, 43102, 10299, 1059840, 1059839, 1059775, 1058291, 1042275, 955069, 707997, 342673, 75905] | -| TriAccRev | [1, 1, 2, 1, 5, 8, 1, 10, 29, 40, 1, 17, 87, 191, 240, 1, 26, 216, 766, 1407, 1664, 1, 37, 462, 2542, 7073, 11453, 12992, 1, 50, 883, 7148, 29639, 69154, 101957, 112256, 1, 65, 1549, 17565, 104771, 351843, 717167, 983935, 1059840] | -| TriDiag | [1, 1, 3, 1, 11, 4, 49, 19, 1, 257, 104, 9, 1539, 641, 70, 1, 10299, 4380, 550, 16] | +StirlingSetB Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [3, 4, 1], [11, 19, 9, 1], [49, 104, 70, 16, 1], [257, 641, 550, 190, 25, 1], [1539, 4380, 4531, 2080, 425, 36, 1], [10299, 32803, 39515, 22491, 6265, 833, 49, 1], [75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1]] | +| RevTabl | [[1], [1, 1], [1, 4, 3], [1, 9, 19, 11], [1, 16, 70, 104, 49], [1, 25, 190, 550, 641, 257], [1, 36, 425, 2080, 4531, 4380, 1539], [1, 49, 833, 6265, 22491, 39515, 32803, 10299], [1, 64, 1484, 16016, 87206, 247072, 365324, 266768, 75905]] | +| AntiDiag | [[1], [1], [3, 1], [11, 4], [49, 19, 1], [257, 104, 9], [1539, 641, 70, 1], [10299, 4380, 550, 16], [75905, 32803, 4531, 190, 1]] | +| AccTabl | [[1], [1, 2], [3, 7, 8], [11, 30, 39, 40], [49, 153, 223, 239, 240], [257, 898, 1448, 1638, 1663, 1664], [1539, 5919, 10450, 12530, 12955, 12991, 12992], [10299, 43102, 82617, 105108, 111373, 112206, 112255, 112256], [75905, 342673, 707997, 955069, 1042275, 1058291, 1059775, 1059839, 1059840]] | +| RevAccTabl | [[1], [2, 1], [8, 7, 3], [40, 39, 30, 11], [240, 239, 223, 153, 49], [1664, 1663, 1638, 1448, 898, 257], [12992, 12991, 12955, 12530, 10450, 5919, 1539], [112256, 112255, 112206, 111373, 105108, 82617, 43102, 10299], [1059840, 1059839, 1059775, 1058291, 1042275, 955069, 707997, 342673, 75905]] | +| AccRevTabl | [[1], [1, 2], [1, 5, 8], [1, 10, 29, 40], [1, 17, 87, 191, 240], [1, 26, 216, 766, 1407, 1664], [1, 37, 462, 2542, 7073, 11453, 12992], [1, 50, 883, 7148, 29639, 69154, 101957, 112256], [1, 65, 1549, 17565, 104771, 351843, 717167, 983935, 1059840]] | StirlingSetB Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840] | -| EvenSum | [1, 1, 4, 20, 120, 832, 6496, 56128, 529920] | -| OddSum | [0, 1, 4, 20, 120, 832, 6496, 56128, 529920] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | -| AccRevSum | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | -| RevAccSum | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | -| DiagSum | [1, 1, 4, 15, 69, 370, 2251, 15245] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840] | +| EvenSum | [1, 1, 4, 20, 120, 832, 6496, 56128, 529920] | +| OddSum | [0, 1, 4, 20, 120, 832, 6496, 56128, 529920] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | +| AccRevSum | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | +| AntiDiagSum | [1, 1, 4, 15, 69, 370, 2251, 15245, 113430] | StirlingSetB Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 12, 1881, 50960, 1721501650, 89999024637600, 2559252346698100095, 2696411135497225545898058560] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 4, 19, 104, 641, 4531, 39515, 365324] | -| Middle | [1, 1, 4, 19, 70, 550, 2080, 22491, 87206] | -| Central | [1, 4, 70, 2080, 87206] | -| LeftSide | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 21, 183, 1929, 23691, 329565, 5095551, 86394513] | -| NegHalf | [1, -1, 5, -29, 201, -1657, 15821, -170389, 2032785] | -| Bin | [1, 2, 12, 96, 950, 10988, 143976, 2094032, 33318534] | -| InvBin | [1, 0, -4, 20, -10, -776, 7784, -25432, -416634] | -| TransSqrs | [0, 1, 8, 64, 544, 4976, 48960, 516160, 5804032] | -| TransNat0 | [0, 1, 6, 40, 296, 2416, 21568, 208832, 2176896] | -| TransNat1 | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1881, 50960, 1721501650, 89999024637600, 2559252346698100095, 2696411135497225545898058560] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 19, 104, 641, 4531, 39515, 365324] | +| ColMiddle | [1, 1, 4, 19, 70, 550, 2080, 22491, 87206] | +| ColECenter | [1, 4, 70, 2080, 87206] | +| ColOCenter | [1, 19, 550, 22491] | +| ColLeft | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 8, 64, 544, 4976, 48960, 516160, 5804032] | +| TransNat0 | [0, 1, 6, 40, 296, 2416, 21568, 208832, 2176896] | +| TransNat1 | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | StirlingSetB Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ StirlingSetB Polynomial values as columns | PolyCol7 | [1, 8, 80, 928, 12096, 173696, 2710784, 45521408, 816215040] | | PolyCol8 | [1, 9, 99, 1251, 17649, 273033, 4574115, 82216467, 1574116353] | -# StirlingSetBInv -True +# StirlingSetB:Inv +[] -StirlingSetBInv Triangle view +StirlingSetB:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ StirlingSetBInv Triangle view | Row7 | [-1, 83033, -97307, 41979, -8715, 931, -49, 1] | | Row8 | [1, -1146656, 1436532, -687232, 166054, -22176, 1652, -64, 1] | -StirlingSetBInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 1, -4, 1, -1, 17, -9, 1, 1, -96, 74, -16, 1, -1, 729, -690, 210, -25, 1, 1, -7060, 7579, -2840, 475, -36, 1, -1, 83033, -97307, 41979, -8715, 931, -49, 1, 1, -1146656, 1436532, -687232, 166054, -22176, 1652, -64, 1] | -| TriRev | [1, 1, -1, 1, -4, 1, 1, -9, 17, -1, 1, -16, 74, -96, 1, 1, -25, 210, -690, 729, -1, 1, -36, 475, -2840, 7579, -7060, 1, 1, -49, 931, -8715, 41979, -97307, 83033, -1, 1, -64, 1652, -22176, 166054, -687232, 1436532, -1146656, 1] | -| TriAcc | [1, -1, 0, 1, -3, -2, -1, 16, 7, 8, 1, -95, -21, -37, -36, -1, 728, 38, 248, 223, 224, 1, -7059, 520, -2320, -1845, -1881, -1880, -1, 83032, -14275, 27704, 18989, 19920, 19871, 19872, 1, -1146655, 289877, -397355, -231301, -253477, -251825, -251889, -251888] | -| TriRevAcc | [1, 0, -1, -2, -3, 1, 8, 7, 16, -1, -36, -37, -21, -95, 1, 224, 223, 248, 38, 728, -1, -1880, -1881, -1845, -2320, 520, -7059, 1, 19872, 19871, 19920, 18989, 27704, -14275, 83032, -1, -251888, -251889, -251825, -253477, -231301, -397355, 289877, -1146655, 1] | -| TriAccRev | [1, 1, 0, 1, -3, -2, 1, -8, 9, 8, 1, -15, 59, -37, -36, 1, -24, 186, -504, 225, 224, 1, -35, 440, -2400, 5179, -1881, -1880, 1, -48, 883, -7832, 34147, -63160, 19873, 19872, 1, -63, 1589, -20587, 145467, -541765, 894767, -251889, -251888] | -| TriDiag | [1, -1, 1, 1, -1, -4, 1, 17, 1, -1, -96, -9, 1, 729, 74, 1, -1, -7060, -690, -16] | - -StirlingSetBInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -2, 8, -36, 224, -1880, 19872, -251888] | -| EvenSum | [1, -1, 2, -10, 76, -716, 8056, -106072, 1604240] | -| OddSum | [0, 1, -4, 18, -112, 940, -9936, 125944, -1856128] | -| AltSum | [1, -2, 6, -28, 188, -1656, 17992, -232016, 3460368] | -| AccSum | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | -| AccRevSum | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | -| RevAccSum | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | -| DiagSum | [1, -1, 2, -5, 19, -106, 805, -7767] | - -StirlingSetBInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -4, 153, 3552, 5868450, 6496374713400, 2674413767201758365, 8381602311980946048] | -| RowGcd | [1, 1, -4, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 4, 17, 96, 729, 7579, 97307, 1436532] | -| Middle | [1, -1, -4, 17, 74, -690, -2840, 41979, 166054] | -| Central | [1, -4, 74, -2840, 166054] | -| LeftSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -3, 43, -487, 6903, -125483, 2805731, -74339151] | -| NegHalf | [1, 3, 13, 95, 1113, 18107, 371941, 9173271, 263542705] | -| Bin | [1, 0, -6, 24, -2, -1280, 21436, -278768, 2992326] | -| InvBin | [1, 2, 10, 80, 894, 12772, 220188, 4418864, 100793542] | -| TransSqrs | [0, 1, 0, -10, 72, -516, 4432, -46264, 573248] | -| TransNat0 | [0, 1, -2, 2, 8, -116, 1304, -16136, 227520] | -| TransNat1 | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | - -StirlingSetBInv Diagonals as rows +StirlingSetB:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -4, 1], [-1, 17, -9, 1], [1, -96, 74, -16, 1], [-1, 729, -690, 210, -25, 1], [1, -7060, 7579, -2840, 475, -36, 1], [-1, 83033, -97307, 41979, -8715, 931, -49, 1], [1, -1146656, 1436532, -687232, 166054, -22176, 1652, -64, 1]] | +| RevTabl | [[1], [1, -1], [1, -4, 1], [1, -9, 17, -1], [1, -16, 74, -96, 1], [1, -25, 210, -690, 729, -1], [1, -36, 475, -2840, 7579, -7060, 1], [1, -49, 931, -8715, 41979, -97307, 83033, -1], [1, -64, 1652, -22176, 166054, -687232, 1436532, -1146656, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -4], [1, 17, 1], [-1, -96, -9], [1, 729, 74, 1], [-1, -7060, -690, -16], [1, 83033, 7579, 210, 1]] | +| AccTabl | [[1], [-1, 0], [1, -3, -2], [-1, 16, 7, 8], [1, -95, -21, -37, -36], [-1, 728, 38, 248, 223, 224], [1, -7059, 520, -2320, -1845, -1881, -1880], [-1, 83032, -14275, 27704, 18989, 19920, 19871, 19872], [1, -1146655, 289877, -397355, -231301, -253477, -251825, -251889, -251888]] | +| RevAccTabl | [[1], [0, -1], [-2, -3, 1], [8, 7, 16, -1], [-36, -37, -21, -95, 1], [224, 223, 248, 38, 728, -1], [-1880, -1881, -1845, -2320, 520, -7059, 1], [19872, 19871, 19920, 18989, 27704, -14275, 83032, -1], [-251888, -251889, -251825, -253477, -231301, -397355, 289877, -1146655, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -3, -2], [1, -8, 9, 8], [1, -15, 59, -37, -36], [1, -24, 186, -504, 225, 224], [1, -35, 440, -2400, 5179, -1881, -1880], [1, -48, 883, -7832, 34147, -63160, 19873, 19872], [1, -63, 1589, -20587, 145467, -541765, 894767, -251889, -251888]] | + +StirlingSetB:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 8, -36, 224, -1880, 19872, -251888] | +| EvenSum | [1, -1, 2, -10, 76, -716, 8056, -106072, 1604240] | +| OddSum | [0, 1, -4, 18, -112, 940, -9936, 125944, -1856128] | +| AltSum | [1, -2, 6, -28, 188, -1656, 17992, -232016, 3460368] | +| AccSum | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | +| AccRevSum | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | +| AntiDiagSum | [1, -1, 2, -5, 19, -106, 805, -7767, 90824] | + +StirlingSetB:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 153, 3552, 5868450, 6496374713400, 2674413767201758365, 8381602311980946048] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 17, 96, 729, 7579, 97307, 1436532] | +| ColMiddle | [1, -1, -4, 17, 74, -690, -2840, 41979, 166054] | +| ColECenter | [1, -4, 74, -2840, 166054] | +| ColOCenter | [-1, 17, -690, 41979] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 0, -10, 72, -516, 4432, -46264, 573248] | +| TransNat0 | [0, 1, -2, 2, 8, -116, 1304, -16136, 227520] | +| TransNat1 | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | + +StirlingSetB:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ StirlingSetBInv Diagonals as rows | DiagRow7 | [-1, -1146656, -24010788, -248149040, -1724283770, -9167835072, -40044216212, -150231628976, -498945642195]| | DiagRow8 | [1, 18164625, 448512021, 5410685621, 43452161071, 264582272383, 1312610750451, 5551818806963, 20649462986438]| -StirlingSetBInv Diagonals as columns +StirlingSetB:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -188,7 +180,7 @@ StirlingSetBInv Diagonals as columns | DiagCol7 | [1, -64, 2724, -100080, 3485658, -120192864, 4196756564, -150231628976, 5551818806963] | | DiagCol8 | [1, -81, 4245, -187605, 7707183, -308369919, 12322357619, -498945642195, 20649462986438] | -StirlingSetBInv Polynomial values as rows +StirlingSetB:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ StirlingSetBInv Polynomial values as rows | PolyRow7 | [-1, 19872, 13, -6448, -141, 4864, 1217, -6672, -7673] | | PolyRow8 | [1, -251888, -15, 66064, 193, -37104, -2063, 40720, 17409] | -StirlingSetBInv Polynomial values as columns +StirlingSetB:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | @@ -214,10 +206,10 @@ StirlingSetBInv Polynomial values as columns | PolyCol7 | [1, 6, 22, 20, -132, 104, 904, -6672, 40720] | | PolyCol8 | [1, 7, 33, 71, -127, -441, 2593, -7673, 17409] | -# StirlingSetBRev -True +# StirlingSetB:Rev +[] -StirlingSetBRev Triangle view +StirlingSetB:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ StirlingSetBRev Triangle view | Row7 | [1, 49, 833, 6265, 22491, 39515, 32803, 10299] | | Row8 | [1, 64, 1484, 16016, 87206, 247072, 365324, 266768, 75905] | -StirlingSetBRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 4, 3, 1, 9, 19, 11, 1, 16, 70, 104, 49, 1, 25, 190, 550, 641, 257, 1, 36, 425, 2080, 4531, 4380, 1539, 1, 49, 833, 6265, 22491, 39515, 32803, 10299, 1, 64, 1484, 16016, 87206, 247072, 365324, 266768, 75905] | -| TriRev | [1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 49, 104, 70, 16, 1, 257, 641, 550, 190, 25, 1, 1539, 4380, 4531, 2080, 425, 36, 1, 10299, 32803, 39515, 22491, 6265, 833, 49, 1, 75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1] | -| TriAcc | [1, 1, 2, 1, 5, 8, 1, 10, 29, 40, 1, 17, 87, 191, 240, 1, 26, 216, 766, 1407, 1664, 1, 37, 462, 2542, 7073, 11453, 12992, 1, 50, 883, 7148, 29639, 69154, 101957, 112256, 1, 65, 1549, 17565, 104771, 351843, 717167, 983935, 1059840] | -| TriRevAcc | [1, 2, 1, 8, 5, 1, 40, 29, 10, 1, 240, 191, 87, 17, 1, 1664, 1407, 766, 216, 26, 1, 12992, 11453, 7073, 2542, 462, 37, 1, 112256, 101957, 69154, 29639, 7148, 883, 50, 1, 1059840, 983935, 717167, 351843, 104771, 17565, 1549, 65, 1] | -| TriAccRev | [1, 1, 2, 3, 7, 8, 11, 30, 39, 40, 49, 153, 223, 239, 240, 257, 898, 1448, 1638, 1663, 1664, 1539, 5919, 10450, 12530, 12955, 12991, 12992, 10299, 43102, 82617, 105108, 111373, 112206, 112255, 112256, 75905, 342673, 707997, 955069, 1042275, 1058291, 1059775, 1059839, 1059840] | -| TriDiag | [1, 1, 1, 1, 1, 4, 1, 9, 3, 1, 16, 19, 1, 25, 70, 11, 1, 36, 190, 104] | - -StirlingSetBRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840] | -| EvenSum | [1, 1, 4, 20, 120, 832, 6496, 56128, 529920] | -| OddSum | [0, 1, 4, 20, 120, 832, 6496, 56128, 529920] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | -| AccRevSum | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | -| RevAccSum | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | -| DiagSum | [1, 1, 2, 5, 13, 36, 107, 331] | - -StirlingSetBRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 12, 1881, 50960, 1721501650, 89999024637600, 2559252346698100095, 2696411135497225545898058560] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 4, 19, 104, 641, 4531, 39515, 365324] | -| Middle | [1, 1, 4, 9, 70, 190, 2080, 6265, 87206] | -| Central | [1, 4, 70, 2080, 87206] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | -| PosHalf | [1, 3, 15, 93, 681, 5691, 53079, 544053, 6058545] | -| NegHalf | [1, -1, -1, 1, 9, 23, -25, -583, -3087] | -| Bin | [1, 2, 12, 96, 950, 10988, 143976, 2094032, 33318534] | -| InvBin | [1, 0, -4, -20, -10, 776, 7784, 25432, -416634] | -| TransSqrs | [0, 1, 16, 184, 2016, 22416, 257856, 3093056, 38803456] | -| TransNat0 | [0, 1, 10, 80, 664, 5904, 56384, 576960, 6301824] | -| TransNat1 | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | - -StirlingSetBRev Diagonals as rows +StirlingSetB:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 4, 3], [1, 9, 19, 11], [1, 16, 70, 104, 49], [1, 25, 190, 550, 641, 257], [1, 36, 425, 2080, 4531, 4380, 1539], [1, 49, 833, 6265, 22491, 39515, 32803, 10299], [1, 64, 1484, 16016, 87206, 247072, 365324, 266768, 75905]] | +| RevTabl | [[1], [1, 1], [3, 4, 1], [11, 19, 9, 1], [49, 104, 70, 16, 1], [257, 641, 550, 190, 25, 1], [1539, 4380, 4531, 2080, 425, 36, 1], [10299, 32803, 39515, 22491, 6265, 833, 49, 1], [75905, 266768, 365324, 247072, 87206, 16016, 1484, 64, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 4], [1, 9, 3], [1, 16, 19], [1, 25, 70, 11], [1, 36, 190, 104], [1, 49, 425, 550, 49]] | +| AccTabl | [[1], [1, 2], [1, 5, 8], [1, 10, 29, 40], [1, 17, 87, 191, 240], [1, 26, 216, 766, 1407, 1664], [1, 37, 462, 2542, 7073, 11453, 12992], [1, 50, 883, 7148, 29639, 69154, 101957, 112256], [1, 65, 1549, 17565, 104771, 351843, 717167, 983935, 1059840]] | +| RevAccTabl | [[1], [2, 1], [8, 5, 1], [40, 29, 10, 1], [240, 191, 87, 17, 1], [1664, 1407, 766, 216, 26, 1], [12992, 11453, 7073, 2542, 462, 37, 1], [112256, 101957, 69154, 29639, 7148, 883, 50, 1], [1059840, 983935, 717167, 351843, 104771, 17565, 1549, 65, 1]] | +| AccRevTabl | [[1], [1, 2], [3, 7, 8], [11, 30, 39, 40], [49, 153, 223, 239, 240], [257, 898, 1448, 1638, 1663, 1664], [1539, 5919, 10450, 12530, 12955, 12991, 12992], [10299, 43102, 82617, 105108, 111373, 112206, 112255, 112256], [75905, 342673, 707997, 955069, 1042275, 1058291, 1059775, 1059839, 1059840]] | + +StirlingSetB:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840] | +| EvenSum | [1, 1, 4, 20, 120, 832, 6496, 56128, 529920] | +| OddSum | [0, 1, 4, 20, 120, 832, 6496, 56128, 529920] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 14, 80, 536, 4080, 34560, 321088, 3236736] | +| AccRevSum | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | +| AntiDiagSum | [1, 1, 2, 5, 13, 36, 107, 331, 1074] | + +StirlingSetB:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 12, 1881, 50960, 1721501650, 89999024637600, 2559252346698100095, 2696411135497225545898058560] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 4, 19, 104, 641, 4531, 39515, 365324] | +| ColMiddle | [1, 1, 4, 9, 70, 190, 2080, 6265, 87206] | +| ColECenter | [1, 4, 70, 2080, 87206] | +| ColOCenter | [1, 9, 190, 6265] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905] | +| TransSqrs | [0, 1, 16, 184, 2016, 22416, 257856, 3093056, 38803456] | +| TransNat0 | [0, 1, 10, 80, 664, 5904, 56384, 576960, 6301824] | +| TransNat1 | [1, 3, 18, 120, 904, 7568, 69376, 689216, 7361664] | + +StirlingSetB:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 3, 11, 49, 257, 1539, 10299, 75905]| @@ -283,7 +271,7 @@ StirlingSetBRev Diagonals as rows | DiagRow7 | [1, 64, 2460, 74400, 1963962, 47710080, 1100247148, 24567605920, 538271502483]| | DiagRow8 | [1, 81, 3855, 141735, 4477407, 128395839, 3453398377, 88980093345, 2228251766598]| -StirlingSetBRev Diagonals as columns +StirlingSetB:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ StirlingSetBRev Diagonals as columns | DiagCol7 | [10299, 266768, 3575820, 32659840, 226854430, 1273177488, 6009565276, 24567605920, 88980093345] | | DiagCol8 | [75905, 2337505, 36971461, 396255541, 3214789391, 20975230991, 114503039651, 538271502483, 2228251766598] | -StirlingSetBRev Polynomial values as rows +StirlingSetB:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ StirlingSetBRev Polynomial values as rows | PolyRow7 | [1, 112256, 5095551, 58038016, 349735445, 1455501696, 4751318131, 13061229056, 31587890121] | | PolyRow8 | [1, 1059840, 86394513, 1415303680, 11117979585, 57028471296, 221251539985, 704619629568, 1937156666625] | -StirlingSetBRev Polynomial values as columns +StirlingSetB:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ StirlingSetBRev Polynomial values as columns | PolyCol7 | [1, 8, 176, 4768, 156864, 6056576, 266289920, 13061229056, 704619629568] | | PolyCol8 | [1, 9, 225, 6921, 258561, 11340873, 566614881, 31587890121, 1937156666625] | -# StirlingSetBInvRev -True +# StirlingSetB:Inv:Rev +[] -StirlingSetBInvRev Triangle view +StirlingSetB:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ StirlingSetBInvRev Triangle view | Row7 | [1, -49, 931, -8715, 41979, -97307, 83033, -1] | | Row8 | [1, -64, 1652, -22176, 166054, -687232, 1436532, -1146656, 1] | -StirlingSetBInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -4, 1, 1, -9, 17, -1, 1, -16, 74, -96, 1, 1, -25, 210, -690, 729, -1, 1, -36, 475, -2840, 7579, -7060, 1, 1, -49, 931, -8715, 41979, -97307, 83033, -1, 1, -64, 1652, -22176, 166054, -687232, 1436532, -1146656, 1] | -| TriRev | [1, -1, 1, 1, -4, 1, -1, 17, -9, 1, 1, -96, 74, -16, 1, -1, 729, -690, 210, -25, 1, 1, -7060, 7579, -2840, 475, -36, 1, -1, 83033, -97307, 41979, -8715, 931, -49, 1, 1, -1146656, 1436532, -687232, 166054, -22176, 1652, -64, 1] | -| TriAcc | [1, 1, 0, 1, -3, -2, 1, -8, 9, 8, 1, -15, 59, -37, -36, 1, -24, 186, -504, 225, 224, 1, -35, 440, -2400, 5179, -1881, -1880, 1, -48, 883, -7832, 34147, -63160, 19873, 19872, 1, -63, 1589, -20587, 145467, -541765, 894767, -251889, -251888] | -| TriRevAcc | [1, 0, 1, -2, -3, 1, 8, 9, -8, 1, -36, -37, 59, -15, 1, 224, 225, -504, 186, -24, 1, -1880, -1881, 5179, -2400, 440, -35, 1, 19872, 19873, -63160, 34147, -7832, 883, -48, 1, -251888, -251889, 894767, -541765, 145467, -20587, 1589, -63, 1] | -| TriAccRev | [1, -1, 0, 1, -3, -2, -1, 16, 7, 8, 1, -95, -21, -37, -36, -1, 728, 38, 248, 223, 224, 1, -7059, 520, -2320, -1845, -1881, -1880, -1, 83032, -14275, 27704, 18989, 19920, 19871, 19872, 1, -1146655, 289877, -397355, -231301, -253477, -251825, -251889, -251888] | -| TriDiag | [1, 1, 1, -1, 1, -4, 1, -9, 1, 1, -16, 17, 1, -25, 74, -1, 1, -36, 210, -96] | - -StirlingSetBInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -2, 8, -36, 224, -1880, 19872, -251888] | -| EvenSum | [1, 1, 2, 18, 76, 940, 8056, 125944, 1604240] | -| OddSum | [0, -1, -4, -10, -112, -716, -9936, -106072, -1856128] | -| AltSum | [1, 2, 6, 28, 188, 1656, 17992, 232016, 3460368] | -| AccSum | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | -| AccRevSum | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | -| RevAccSum | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | -| DiagSum | [1, 1, 0, -3, -7, 2, 49, 79] | - -StirlingSetBInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -4, 153, 3552, 5868450, 6496374713400, 2674413767201758365, 8381602311980946048] | -| RowGcd | [1, 1, -4, 1, 2, 1, 1, 1, 2] | -| RowMax | [1, 1, 4, 17, 96, 729, 7579, 97307, 1436532] | -| Middle | [1, 1, -4, -9, 74, 210, -2840, -8715, 166054] | -| Central | [1, -4, 74, -2840, 166054] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 1, -1, 1, -1, 1, -1, 1] | -| PosHalf | [1, 1, -3, 5, -7, 9, -11, 13, -15] | -| NegHalf | [1, -3, 13, -79, 633, -6331, 75973, -1063623, 17017969] | -| Bin | [1, 0, -6, 24, -2, -1280, 21436, -278768, 2992326] | -| InvBin | [1, -2, 10, -80, 894, -12772, 220188, -4418864, 100793542] | -| TransSqrs | [0, -1, 0, 50, -568, 6244, -78896, 1153368, -19187904] | -| TransNat0 | [0, -1, -2, 22, -152, 1236, -12584, 155240, -2242624] | -| TransNat1 | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | - -StirlingSetBInvRev Diagonals as rows +StirlingSetB:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -4, 1], [1, -9, 17, -1], [1, -16, 74, -96, 1], [1, -25, 210, -690, 729, -1], [1, -36, 475, -2840, 7579, -7060, 1], [1, -49, 931, -8715, 41979, -97307, 83033, -1], [1, -64, 1652, -22176, 166054, -687232, 1436532, -1146656, 1]] | +| RevTabl | [[1], [-1, 1], [1, -4, 1], [-1, 17, -9, 1], [1, -96, 74, -16, 1], [-1, 729, -690, 210, -25, 1], [1, -7060, 7579, -2840, 475, -36, 1], [-1, 83033, -97307, 41979, -8715, 931, -49, 1], [1, -1146656, 1436532, -687232, 166054, -22176, 1652, -64, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -4], [1, -9, 1], [1, -16, 17], [1, -25, 74, -1], [1, -36, 210, -96], [1, -49, 475, -690, 1]] | +| AccTabl | [[1], [1, 0], [1, -3, -2], [1, -8, 9, 8], [1, -15, 59, -37, -36], [1, -24, 186, -504, 225, 224], [1, -35, 440, -2400, 5179, -1881, -1880], [1, -48, 883, -7832, 34147, -63160, 19873, 19872], [1, -63, 1589, -20587, 145467, -541765, 894767, -251889, -251888]] | +| RevAccTabl | [[1], [0, 1], [-2, -3, 1], [8, 9, -8, 1], [-36, -37, 59, -15, 1], [224, 225, -504, 186, -24, 1], [-1880, -1881, 5179, -2400, 440, -35, 1], [19872, 19873, -63160, 34147, -7832, 883, -48, 1], [-251888, -251889, 894767, -541765, 145467, -20587, 1589, -63, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -3, -2], [-1, 16, 7, 8], [1, -95, -21, -37, -36], [-1, 728, 38, 248, 223, 224], [1, -7059, 520, -2320, -1845, -1881, -1880], [-1, 83032, -14275, 27704, 18989, 19920, 19871, 19872], [1, -1146655, 289877, -397355, -231301, -253477, -251825, -251889, -251888]] | + +StirlingSetB:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -2, 8, -36, 224, -1880, 19872, -251888] | +| EvenSum | [1, 1, 2, 18, 76, 940, 8056, 125944, 1604240] | +| OddSum | [0, -1, -4, -10, -112, -716, -9936, -106072, -1856128] | +| AltSum | [1, 2, 6, 28, 188, 1656, 17992, 232016, 3460368] | +| AccSum | [1, 1, -4, 10, -28, 108, -576, 3736, -24368] | +| AccRevSum | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | +| AntiDiagSum | [1, 1, 0, -3, -7, 2, 49, 79, -262] | + +StirlingSetB:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 4, 153, 3552, 5868450, 6496374713400, 2674413767201758365, 8381602311980946048] | +| RowGcd | [1, 1, 4, 1, 2, 1, 1, 1, 2] | +| RowMax | [1, 1, 4, 17, 96, 729, 7579, 97307, 1436532] | +| ColMiddle | [1, 1, -4, -9, 74, 210, -2840, -8715, 166054] | +| ColECenter | [1, -4, 74, -2840, 166054] | +| ColOCenter | [1, -9, 210, -8715] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| TransSqrs | [0, -1, 0, 50, -568, 6244, -78896, 1153368, -19187904] | +| TransNat0 | [0, -1, -2, 22, -152, 1236, -12584, 155240, -2242624] | +| TransNat1 | [1, -1, -4, 30, -188, 1460, -14464, 175112, -2494512] | + +StirlingSetB:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 1, -1, 1, -1, 1, -1, 1]| @@ -391,7 +375,7 @@ StirlingSetBInvRev Diagonals as rows | DiagRow7 | [1, -64, 2724, -100080, 3485658, -120192864, 4196756564, -150231628976, 5551818806963]| | DiagRow8 | [1, -81, 4245, -187605, 7707183, -308369919, 12322357619, -498945642195, 20649462986438]| -StirlingSetBInvRev Diagonals as columns +StirlingSetB:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ StirlingSetBInvRev Diagonals as columns | DiagCol7 | [-1, -1146656, -24010788, -248149040, -1724283770, -9167835072, -40044216212, -150231628976, -498945642195] | | DiagCol8 | [1, 18164625, 448512021, 5410685621, 43452161071, 264582272383, 1312610750451, 5551818806963, 20649462986438] | -StirlingSetBInvRev Polynomial values as rows +StirlingSetB:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ StirlingSetBInvRev Polynomial values as rows | PolyRow7 | [1, 19872, 2805731, 40056496, 250647981, 1018398656, 3169604047, 8230334736, 18743492921] | | PolyRow8 | [1, -251888, -74339151, -1614629360, -13565320127, -69182844144, -259099218383, -786467342576, -2049967211775] | -StirlingSetBInvRev Polynomial values as columns +StirlingSetB:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/Sylvester.md b/data/md/Sylvester.md index 5e840f7..8879ad2 100644 --- a/data/md/Sylvester.md +++ b/data/md/Sylvester.md @@ -14,45 +14,41 @@ Sylvester Triangle view | Row7 | [0, 720, 2604, 4046, 3570, 1960, 672, 128] | | Row8 | [0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256] | -Sylvester Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 2, 0, 1, 4, 0, 2, 6, 8, 0, 6, 19, 24, 16, 0, 24, 80, 110, 80, 32, 0, 120, 418, 615, 500, 240, 64, 0, 720, 2604, 4046, 3570, 1960, 672, 128, 0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256] | -| TriRev | [1, 2, 0, 4, 1, 0, 8, 6, 2, 0, 16, 24, 19, 6, 0, 32, 80, 110, 80, 24, 0, 64, 240, 500, 615, 418, 120, 0, 128, 672, 1960, 3570, 4046, 2604, 720, 0, 256, 1792, 6944, 17360, 28777, 30604, 18828, 5040, 0] | -| TriAcc | [1, 0, 2, 0, 1, 5, 0, 2, 8, 16, 0, 6, 25, 49, 65, 0, 24, 104, 214, 294, 326, 0, 120, 538, 1153, 1653, 1893, 1957, 0, 720, 3324, 7370, 10940, 12900, 13572, 13700, 0, 5040, 23868, 54472, 83249, 100609, 107553, 109345, 109601] | -| TriRevAcc | [1, 2, 0, 5, 1, 0, 16, 8, 2, 0, 65, 49, 25, 6, 0, 326, 294, 214, 104, 24, 0, 1957, 1893, 1653, 1153, 538, 120, 0, 13700, 13572, 12900, 10940, 7370, 3324, 720, 0, 109601, 109345, 107553, 100609, 83249, 54472, 23868, 5040, 0] | -| TriAccRev | [1, 2, 2, 4, 5, 5, 8, 14, 16, 16, 16, 40, 59, 65, 65, 32, 112, 222, 302, 326, 326, 64, 304, 804, 1419, 1837, 1957, 1957, 128, 800, 2760, 6330, 10376, 12980, 13700, 13700, 256, 2048, 8992, 26352, 55129, 85733, 104561, 109601, 109601] | -| TriDiag | [1, 0, 0, 2, 0, 1, 0, 2, 4, 0, 6, 6, 0, 24, 19, 8, 0, 120, 80, 24] | +Sylvester Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 2], [0, 1, 4], [0, 2, 6, 8], [0, 6, 19, 24, 16], [0, 24, 80, 110, 80, 32], [0, 120, 418, 615, 500, 240, 64], [0, 720, 2604, 4046, 3570, 1960, 672, 128], [0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256]] | +| RevTabl | [[1], [2, 0], [4, 1, 0], [8, 6, 2, 0], [16, 24, 19, 6, 0], [32, 80, 110, 80, 24, 0], [64, 240, 500, 615, 418, 120, 0], [128, 672, 1960, 3570, 4046, 2604, 720, 0], [256, 1792, 6944, 17360, 28777, 30604, 18828, 5040, 0]] | +| AntiDiag | [[1], [0], [0, 2], [0, 1], [0, 2, 4], [0, 6, 6], [0, 24, 19, 8], [0, 120, 80, 24], [0, 720, 418, 110, 16]] | +| AccTabl | [[1], [0, 2], [0, 1, 5], [0, 2, 8, 16], [0, 6, 25, 49, 65], [0, 24, 104, 214, 294, 326], [0, 120, 538, 1153, 1653, 1893, 1957], [0, 720, 3324, 7370, 10940, 12900, 13572, 13700], [0, 5040, 23868, 54472, 83249, 100609, 107553, 109345, 109601]] | +| RevAccTabl | [[1], [2, 0], [5, 1, 0], [16, 8, 2, 0], [65, 49, 25, 6, 0], [326, 294, 214, 104, 24, 0], [1957, 1893, 1653, 1153, 538, 120, 0], [13700, 13572, 12900, 10940, 7370, 3324, 720, 0], [109601, 109345, 107553, 100609, 83249, 54472, 23868, 5040, 0]] | +| AccRevTabl | [[1], [2, 2], [4, 5, 5], [8, 14, 16, 16], [16, 40, 59, 65, 65], [32, 112, 222, 302, 326, 326], [64, 304, 804, 1419, 1837, 1957, 1957], [128, 800, 2760, 6330, 10376, 12980, 13700, 13700], [256, 2048, 8992, 26352, 55129, 85733, 104561, 109601, 109601]] | Sylvester Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 0, 4, 6, 35, 160, 982, 6846, 54805] | -| OddSum | [0, 2, 1, 10, 30, 166, 975, 6854, 54796] | -| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | -| AccSum | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | -| AccRevSum | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | -| RevAccSum | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | -| DiagSum | [1, 0, 2, 1, 6, 12, 51, 224] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 0, 4, 6, 35, 160, 982, 6846, 54805] | +| OddSum | [0, 2, 1, 10, 30, 166, 975, 6854, 54796] | +| AltSum | [1, -2, 3, -4, 5, -6, 7, -8, 9] | +| AccSum | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | +| AccRevSum | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | +| AntiDiagSum | [1, 0, 2, 1, 6, 12, 51, 224, 1264] | Sylvester Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 24, 912, 5280, 205656000, 2528588160, 5874643821231360] | -| RowGcd | [1, 2, 4, 2, 1, 2, 1, 2, 1] | -| RowMax | [1, 2, 4, 8, 24, 110, 615, 4046, 30604] | -| Middle | [1, 0, 1, 2, 19, 80, 615, 4046, 28777] | -| Central | [1, 1, 19, 615, 28777] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| PosHalf | [1, 2, 6, 28, 188, 1656, 17992, 232016, 3460368] | -| NegHalf | [1, 2, 2, 4, -4, 56, -488, 5552, -73456] | -| Bin | [1, 2, 6, 32, 250, 2452, 28294, 372276, 5476902] | -| InvBin | [1, 2, 2, -4, 10, 52, -626, 3600, -4378] | -| TransSqrs | [0, 2, 17, 98, 554, 3414, 23631, 184134, 1604396] | -| TransNat0 | [0, 2, 9, 38, 180, 994, 6385, 47074, 392672] | -| TransNat1 | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 24, 912, 5280, 205656000, 2528588160, 5874643821231360] | +| RowGcd | [1, 2, 4, 2, 1, 2, 1, 2, 1] | +| RowMax | [1, 2, 4, 8, 24, 110, 615, 4046, 30604] | +| ColMiddle | [1, 0, 1, 2, 19, 80, 615, 4046, 28777] | +| ColECenter | [1, 1, 19, 615, 28777] | +| ColOCenter | [0, 2, 80, 4046] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| TransSqrs | [0, 2, 17, 98, 554, 3414, 23631, 184134, 1604396] | +| TransNat0 | [0, 2, 9, 38, 180, 994, 6385, 47074, 392672] | +| TransNat1 | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | Sylvester Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Sylvester Polynomial values as columns | PolyCol7 | [1, 14, 203, 3052, 47621, 771722, 12995983, 227507336, 4140850889] | | PolyCol8 | [1, 16, 264, 4496, 79088, 1437888, 27032128, 525688192, 10577029248] | -# SylvesterRev -True +# Sylvester:Rev +[] -SylvesterRev Triangle view +Sylvester:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SylvesterRev Triangle view | Row7 | [128, 672, 1960, 3570, 4046, 2604, 720, 0] | | Row8 | [256, 1792, 6944, 17360, 28777, 30604, 18828, 5040, 0] | -SylvesterRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 2, 0, 4, 1, 0, 8, 6, 2, 0, 16, 24, 19, 6, 0, 32, 80, 110, 80, 24, 0, 64, 240, 500, 615, 418, 120, 0, 128, 672, 1960, 3570, 4046, 2604, 720, 0, 256, 1792, 6944, 17360, 28777, 30604, 18828, 5040, 0] | -| TriRev | [1, 0, 2, 0, 1, 4, 0, 2, 6, 8, 0, 6, 19, 24, 16, 0, 24, 80, 110, 80, 32, 0, 120, 418, 615, 500, 240, 64, 0, 720, 2604, 4046, 3570, 1960, 672, 128, 0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256] | -| TriAcc | [1, 2, 2, 4, 5, 5, 8, 14, 16, 16, 16, 40, 59, 65, 65, 32, 112, 222, 302, 326, 326, 64, 304, 804, 1419, 1837, 1957, 1957, 128, 800, 2760, 6330, 10376, 12980, 13700, 13700, 256, 2048, 8992, 26352, 55129, 85733, 104561, 109601, 109601] | -| TriRevAcc | [1, 2, 2, 5, 5, 4, 16, 16, 14, 8, 65, 65, 59, 40, 16, 326, 326, 302, 222, 112, 32, 1957, 1957, 1837, 1419, 804, 304, 64, 13700, 13700, 12980, 10376, 6330, 2760, 800, 128, 109601, 109601, 104561, 85733, 55129, 26352, 8992, 2048, 256] | -| TriAccRev | [1, 0, 2, 0, 1, 5, 0, 2, 8, 16, 0, 6, 25, 49, 65, 0, 24, 104, 214, 294, 326, 0, 120, 538, 1153, 1653, 1893, 1957, 0, 720, 3324, 7370, 10940, 12900, 13572, 13700, 0, 5040, 23868, 54472, 83249, 100609, 107553, 109345, 109601] | -| TriDiag | [1, 2, 4, 0, 8, 1, 16, 6, 0, 32, 24, 2, 64, 80, 19, 0, 128, 240, 110, 6] | - -SylvesterRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | -| EvenSum | [1, 2, 4, 10, 35, 166, 982, 6854, 54805] | -| OddSum | [0, 0, 1, 6, 30, 160, 975, 6846, 54796] | -| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | -| AccSum | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | -| AccRevSum | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | -| RevAccSum | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | -| DiagSum | [1, 2, 4, 9, 22, 58, 163, 484] | - -SylvesterRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 2, 4, 24, 912, 5280, 205656000, 2528588160, 5874643821231360] | -| RowGcd | [1, 2, 4, 2, 1, 2, 1, 2, 1] | -| RowMax | [1, 2, 4, 8, 24, 110, 615, 4046, 30604] | -| Middle | [1, 2, 1, 6, 19, 110, 615, 3570, 28777] | -| Central | [1, 1, 19, 615, 28777] | -| LeftSide | [1, 2, 4, 8, 16, 32, 64, 128, 256] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 4, 18, 92, 536, 3552, 26608, 223456, 2085504] | -| NegHalf | [1, -4, 14, -44, 128, -352, 928, -2368, 5888] | -| Bin | [1, 2, 6, 32, 250, 2452, 28294, 372276, 5476902] | -| InvBin | [1, -2, 2, 4, 10, -52, -626, -3600, -4378] | -| TransSqrs | [0, 0, 1, 14, 154, 1624, 17463, 196398, 2336108] | -| TransNat0 | [0, 0, 1, 10, 80, 636, 5357, 48826, 484136] | -| TransNat1 | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | - -SylvesterRev Diagonals as rows +Sylvester:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [2, 0], [4, 1, 0], [8, 6, 2, 0], [16, 24, 19, 6, 0], [32, 80, 110, 80, 24, 0], [64, 240, 500, 615, 418, 120, 0], [128, 672, 1960, 3570, 4046, 2604, 720, 0], [256, 1792, 6944, 17360, 28777, 30604, 18828, 5040, 0]] | +| RevTabl | [[1], [0, 2], [0, 1, 4], [0, 2, 6, 8], [0, 6, 19, 24, 16], [0, 24, 80, 110, 80, 32], [0, 120, 418, 615, 500, 240, 64], [0, 720, 2604, 4046, 3570, 1960, 672, 128], [0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256]] | +| AntiDiag | [[1], [2], [4, 0], [8, 1], [16, 6, 0], [32, 24, 2], [64, 80, 19, 0], [128, 240, 110, 6], [256, 672, 500, 80, 0]] | +| AccTabl | [[1], [2, 2], [4, 5, 5], [8, 14, 16, 16], [16, 40, 59, 65, 65], [32, 112, 222, 302, 326, 326], [64, 304, 804, 1419, 1837, 1957, 1957], [128, 800, 2760, 6330, 10376, 12980, 13700, 13700], [256, 2048, 8992, 26352, 55129, 85733, 104561, 109601, 109601]] | +| RevAccTabl | [[1], [2, 2], [5, 5, 4], [16, 16, 14, 8], [65, 65, 59, 40, 16], [326, 326, 302, 222, 112, 32], [1957, 1957, 1837, 1419, 804, 304, 64], [13700, 13700, 12980, 10376, 6330, 2760, 800, 128], [109601, 109601, 104561, 85733, 55129, 26352, 8992, 2048, 256]] | +| AccRevTabl | [[1], [0, 2], [0, 1, 5], [0, 2, 8, 16], [0, 6, 25, 49, 65], [0, 24, 104, 214, 294, 326], [0, 120, 538, 1153, 1653, 1893, 1957], [0, 720, 3324, 7370, 10940, 12900, 13572, 13700], [0, 5040, 23868, 54472, 83249, 100609, 107553, 109345, 109601]] | + +Sylvester:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 5, 16, 65, 326, 1957, 13700, 109601] | +| EvenSum | [1, 2, 4, 10, 35, 166, 982, 6854, 54805] | +| OddSum | [0, 0, 1, 6, 30, 160, 975, 6846, 54796] | +| AltSum | [1, 2, 3, 4, 5, 6, 7, 8, 9] | +| AccSum | [1, 4, 14, 54, 245, 1320, 8342, 60774, 502273] | +| AccRevSum | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | +| AntiDiagSum | [1, 2, 4, 9, 22, 58, 163, 484, 1508] | + +Sylvester:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 2, 4, 24, 912, 5280, 205656000, 2528588160, 5874643821231360] | +| RowGcd | [1, 2, 4, 2, 1, 2, 1, 2, 1] | +| RowMax | [1, 2, 4, 8, 24, 110, 615, 4046, 30604] | +| ColMiddle | [1, 2, 1, 6, 19, 110, 615, 3570, 28777] | +| ColECenter | [1, 1, 19, 615, 28777] | +| ColOCenter | [2, 6, 110, 3570] | +| ColLeft | [1, 2, 4, 8, 16, 32, 64, 128, 256] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 14, 154, 1624, 17463, 196398, 2336108] | +| TransNat0 | [0, 0, 1, 10, 80, 636, 5357, 48826, 484136] | +| TransNat1 | [1, 2, 6, 26, 145, 962, 7314, 62526, 593737] | + +Sylvester:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ SylvesterRev Diagonals as rows | DiagRow7 | [128, 1792, 22848, 292320, 3878952, 54048456, 794409902, 12330989671, 202043487646]| | DiagRow8 | [256, 4608, 71040, 1066560, 16274544, 256844016, 4225162084, 72692509890, 1309509778577]| -SylvesterRev Diagonals as columns +Sylvester:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | @@ -188,7 +180,7 @@ SylvesterRev Diagonals as columns | DiagCol7 | [0, 5040, 154944, 2495340, 28167920, 250911210, 1880186880, 12330989671, 72692509890] | | DiagCol8 | [0, 40320, 1429776, 26263512, 334925404, 3342770288, 27866124286, 202043487646, 1309509778577] | -SylvesterRev Polynomial values as rows +Sylvester:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SylvesterRev Polynomial values as rows | PolyRow7 | [128, 13700, 232016, 1601552, 6914048, 22414988, 59930480, 139512536, 292602752] | | PolyRow8 | [256, 109601, 3460368, 35052649, 199629824, 803913441, 2568599056, 6955514633, 16635343104] | -SylvesterRev Polynomial values as columns +Sylvester:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 2, 4, 8, 16, 32, 64, 128, 256] | diff --git a/data/md/SymPoly.md b/data/md/SymPoly.md index 74fb49d..0c7ffbf 100644 --- a/data/md/SymPoly.md +++ b/data/md/SymPoly.md @@ -1,5 +1,5 @@ # SymPoly -['A093905', 'A105954', 'A165674', 'A165675'] +['A165675', 'A093905', 'A105954', 'A165674'] SymPoly Triangle view | Row | Seq | @@ -14,45 +14,41 @@ SymPoly Triangle view | Row7 | [5040, 13068, 8028, 2754, 638, 107, 13, 1] | | Row8 | [40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1] | -SymPoly Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 3, 1, 6, 11, 5, 1, 24, 50, 26, 7, 1, 120, 274, 154, 47, 9, 1, 720, 1764, 1044, 342, 74, 11, 1, 5040, 13068, 8028, 2754, 638, 107, 13, 1, 40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1] | -| TriRev | [1, 1, 1, 1, 3, 2, 1, 5, 11, 6, 1, 7, 26, 50, 24, 1, 9, 47, 154, 274, 120, 1, 11, 74, 342, 1044, 1764, 720, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320] | -| TriAcc | [1, 1, 2, 2, 5, 6, 6, 17, 22, 23, 24, 74, 100, 107, 108, 120, 394, 548, 595, 604, 605, 720, 2484, 3528, 3870, 3944, 3955, 3956, 5040, 18108, 26136, 28890, 29528, 29635, 29648, 29649, 40320, 149904, 219168, 243720, 249664, 250730, 250876, 250891, 250892] | -| TriRevAcc | [1, 2, 1, 6, 5, 2, 23, 22, 17, 6, 108, 107, 100, 74, 24, 605, 604, 595, 548, 394, 120, 3956, 3955, 3944, 3870, 3528, 2484, 720, 29649, 29648, 29635, 29528, 28890, 26136, 18108, 5040, 250892, 250891, 250876, 250730, 249664, 243720, 219168, 149904, 40320] | -| TriAccRev | [1, 1, 2, 1, 4, 6, 1, 6, 17, 23, 1, 8, 34, 84, 108, 1, 10, 57, 211, 485, 605, 1, 12, 86, 428, 1472, 3236, 3956, 1, 14, 121, 759, 3513, 11541, 24609, 29649, 1, 16, 162, 1228, 7172, 31724, 100988, 210572, 250892] | -| TriDiag | [1, 1, 2, 1, 6, 3, 24, 11, 1, 120, 50, 5, 720, 274, 26, 1, 5040, 1764, 154, 7] | +SymPoly Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 3, 1], [6, 11, 5, 1], [24, 50, 26, 7, 1], [120, 274, 154, 47, 9, 1], [720, 1764, 1044, 342, 74, 11, 1], [5040, 13068, 8028, 2754, 638, 107, 13, 1], [40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 2], [1, 5, 11, 6], [1, 7, 26, 50, 24], [1, 9, 47, 154, 274, 120], [1, 11, 74, 342, 1044, 1764, 720], [1, 13, 107, 638, 2754, 8028, 13068, 5040], [1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 3], [24, 11, 1], [120, 50, 5], [720, 274, 26, 1], [5040, 1764, 154, 7], [40320, 13068, 1044, 47, 1]] | +| AccTabl | [[1], [1, 2], [2, 5, 6], [6, 17, 22, 23], [24, 74, 100, 107, 108], [120, 394, 548, 595, 604, 605], [720, 2484, 3528, 3870, 3944, 3955, 3956], [5040, 18108, 26136, 28890, 29528, 29635, 29648, 29649], [40320, 149904, 219168, 243720, 249664, 250730, 250876, 250891, 250892]] | +| RevAccTabl | [[1], [2, 1], [6, 5, 2], [23, 22, 17, 6], [108, 107, 100, 74, 24], [605, 604, 595, 548, 394, 120], [3956, 3955, 3944, 3870, 3528, 2484, 720], [29649, 29648, 29635, 29528, 28890, 26136, 18108, 5040], [250892, 250891, 250876, 250730, 249664, 243720, 219168, 149904, 40320]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 6], [1, 6, 17, 23], [1, 8, 34, 84, 108], [1, 10, 57, 211, 485, 605], [1, 12, 86, 428, 1472, 3236, 3956], [1, 14, 121, 759, 3513, 11541, 24609, 29649], [1, 16, 162, 1228, 7172, 31724, 100988, 210572, 250892]] | SymPoly Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 23, 108, 605, 3956, 29649, 250892] | -| EvenSum | [1, 1, 3, 11, 51, 283, 1839, 13719, 115675] | -| OddSum | [0, 1, 3, 12, 57, 322, 2117, 15930, 135217] | -| AltSum | [1, 0, 0, -1, -6, -39, -278, -2211, -19542] | -| AccSum | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | -| AccRevSum | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | -| RevAccSum | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | -| DiagSum | [1, 1, 3, 9, 36, 175, 1021, 6965] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 23, 108, 605, 3956, 29649, 250892] | +| EvenSum | [1, 1, 3, 11, 51, 283, 1839, 13719, 115675] | +| OddSum | [0, 1, 3, 12, 57, 322, 2117, 15930, 135217] | +| AltSum | [1, 0, 0, -1, -6, -39, -278, -2211, -19542] | +| AccSum | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | +| AccRevSum | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | +| AntiDiagSum | [1, 1, 3, 9, 36, 175, 1021, 6965, 54480] | SymPoly Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 330, 54600, 178489080, 7911786960, 839338855795440, 11191805812769662080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | -| Middle | [1, 1, 3, 11, 26, 154, 342, 2754, 5944] | -| Central | [1, 3, 26, 342, 5944] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 15, 103, 903, 9663, 122287, 1787991, 29671479] | -| NegHalf | [1, -1, 3, -13, 75, -517, 3875, -26301, 29531] | -| Bin | [1, 2, 9, 55, 409, 3546, 34981, 386163, 4711281] | -| InvBin | [1, 0, -3, 13, -47, 136, 1, -5935, 88481] | -| TransSqrs | [0, 1, 7, 40, 233, 1482, 10513, 83366, 735417] | -| TransNat0 | [0, 1, 5, 24, 127, 764, 5235, 40558, 351863] | -| TransNat1 | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 330, 54600, 178489080, 7911786960, 839338855795440, 11191805812769662080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | +| ColMiddle | [1, 1, 3, 11, 26, 154, 342, 2754, 5944] | +| ColECenter | [1, 3, 26, 342, 5944] | +| ColOCenter | [1, 11, 154, 2754] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 40, 233, 1482, 10513, 83366, 735417] | +| TransNat0 | [0, 1, 5, 24, 127, 764, 5235, 40558, 351863] | +| TransNat1 | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | SymPoly Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ SymPoly Polynomial values as columns | PolyCol7 | [1, 8, 72, 671, 6450, 64121, 661730, 7117677, 80105186] | | PolyCol8 | [1, 9, 90, 926, 9768, 105864, 1182448, 13657872, 163705344] | -# SymPolyInv -True +# SymPoly:Inv +[] -SymPolyInv Triangle view +SymPoly:Inv Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ SymPolyInv Triangle view | Row7 | [0, 0, 0, 0, 0, 36, -13, 1] | | Row8 | [0, 0, 0, 0, 0, 0, 49, -15, 1] | -SymPolyInv Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, -1, 1, 1, -3, 1, 0, 4, -5, 1, 0, 0, 9, -7, 1, 0, 0, 0, 16, -9, 1, 0, 0, 0, 0, 25, -11, 1, 0, 0, 0, 0, 0, 36, -13, 1, 0, 0, 0, 0, 0, 0, 49, -15, 1] | -| TriRev | [1, 1, -1, 1, -3, 1, 1, -5, 4, 0, 1, -7, 9, 0, 0, 1, -9, 16, 0, 0, 0, 1, -11, 25, 0, 0, 0, 0, 1, -13, 36, 0, 0, 0, 0, 0, 1, -15, 49, 0, 0, 0, 0, 0, 0] | -| TriAcc | [1, -1, 0, 1, -2, -1, 0, 4, -1, 0, 0, 0, 9, 2, 3, 0, 0, 0, 16, 7, 8, 0, 0, 0, 0, 25, 14, 15, 0, 0, 0, 0, 0, 36, 23, 24, 0, 0, 0, 0, 0, 0, 49, 34, 35] | -| TriRevAcc | [1, 0, -1, -1, -2, 1, 0, -1, 4, 0, 3, 2, 9, 0, 0, 8, 7, 16, 0, 0, 0, 15, 14, 25, 0, 0, 0, 0, 24, 23, 36, 0, 0, 0, 0, 0, 35, 34, 49, 0, 0, 0, 0, 0, 0] | -| TriAccRev | [1, 1, 0, 1, -2, -1, 1, -4, 0, 0, 1, -6, 3, 3, 3, 1, -8, 8, 8, 8, 8, 1, -10, 15, 15, 15, 15, 15, 1, -12, 24, 24, 24, 24, 24, 24, 1, -14, 35, 35, 35, 35, 35, 35, 35] | -| TriDiag | [1, -1, 1, 1, 0, -3, 0, 4, 1, 0, 0, -5, 0, 0, 9, 1, 0, 0, 0, -7] | - -SymPolyInv Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 0, 3, 8, 15, 24, 35] | -| EvenSum | [1, -1, 2, -5, 10, -9, 26, -13, 50] | -| OddSum | [0, 1, -3, 5, -7, 17, -11, 37, -15] | -| AltSum | [1, -2, 5, -10, 17, -26, 37, -50, 65] | -| AccSum | [1, -1, -2, 3, 14, 31, 54, 83, 118] | -| AccRevSum | [1, -1, -2, 3, 14, 31, 54, 83, 118] | -| RevAccSum | [1, 1, -2, -3, 4, 25, 66, 133, 232] | -| DiagSum | [1, -1, 2, -3, 5, -5, 10, -7] | - -SymPolyInv Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -3, 20, 63, 144, 275, 468, 735] | -| RowGcd | [1, 1, -3, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 5, 9, 16, 25, 36, 49] | -| Middle | [1, -1, -3, 4, 9, 0, 0, 0, 0] | -| Central | [1, -3, 9, 0, 0] | -| LeftSide | [1, -1, 1, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, -1, -1, 7, 23, 47, 79, 119, 167] | -| NegHalf | [1, 3, 11, 27, 51, 83, 123, 171, 227] | -| Bin | [1, 0, -4, -2, 27, 116, 310, 666, 1253] | -| InvBin | [1, 2, 8, 28, 83, 206, 442, 848, 1493] | -| TransSqrs | [0, 1, 1, -7, -11, 25, 161, 481, 1093] | -| TransNat0 | [0, 1, -1, -3, 1, 17, 51, 109, 197] | -| TransNat1 | [1, 1, -2, -3, 4, 25, 66, 133, 232] | - -SymPolyInv Diagonals as rows +SymPoly:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -3, 1], [0, 4, -5, 1], [0, 0, 9, -7, 1], [0, 0, 0, 16, -9, 1], [0, 0, 0, 0, 25, -11, 1], [0, 0, 0, 0, 0, 36, -13, 1], [0, 0, 0, 0, 0, 0, 49, -15, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 1], [1, -5, 4, 0], [1, -7, 9, 0, 0], [1, -9, 16, 0, 0, 0], [1, -11, 25, 0, 0, 0, 0], [1, -13, 36, 0, 0, 0, 0, 0], [1, -15, 49, 0, 0, 0, 0, 0, 0]] | +| AntiDiag | [[1], [-1], [1, 1], [0, -3], [0, 4, 1], [0, 0, -5], [0, 0, 9, 1], [0, 0, 0, -7], [0, 0, 0, 16, 1]] | +| AccTabl | [[1], [-1, 0], [1, -2, -1], [0, 4, -1, 0], [0, 0, 9, 2, 3], [0, 0, 0, 16, 7, 8], [0, 0, 0, 0, 25, 14, 15], [0, 0, 0, 0, 0, 36, 23, 24], [0, 0, 0, 0, 0, 0, 49, 34, 35]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 1], [0, -1, 4, 0], [3, 2, 9, 0, 0], [8, 7, 16, 0, 0, 0], [15, 14, 25, 0, 0, 0, 0], [24, 23, 36, 0, 0, 0, 0, 0], [35, 34, 49, 0, 0, 0, 0, 0, 0]] | +| AccRevTabl | [[1], [1, 0], [1, -2, -1], [1, -4, 0, 0], [1, -6, 3, 3, 3], [1, -8, 8, 8, 8, 8], [1, -10, 15, 15, 15, 15, 15], [1, -12, 24, 24, 24, 24, 24, 24], [1, -14, 35, 35, 35, 35, 35, 35, 35]] | + +SymPoly:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 0, 3, 8, 15, 24, 35] | +| EvenSum | [1, -1, 2, -5, 10, -9, 26, -13, 50] | +| OddSum | [0, 1, -3, 5, -7, 17, -11, 37, -15] | +| AltSum | [1, -2, 5, -10, 17, -26, 37, -50, 65] | +| AccSum | [1, -1, -2, 3, 14, 31, 54, 83, 118] | +| AccRevSum | [1, 1, -2, -3, 4, 25, 66, 133, 232] | +| AntiDiagSum | [1, -1, 2, -3, 5, -5, 10, -7, 17] | + +SymPoly:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 20, 63, 144, 275, 468, 735] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 5, 9, 16, 25, 36, 49] | +| ColMiddle | [1, -1, -3, 4, 9, 0, 0, 0, 0] | +| ColECenter | [1, -3, 9, 0, 0] | +| ColOCenter | [-1, 4, 0, 0] | +| ColLeft | [1, -1, 1, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -7, -11, 25, 161, 481, 1093] | +| TransNat0 | [0, 1, -1, -3, 1, 17, 51, 109, 197] | +| TransNat1 | [1, 1, -2, -3, 4, 25, 66, 133, 232] | + +SymPoly:Inv Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ SymPolyInv Diagonals as rows | DiagRow7 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [0, 0, 0, 0, 0, 0, 0, 0, 0]| -SymPolyInv Diagonals as columns +SymPoly:Inv Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, -1, 1, 0, 0, 0, 0, 0, 0] | @@ -188,7 +180,7 @@ SymPolyInv Diagonals as columns | DiagCol7 | [1, -15, 64, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [1, -17, 81, 0, 0, 0, 0, 0, 0] | -SymPolyInv Polynomial values as rows +SymPoly:Inv Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ SymPolyInv Polynomial values as rows | PolyRow7 | [0, 24, 448, 1458, 0, -12500, -46656, -100842, -131072] | | PolyRow8 | [0, 35, 1472, 9477, 20480, -15625, -233280, -823543, -1835008] | -SymPolyInv Polynomial values as columns +SymPoly:Inv Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, -1, 1, 0, 0, 0, 0, 0, 0] | @@ -214,10 +206,10 @@ SymPolyInv Polynomial values as columns | PolyCol7 | [1, 6, 29, 126, 441, 686, -7203, -100842, -823543] | | PolyCol8 | [1, 7, 41, 224, 1088, 4096, 4096, -131072, -1835008] | -# SymPolyRev -True +# SymPoly:Rev +[] -SymPolyRev Triangle view +SymPoly:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -230,47 +222,43 @@ SymPolyRev Triangle view | Row7 | [1, 13, 107, 638, 2754, 8028, 13068, 5040] | | Row8 | [1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320] | -SymPolyRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 2, 1, 5, 11, 6, 1, 7, 26, 50, 24, 1, 9, 47, 154, 274, 120, 1, 11, 74, 342, 1044, 1764, 720, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320] | -| TriRev | [1, 1, 1, 2, 3, 1, 6, 11, 5, 1, 24, 50, 26, 7, 1, 120, 274, 154, 47, 9, 1, 720, 1764, 1044, 342, 74, 11, 1, 5040, 13068, 8028, 2754, 638, 107, 13, 1, 40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1] | -| TriAcc | [1, 1, 2, 1, 4, 6, 1, 6, 17, 23, 1, 8, 34, 84, 108, 1, 10, 57, 211, 485, 605, 1, 12, 86, 428, 1472, 3236, 3956, 1, 14, 121, 759, 3513, 11541, 24609, 29649, 1, 16, 162, 1228, 7172, 31724, 100988, 210572, 250892] | -| TriRevAcc | [1, 2, 1, 6, 4, 1, 23, 17, 6, 1, 108, 84, 34, 8, 1, 605, 485, 211, 57, 10, 1, 3956, 3236, 1472, 428, 86, 12, 1, 29649, 24609, 11541, 3513, 759, 121, 14, 1, 250892, 210572, 100988, 31724, 7172, 1228, 162, 16, 1] | -| TriAccRev | [1, 1, 2, 2, 5, 6, 6, 17, 22, 23, 24, 74, 100, 107, 108, 120, 394, 548, 595, 604, 605, 720, 2484, 3528, 3870, 3944, 3955, 3956, 5040, 18108, 26136, 28890, 29528, 29635, 29648, 29649, 40320, 149904, 219168, 243720, 249664, 250730, 250876, 250891, 250892] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 5, 2, 1, 7, 11, 1, 9, 26, 6, 1, 11, 47, 50] | - -SymPolyRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 23, 108, 605, 3956, 29649, 250892] | -| EvenSum | [1, 1, 3, 12, 51, 322, 1839, 15930, 115675] | -| OddSum | [0, 1, 3, 11, 57, 283, 2117, 13719, 135217] | -| AltSum | [1, 0, 0, 1, -6, 39, -278, 2211, -19542] | -| AccSum | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | -| AccRevSum | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | -| RevAccSum | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | -| DiagSum | [1, 1, 2, 4, 8, 19, 42, 109] | - -SymPolyRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 330, 54600, 178489080, 7911786960, 839338855795440, 11191805812769662080] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | -| Middle | [1, 1, 3, 5, 26, 47, 342, 638, 5944] | -| Central | [1, 3, 26, 342, 5944] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 12, 56, 300, 1836, 12760, 99912, 873696] | -| NegHalf | [1, -1, 0, -4, -12, -76, -472, -3528, -29536] | -| Bin | [1, 2, 9, 55, 409, 3546, 34981, 386163, 4711281] | -| InvBin | [1, 0, -3, -13, -47, -136, 1, 5935, 88481] | -| TransSqrs | [0, 1, 11, 103, 945, 8967, 90109, 968355, 11162697] | -| TransNat0 | [0, 1, 7, 45, 305, 2261, 18501, 166985, 1655273] | -| TransNat1 | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | - -SymPolyRev Diagonals as rows +SymPoly:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 2], [1, 5, 11, 6], [1, 7, 26, 50, 24], [1, 9, 47, 154, 274, 120], [1, 11, 74, 342, 1044, 1764, 720], [1, 13, 107, 638, 2754, 8028, 13068, 5040], [1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320]] | +| RevTabl | [[1], [1, 1], [2, 3, 1], [6, 11, 5, 1], [24, 50, 26, 7, 1], [120, 274, 154, 47, 9, 1], [720, 1764, 1044, 342, 74, 11, 1], [5040, 13068, 8028, 2754, 638, 107, 13, 1], [40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 5, 2], [1, 7, 11], [1, 9, 26, 6], [1, 11, 47, 50], [1, 13, 74, 154, 24]] | +| AccTabl | [[1], [1, 2], [1, 4, 6], [1, 6, 17, 23], [1, 8, 34, 84, 108], [1, 10, 57, 211, 485, 605], [1, 12, 86, 428, 1472, 3236, 3956], [1, 14, 121, 759, 3513, 11541, 24609, 29649], [1, 16, 162, 1228, 7172, 31724, 100988, 210572, 250892]] | +| RevAccTabl | [[1], [2, 1], [6, 4, 1], [23, 17, 6, 1], [108, 84, 34, 8, 1], [605, 485, 211, 57, 10, 1], [3956, 3236, 1472, 428, 86, 12, 1], [29649, 24609, 11541, 3513, 759, 121, 14, 1], [250892, 210572, 100988, 31724, 7172, 1228, 162, 16, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 5, 6], [6, 17, 22, 23], [24, 74, 100, 107, 108], [120, 394, 548, 595, 604, 605], [720, 2484, 3528, 3870, 3944, 3955, 3956], [5040, 18108, 26136, 28890, 29528, 29635, 29648, 29649], [40320, 149904, 219168, 243720, 249664, 250730, 250876, 250891, 250892]] | + +SymPoly:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 23, 108, 605, 3956, 29649, 250892] | +| EvenSum | [1, 1, 3, 12, 51, 322, 1839, 15930, 115675] | +| OddSum | [0, 1, 3, 11, 57, 283, 2117, 13719, 135217] | +| AltSum | [1, 0, 0, 1, -6, 39, -278, 2211, -19542] | +| AccSum | [1, 3, 11, 47, 235, 1369, 9191, 70207, 602755] | +| AccRevSum | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | +| AntiDiagSum | [1, 1, 2, 4, 8, 19, 42, 109, 266] | + +SymPoly:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 330, 54600, 178489080, 7911786960, 839338855795440, 11191805812769662080] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 11, 50, 274, 1764, 13068, 109584] | +| ColMiddle | [1, 1, 3, 5, 26, 47, 342, 638, 5944] | +| ColECenter | [1, 3, 26, 342, 5944] | +| ColOCenter | [1, 5, 47, 638] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 11, 103, 945, 8967, 90109, 968355, 11162697] | +| TransNat0 | [0, 1, 7, 45, 305, 2261, 18501, 166985, 1655273] | +| TransNat1 | [1, 3, 13, 68, 413, 2866, 22457, 196634, 1906165] | + +SymPoly:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320]| @@ -283,7 +271,7 @@ SymPolyRev Diagonals as rows | DiagRow7 | [1, 15, 191, 2414, 31594, 434568, 6314664, 97053936, 1576890000]| | DiagRow8 | [1, 17, 242, 3382, 48504, 725592, 11393808, 188204400, 3270729600]| -SymPolyRev Diagonals as columns +SymPoly:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -296,7 +284,7 @@ SymPolyRev Diagonals as columns | DiagCol7 | [5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, 97053936, 188204400] | | DiagCol8 | [40320, 1026576, 6999840, 30334320, 101378880, 284574960, 703404576, 1576890000, 3270729600] | -SymPolyRev Polynomial values as rows +SymPoly:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -309,7 +297,7 @@ SymPolyRev Polynomial values as rows | PolyRow7 | [1, 29649, 1787991, 22741159, 145070181, 624828741, 2086714699, 5829856971, 14270019369] | | PolyRow8 | [1, 250892, 29671479, 561170926, 4748274653, 25474076976, 101829060307, 331260042674, 925258217721] | -SymPolyRev Polynomial values as columns +SymPoly:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -322,10 +310,10 @@ SymPolyRev Polynomial values as columns | PolyCol7 | [1, 8, 120, 2633, 76098, 2729903, 116982482, 5829856971, 331260042674] | | PolyCol8 | [1, 9, 153, 3817, 125625, 5136393, 251002585, 14270019369, 925258217721] | -# SymPolyInvRev -True +# SymPoly:Inv:Rev +[] -SymPolyInvRev Triangle view +SymPoly:Inv:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -338,47 +326,43 @@ SymPolyInvRev Triangle view | Row7 | [1, -13, 36, 0, 0, 0, 0, 0] | | Row8 | [1, -15, 49, 0, 0, 0, 0, 0, 0] | -SymPolyInvRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, -1, 1, -3, 1, 1, -5, 4, 0, 1, -7, 9, 0, 0, 1, -9, 16, 0, 0, 0, 1, -11, 25, 0, 0, 0, 0, 1, -13, 36, 0, 0, 0, 0, 0, 1, -15, 49, 0, 0, 0, 0, 0, 0] | -| TriRev | [1, -1, 1, 1, -3, 1, 0, 4, -5, 1, 0, 0, 9, -7, 1, 0, 0, 0, 16, -9, 1, 0, 0, 0, 0, 25, -11, 1, 0, 0, 0, 0, 0, 36, -13, 1, 0, 0, 0, 0, 0, 0, 49, -15, 1] | -| TriAcc | [1, 1, 0, 1, -2, -1, 1, -4, 0, 0, 1, -6, 3, 3, 3, 1, -8, 8, 8, 8, 8, 1, -10, 15, 15, 15, 15, 15, 1, -12, 24, 24, 24, 24, 24, 24, 1, -14, 35, 35, 35, 35, 35, 35, 35] | -| TriRevAcc | [1, 0, 1, -1, -2, 1, 0, 0, -4, 1, 3, 3, 3, -6, 1, 8, 8, 8, 8, -8, 1, 15, 15, 15, 15, 15, -10, 1, 24, 24, 24, 24, 24, 24, -12, 1, 35, 35, 35, 35, 35, 35, 35, -14, 1] | -| TriAccRev | [1, -1, 0, 1, -2, -1, 0, 4, -1, 0, 0, 0, 9, 2, 3, 0, 0, 0, 16, 7, 8, 0, 0, 0, 0, 25, 14, 15, 0, 0, 0, 0, 0, 36, 23, 24, 0, 0, 0, 0, 0, 0, 49, 34, 35] | -| TriDiag | [1, 1, 1, -1, 1, -3, 1, -5, 1, 1, -7, 4, 1, -9, 9, 0, 1, -11, 16, 0] | - -SymPolyInvRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 0, -1, 0, 3, 8, 15, 24, 35] | -| EvenSum | [1, 1, 2, 5, 10, 17, 26, 37, 50] | -| OddSum | [0, -1, -3, -5, -7, -9, -11, -13, -15] | -| AltSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | -| AccSum | [1, 1, -2, -3, 4, 25, 66, 133, 232] | -| AccRevSum | [1, 1, -2, -3, 4, 25, 66, 133, 232] | -| RevAccSum | [1, -1, -2, 3, 14, 31, 54, 83, 118] | -| DiagSum | [1, 1, 0, -2, -3, -2, 1, 6] | - -SymPolyInvRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, -3, 20, 63, 144, 275, 468, 735] | -| RowGcd | [1, 1, -3, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 5, 9, 16, 25, 36, 49] | -| Middle | [1, 1, -3, -5, 9, 16, 0, 0, 0] | -| Central | [1, -3, 9, 0, 0] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, -1, 1, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 1, -1, -4, -4, 16, 112, 448, 1472] | -| NegHalf | [1, -3, 11, -36, 108, -304, 816, -2112, 5312] | -| Bin | [1, 0, -4, -2, 27, 116, 310, 666, 1253] | -| InvBin | [1, -2, 8, -28, 83, -206, 442, -848, 1493] | -| TransSqrs | [0, -1, 1, 11, 29, 55, 89, 131, 181] | -| TransNat0 | [0, -1, -1, 3, 11, 23, 39, 59, 83] | -| TransNat1 | [1, -1, -2, 3, 14, 31, 54, 83, 118] | - -SymPolyInvRev Diagonals as rows +SymPoly:Inv:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, -1], [1, -3, 1], [1, -5, 4, 0], [1, -7, 9, 0, 0], [1, -9, 16, 0, 0, 0], [1, -11, 25, 0, 0, 0, 0], [1, -13, 36, 0, 0, 0, 0, 0], [1, -15, 49, 0, 0, 0, 0, 0, 0]] | +| RevTabl | [[1], [-1, 1], [1, -3, 1], [0, 4, -5, 1], [0, 0, 9, -7, 1], [0, 0, 0, 16, -9, 1], [0, 0, 0, 0, 25, -11, 1], [0, 0, 0, 0, 0, 36, -13, 1], [0, 0, 0, 0, 0, 0, 49, -15, 1]] | +| AntiDiag | [[1], [1], [1, -1], [1, -3], [1, -5, 1], [1, -7, 4], [1, -9, 9, 0], [1, -11, 16, 0], [1, -13, 25, 0, 0]] | +| AccTabl | [[1], [1, 0], [1, -2, -1], [1, -4, 0, 0], [1, -6, 3, 3, 3], [1, -8, 8, 8, 8, 8], [1, -10, 15, 15, 15, 15, 15], [1, -12, 24, 24, 24, 24, 24, 24], [1, -14, 35, 35, 35, 35, 35, 35, 35]] | +| RevAccTabl | [[1], [0, 1], [-1, -2, 1], [0, 0, -4, 1], [3, 3, 3, -6, 1], [8, 8, 8, 8, -8, 1], [15, 15, 15, 15, 15, -10, 1], [24, 24, 24, 24, 24, 24, -12, 1], [35, 35, 35, 35, 35, 35, 35, -14, 1]] | +| AccRevTabl | [[1], [-1, 0], [1, -2, -1], [0, 4, -1, 0], [0, 0, 9, 2, 3], [0, 0, 0, 16, 7, 8], [0, 0, 0, 0, 25, 14, 15], [0, 0, 0, 0, 0, 36, 23, 24], [0, 0, 0, 0, 0, 0, 49, 34, 35]] | + +SymPoly:Inv:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 0, 3, 8, 15, 24, 35] | +| EvenSum | [1, 1, 2, 5, 10, 17, 26, 37, 50] | +| OddSum | [0, -1, -3, -5, -7, -9, -11, -13, -15] | +| AltSum | [1, 2, 5, 10, 17, 26, 37, 50, 65] | +| AccSum | [1, 1, -2, -3, 4, 25, 66, 133, 232] | +| AccRevSum | [1, -1, -2, 3, 14, 31, 54, 83, 118] | +| AntiDiagSum | [1, 1, 0, -2, -3, -2, 1, 6, 13] | + +SymPoly:Inv:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 20, 63, 144, 275, 468, 735] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 5, 9, 16, 25, 36, 49] | +| ColMiddle | [1, 1, -3, -5, 9, 16, 0, 0, 0] | +| ColECenter | [1, -3, 9, 0, 0] | +| ColOCenter | [1, -5, 16, 0] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, -1, 1, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, -1, 1, 11, 29, 55, 89, 131, 181] | +| TransNat0 | [0, -1, -1, 3, 11, 23, 39, 59, 83] | +| TransNat1 | [1, -1, -2, 3, 14, 31, 54, 83, 118] | + +SymPoly:Inv:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, -1, 1, 0, 0, 0, 0, 0, 0]| @@ -391,7 +375,7 @@ SymPolyInvRev Diagonals as rows | DiagRow7 | [1, -15, 64, 0, 0, 0, 0, 0, 0]| | DiagRow8 | [1, -17, 81, 0, 0, 0, 0, 0, 0]| -SymPolyInvRev Diagonals as columns +SymPoly:Inv:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -404,7 +388,7 @@ SymPolyInvRev Diagonals as columns | DiagCol7 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | | DiagCol8 | [0, 0, 0, 0, 0, 0, 0, 0, 0] | -SymPolyInvRev Polynomial values as rows +SymPoly:Inv:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -417,7 +401,7 @@ SymPolyInvRev Polynomial values as rows | PolyRow7 | [1, 24, 119, 286, 525, 836, 1219, 1674, 2201] | | PolyRow8 | [1, 35, 167, 397, 725, 1151, 1675, 2297, 3017] | -SymPolyInvRev Polynomial values as columns +SymPoly:Inv:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | diff --git a/data/md/TernaryTrees.md b/data/md/TernaryTrees.md index c302f97..f3be7e6 100644 --- a/data/md/TernaryTrees.md +++ b/data/md/TernaryTrees.md @@ -14,45 +14,41 @@ TernaryTrees Triangle view | Row7 | [0, 1, 13, 88, 408, 1428, 3876, 7752] | | Row8 | [0, 1, 15, 117, 627, 2565, 8379, 21945, 43263] | -TernaryTrees Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263] | -| TriRev | [1, 1, 0, 3, 1, 0, 12, 5, 1, 0, 55, 25, 7, 1, 0, 273, 130, 42, 9, 1, 0, 1428, 700, 245, 63, 11, 1, 0, 7752, 3876, 1428, 408, 88, 13, 1, 0, 43263, 21945, 8379, 2565, 627, 117, 15, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 4, 0, 1, 6, 18, 0, 1, 8, 33, 88, 0, 1, 10, 52, 182, 455, 0, 1, 12, 75, 320, 1020, 2448, 0, 1, 14, 102, 510, 1938, 5814, 13566, 0, 1, 16, 133, 760, 3325, 11704, 33649, 76912] | -| TriRevAcc | [1, 1, 0, 4, 1, 0, 18, 6, 1, 0, 88, 33, 8, 1, 0, 455, 182, 52, 10, 1, 0, 2448, 1020, 320, 75, 12, 1, 0, 13566, 5814, 1938, 510, 102, 14, 1, 0, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 0] | -| TriAccRev | [1, 1, 1, 3, 4, 4, 12, 17, 18, 18, 55, 80, 87, 88, 88, 273, 403, 445, 454, 455, 455, 1428, 2128, 2373, 2436, 2447, 2448, 2448, 7752, 11628, 13056, 13464, 13552, 13565, 13566, 13566, 43263, 65208, 73587, 76152, 76779, 76896, 76911, 76912, 76912] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 3, 0, 1, 5, 0, 1, 7, 12, 0, 1, 9, 25] | +TernaryTrees Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 3], [0, 1, 5, 12], [0, 1, 7, 25, 55], [0, 1, 9, 42, 130, 273], [0, 1, 11, 63, 245, 700, 1428], [0, 1, 13, 88, 408, 1428, 3876, 7752], [0, 1, 15, 117, 627, 2565, 8379, 21945, 43263]] | +| RevTabl | [[1], [1, 0], [3, 1, 0], [12, 5, 1, 0], [55, 25, 7, 1, 0], [273, 130, 42, 9, 1, 0], [1428, 700, 245, 63, 11, 1, 0], [7752, 3876, 1428, 408, 88, 13, 1, 0], [43263, 21945, 8379, 2565, 627, 117, 15, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 3], [0, 1, 5], [0, 1, 7, 12], [0, 1, 9, 25], [0, 1, 11, 42, 55]] | +| AccTabl | [[1], [0, 1], [0, 1, 4], [0, 1, 6, 18], [0, 1, 8, 33, 88], [0, 1, 10, 52, 182, 455], [0, 1, 12, 75, 320, 1020, 2448], [0, 1, 14, 102, 510, 1938, 5814, 13566], [0, 1, 16, 133, 760, 3325, 11704, 33649, 76912]] | +| RevAccTabl | [[1], [1, 0], [4, 1, 0], [18, 6, 1, 0], [88, 33, 8, 1, 0], [455, 182, 52, 10, 1, 0], [2448, 1020, 320, 75, 12, 1, 0], [13566, 5814, 1938, 510, 102, 14, 1, 0], [76912, 33649, 11704, 3325, 760, 133, 16, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [3, 4, 4], [12, 17, 18, 18], [55, 80, 87, 88, 88], [273, 403, 445, 454, 455, 455], [1428, 2128, 2373, 2436, 2447, 2448, 2448], [7752, 11628, 13056, 13464, 13552, 13565, 13566, 13566], [43263, 65208, 73587, 76152, 76779, 76896, 76911, 76912, 76912]] | TernaryTrees Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 4, 18, 88, 455, 2448, 13566, 76912] | -| EvenSum | [1, 0, 3, 5, 62, 139, 1684, 4297, 52284] | -| OddSum | [0, 1, 1, 13, 26, 316, 764, 9269, 24628] | -| AltSum | [1, -1, 2, -8, 36, -177, 920, -4972, 27656] | -| AccSum | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | -| AccRevSum | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | -| RevAccSum | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | -| DiagSum | [1, 0, 1, 1, 4, 6, 20, 35] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 4, 18, 88, 455, 2448, 13566, 76912] | +| EvenSum | [1, 0, 3, 5, 62, 139, 1684, 4297, 52284] | +| OddSum | [0, 1, 1, 13, 26, 316, 764, 9269, 24628] | +| AltSum | [1, -1, 2, -8, 36, -177, 920, -4972, 27656] | +| AccSum | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | +| AccRevSum | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | +| AntiDiagSum | [1, 0, 1, 1, 4, 6, 20, 35, 109] | TernaryTrees Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 60, 1925, 8190, 8246700, 7759752, 413377965] | -| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 3] | -| RowMax | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | -| Middle | [1, 0, 1, 1, 7, 9, 63, 88, 627] | -| Central | [1, 1, 7, 63, 627] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | -| PosHalf | [1, 1, 5, 26, 141, 789, 4520, 26368, 156053] | -| NegHalf | [1, 1, 1, 6, 25, 125, 648, 3504, 19489] | -| Bin | [1, 1, 5, 30, 201, 1438, 10734, 82512, 647945] | -| InvBin | [1, 1, 1, 0, -7, -42, -198, -858, -3575] | -| TransSqrs | [0, 1, 13, 129, 1134, 9320, 73440, 562457, 4221052] | -| TransNat0 | [0, 1, 7, 47, 310, 2030, 13260, 86583, 565708] | -| TransNat1 | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 60, 1925, 8190, 8246700, 7759752, 413377965] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 3] | +| RowMax | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | +| ColMiddle | [1, 0, 1, 1, 7, 9, 63, 88, 627] | +| ColECenter | [1, 1, 7, 63, 627] | +| ColOCenter | [0, 1, 9, 88] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | +| TransSqrs | [0, 1, 13, 129, 1134, 9320, 73440, 562457, 4221052] | +| TransNat0 | [0, 1, 7, 47, 310, 2030, 13260, 86583, 565708] | +| TransNat1 | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | TernaryTrees Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ TernaryTrees Polynomial values as columns | PolyCol7 | [1, 7, 154, 4368, 140980, 4915295, 180378072, 6865123692, 268505674024] | | PolyCol8 | [1, 8, 200, 6472, 238536, 9500232, 398315720, 17321702216, 774137880008] | -# TernaryTreesRev -True +# TernaryTrees:Rev +[] -TernaryTreesRev Triangle view +TernaryTrees:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ TernaryTreesRev Triangle view | Row7 | [7752, 3876, 1428, 408, 88, 13, 1, 0] | | Row8 | [43263, 21945, 8379, 2565, 627, 117, 15, 1, 0] | -TernaryTreesRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 3, 1, 0, 12, 5, 1, 0, 55, 25, 7, 1, 0, 273, 130, 42, 9, 1, 0, 1428, 700, 245, 63, 11, 1, 0, 7752, 3876, 1428, 408, 88, 13, 1, 0, 43263, 21945, 8379, 2565, 627, 117, 15, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263] | -| TriAcc | [1, 1, 1, 3, 4, 4, 12, 17, 18, 18, 55, 80, 87, 88, 88, 273, 403, 445, 454, 455, 455, 1428, 2128, 2373, 2436, 2447, 2448, 2448, 7752, 11628, 13056, 13464, 13552, 13565, 13566, 13566, 43263, 65208, 73587, 76152, 76779, 76896, 76911, 76912, 76912] | -| TriRevAcc | [1, 1, 1, 4, 4, 3, 18, 18, 17, 12, 88, 88, 87, 80, 55, 455, 455, 454, 445, 403, 273, 2448, 2448, 2447, 2436, 2373, 2128, 1428, 13566, 13566, 13565, 13552, 13464, 13056, 11628, 7752, 76912, 76912, 76911, 76896, 76779, 76152, 73587, 65208, 43263] | -| TriAccRev | [1, 0, 1, 0, 1, 4, 0, 1, 6, 18, 0, 1, 8, 33, 88, 0, 1, 10, 52, 182, 455, 0, 1, 12, 75, 320, 1020, 2448, 0, 1, 14, 102, 510, 1938, 5814, 13566, 0, 1, 16, 133, 760, 3325, 11704, 33649, 76912] | -| TriDiag | [1, 1, 3, 0, 12, 1, 55, 5, 0, 273, 25, 1, 1428, 130, 7, 0, 7752, 700, 42, 1] | - -TernaryTreesRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 4, 18, 88, 455, 2448, 13566, 76912] | -| EvenSum | [1, 1, 3, 13, 62, 316, 1684, 9269, 52284] | -| OddSum | [0, 0, 1, 5, 26, 139, 764, 4297, 24628] | -| AltSum | [1, 1, 2, 8, 36, 177, 920, 4972, 27656] | -| AccSum | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | -| AccRevSum | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | -| RevAccSum | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | -| DiagSum | [1, 1, 3, 13, 60, 299, 1565, 8495] | - -TernaryTreesRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 60, 1925, 8190, 8246700, 7759752, 413377965] | -| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 3] | -| RowMax | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | -| Middle | [1, 1, 1, 5, 7, 42, 63, 408, 627] | -| Central | [1, 1, 7, 63, 627] | -| LeftSide | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 14, 118, 1110, 11190, 118262, 1293302, 14513654] | -| NegHalf | [1, -2, 10, -78, 706, -6958, 72450, -784014, 8729698] | -| Bin | [1, 1, 5, 30, 201, 1438, 10734, 82512, 647945] | -| InvBin | [1, -1, 1, 0, -7, 42, -198, 858, -3575] | -| TransSqrs | [0, 0, 1, 9, 62, 395, 2448, 15029, 92092] | -| TransNat0 | [0, 0, 1, 7, 42, 245, 1428, 8379, 49588] | -| TransNat1 | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | - -TernaryTreesRev Diagonals as rows +TernaryTrees:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [3, 1, 0], [12, 5, 1, 0], [55, 25, 7, 1, 0], [273, 130, 42, 9, 1, 0], [1428, 700, 245, 63, 11, 1, 0], [7752, 3876, 1428, 408, 88, 13, 1, 0], [43263, 21945, 8379, 2565, 627, 117, 15, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 5, 12], [0, 1, 7, 25, 55], [0, 1, 9, 42, 130, 273], [0, 1, 11, 63, 245, 700, 1428], [0, 1, 13, 88, 408, 1428, 3876, 7752], [0, 1, 15, 117, 627, 2565, 8379, 21945, 43263]] | +| AntiDiag | [[1], [1], [3, 0], [12, 1], [55, 5, 0], [273, 25, 1], [1428, 130, 7, 0], [7752, 700, 42, 1], [43263, 3876, 245, 9, 0]] | +| AccTabl | [[1], [1, 1], [3, 4, 4], [12, 17, 18, 18], [55, 80, 87, 88, 88], [273, 403, 445, 454, 455, 455], [1428, 2128, 2373, 2436, 2447, 2448, 2448], [7752, 11628, 13056, 13464, 13552, 13565, 13566, 13566], [43263, 65208, 73587, 76152, 76779, 76896, 76911, 76912, 76912]] | +| RevAccTabl | [[1], [1, 1], [4, 4, 3], [18, 18, 17, 12], [88, 88, 87, 80, 55], [455, 455, 454, 445, 403, 273], [2448, 2448, 2447, 2436, 2373, 2128, 1428], [13566, 13566, 13565, 13552, 13464, 13056, 11628, 7752], [76912, 76912, 76911, 76896, 76779, 76152, 73587, 65208, 43263]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 4], [0, 1, 6, 18], [0, 1, 8, 33, 88], [0, 1, 10, 52, 182, 455], [0, 1, 12, 75, 320, 1020, 2448], [0, 1, 14, 102, 510, 1938, 5814, 13566], [0, 1, 16, 133, 760, 3325, 11704, 33649, 76912]] | + +TernaryTrees:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 4, 18, 88, 455, 2448, 13566, 76912] | +| EvenSum | [1, 1, 3, 13, 62, 316, 1684, 9269, 52284] | +| OddSum | [0, 0, 1, 5, 26, 139, 764, 4297, 24628] | +| AltSum | [1, 1, 2, 8, 36, 177, 920, 4972, 27656] | +| AccSum | [1, 2, 11, 65, 398, 2485, 15708, 100149, 642620] | +| AccRevSum | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | +| AntiDiagSum | [1, 1, 3, 13, 60, 299, 1565, 8495, 47393] | + +TernaryTrees:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 60, 1925, 8190, 8246700, 7759752, 413377965] | +| RowGcd | [1, 1, 3, 1, 1, 1, 1, 1, 3] | +| RowMax | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | +| ColMiddle | [1, 1, 1, 5, 7, 42, 63, 408, 627] | +| ColECenter | [1, 1, 7, 63, 627] | +| ColOCenter | [1, 5, 42, 408] | +| ColLeft | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 9, 62, 395, 2448, 15029, 92092] | +| TransNat0 | [0, 0, 1, 7, 42, 245, 1428, 8379, 49588] | +| TransNat1 | [1, 1, 5, 25, 130, 700, 3876, 21945, 126500] | + +TernaryTrees:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ TernaryTreesRev Diagonals as rows | DiagRow7 | [7752, 21945, 49588, 98670, 180180, 308763, 503440, 788392, 1193808]| | DiagRow8 | [43263, 126500, 296010, 610740, 1157013, 2056912, 3478200, 5645904, 8855691]| -TernaryTreesRev Diagonals as columns +TernaryTrees:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | @@ -188,7 +180,7 @@ TernaryTreesRev Diagonals as columns | DiagCol7 | [0, 1, 17, 187, 1700, 13923, 106981, 788392, 5645904] | | DiagCol8 | [0, 1, 19, 228, 2223, 19285, 155496, 1193808, 8855691] | -TernaryTreesRev Polynomial values as rows +TernaryTrees:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ TernaryTreesRev Polynomial values as rows | PolyRow7 | [7752, 13566, 26368, 54264, 112152, 225082, 432336, 792228, 1387624] | | PolyRow8 | [43263, 76912, 156053, 346104, 787411, 1753088, 3732777, 7547368, 14499719] | -TernaryTreesRev Polynomial values as columns +TernaryTrees:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 3, 12, 55, 273, 1428, 7752, 43263] | diff --git a/data/md/WardSet.md b/data/md/WardSet.md index eb376aa..12c2027 100644 --- a/data/md/WardSet.md +++ b/data/md/WardSet.md @@ -1,5 +1,5 @@ # WardSet -['A134991', 'A269939'] +['A269939', 'A134991'] WardSet Triangle view | Row | Seq | @@ -14,45 +14,41 @@ WardSet Triangle view | Row7 | [0, 1, 246, 6825, 56980, 190575, 270270, 135135] | | Row8 | [0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025] | -WardSet Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025] | -| TriRev | [1, 1, 0, 3, 1, 0, 15, 10, 1, 0, 105, 105, 25, 1, 0, 945, 1260, 490, 56, 1, 0, 10395, 17325, 9450, 1918, 119, 1, 0, 135135, 270270, 190575, 56980, 6825, 246, 1, 0, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 0] | -| TriAcc | [1, 0, 1, 0, 1, 4, 0, 1, 11, 26, 0, 1, 26, 131, 236, 0, 1, 57, 547, 1807, 2752, 0, 1, 120, 2038, 11488, 28813, 39208, 0, 1, 247, 7072, 64052, 254627, 524897, 660032, 0, 1, 502, 23437, 326432, 1963067, 6062162, 10791887, 12818912] | -| TriRevAcc | [1, 1, 0, 4, 1, 0, 26, 11, 1, 0, 236, 131, 26, 1, 0, 2752, 1807, 547, 57, 1, 0, 39208, 28813, 11488, 2038, 120, 1, 0, 660032, 524897, 254627, 64052, 7072, 247, 1, 0, 12818912, 10791887, 6062162, 1963067, 326432, 23437, 502, 1, 0] | -| TriAccRev | [1, 1, 1, 3, 4, 4, 15, 25, 26, 26, 105, 210, 235, 236, 236, 945, 2205, 2695, 2751, 2752, 2752, 10395, 27720, 37170, 39088, 39207, 39208, 39208, 135135, 405405, 595980, 652960, 659785, 660031, 660032, 660032, 2027025, 6756750, 10855845, 12492480, 12795475, 12818410, 12818911, 12818912, 12818912] | -| TriDiag | [1, 0, 0, 1, 0, 1, 0, 1, 3, 0, 1, 10, 0, 1, 25, 15, 0, 1, 56, 105] | +WardSet Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [0, 1], [0, 1, 3], [0, 1, 10, 15], [0, 1, 25, 105, 105], [0, 1, 56, 490, 1260, 945], [0, 1, 119, 1918, 9450, 17325, 10395], [0, 1, 246, 6825, 56980, 190575, 270270, 135135], [0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025]] | +| RevTabl | [[1], [1, 0], [3, 1, 0], [15, 10, 1, 0], [105, 105, 25, 1, 0], [945, 1260, 490, 56, 1, 0], [10395, 17325, 9450, 1918, 119, 1, 0], [135135, 270270, 190575, 56980, 6825, 246, 1, 0], [2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 0]] | +| AntiDiag | [[1], [0], [0, 1], [0, 1], [0, 1, 3], [0, 1, 10], [0, 1, 25, 15], [0, 1, 56, 105], [0, 1, 119, 490, 105]] | +| AccTabl | [[1], [0, 1], [0, 1, 4], [0, 1, 11, 26], [0, 1, 26, 131, 236], [0, 1, 57, 547, 1807, 2752], [0, 1, 120, 2038, 11488, 28813, 39208], [0, 1, 247, 7072, 64052, 254627, 524897, 660032], [0, 1, 502, 23437, 326432, 1963067, 6062162, 10791887, 12818912]] | +| RevAccTabl | [[1], [1, 0], [4, 1, 0], [26, 11, 1, 0], [236, 131, 26, 1, 0], [2752, 1807, 547, 57, 1, 0], [39208, 28813, 11488, 2038, 120, 1, 0], [660032, 524897, 254627, 64052, 7072, 247, 1, 0], [12818912, 10791887, 6062162, 1963067, 326432, 23437, 502, 1, 0]] | +| AccRevTabl | [[1], [1, 1], [3, 4, 4], [15, 25, 26, 26], [105, 210, 235, 236, 236], [945, 2205, 2695, 2751, 2752, 2752], [10395, 27720, 37170, 39088, 39207, 39208, 39208], [135135, 405405, 595980, 652960, 659785, 660031, 660032, 660032], [2027025, 6756750, 10855845, 12492480, 12795475, 12818410, 12818911, 12818912, 12818912]] | WardSet Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912] | -| EvenSum | [1, 0, 3, 10, 130, 1316, 19964, 327496, 6429616] | -| OddSum | [0, 1, 1, 16, 106, 1436, 19244, 332536, 6389296] | -| AltSum | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | -| AccSum | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | -| AccRevSum | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | -| RevAccSum | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | -| DiagSum | [1, 0, 1, 1, 4, 11, 41, 162] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912] | +| EvenSum | [1, 0, 3, 10, 130, 1316, 19964, 327496, 6429616] | +| OddSum | [0, 1, 1, 16, 106, 1436, 19244, 332536, 6389296] | +| AltSum | [1, -1, 2, -6, 24, -120, 720, -5040, 40320] | +| AccSum | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | +| AccRevSum | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | +| AntiDiagSum | [1, 0, 1, 1, 4, 11, 41, 162, 715] | WardSet Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 30, 525, 52920, 242099550, 45099954900, 367310208815775225] | -| RowGcd | [1, 1, 3, 5, 5, 7, 7, 1, 1] | -| RowMax | [1, 1, 3, 15, 105, 1260, 17325, 270270, 4729725] | -| Middle | [1, 0, 1, 1, 25, 56, 1918, 6825, 302995] | -| Central | [1, 1, 25, 1918, 302995] | -| LeftSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| RightSide | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| PosHalf | [1, 1, 5, 39, 423, 5889, 100125, 2010951, 46589967] | -| NegHalf | [1, 1, 1, -1, -13, -47, 73, 2447, 16811] | -| Bin | [1, 1, 5, 48, 679, 12710, 296246, 8267448, 268799091] | -| InvBin | [1, 1, 1, -12, -169, -1010, 11614, 484736, 7251635] | -| TransSqrs | [0, 1, 13, 176, 2726, 48420, 976284, 22089800, 555025760] | -| TransNat0 | [0, 1, 7, 66, 786, 11348, 192788, 3769328, 83383808] | -| TransNat1 | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 30, 525, 52920, 242099550, 45099954900, 367310208815775225] | +| RowGcd | [1, 1, 3, 5, 5, 7, 7, 1, 1] | +| RowMax | [1, 1, 3, 15, 105, 1260, 17325, 270270, 4729725] | +| ColMiddle | [1, 0, 1, 1, 25, 56, 1918, 6825, 302995] | +| ColECenter | [1, 1, 25, 1918, 302995] | +| ColOCenter | [0, 1, 56, 6825] | +| ColLeft | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| ColRight | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| TransSqrs | [0, 1, 13, 176, 2726, 48420, 976284, 22089800, 555025760] | +| TransNat0 | [0, 1, 7, 66, 786, 11348, 192788, 3769328, 83383808] | +| TransNat1 | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | WardSet Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ WardSet Polynomial values as columns | PolyCol7 | [1, 7, 154, 5642, 289352, 19078696, 1537495792, 146428634576, 16091024397056] | | PolyCol8 | [1, 8, 200, 8328, 485448, 36381192, 3332389320, 360729956232, 45056223763272] | -# WardSetRev -True +# WardSet:Rev +[] -WardSetRev Triangle view +WardSet:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ WardSetRev Triangle view | Row7 | [135135, 270270, 190575, 56980, 6825, 246, 1, 0] | | Row8 | [2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 0] | -WardSetRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 0, 3, 1, 0, 15, 10, 1, 0, 105, 105, 25, 1, 0, 945, 1260, 490, 56, 1, 0, 10395, 17325, 9450, 1918, 119, 1, 0, 135135, 270270, 190575, 56980, 6825, 246, 1, 0, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 0] | -| TriRev | [1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025] | -| TriAcc | [1, 1, 1, 3, 4, 4, 15, 25, 26, 26, 105, 210, 235, 236, 236, 945, 2205, 2695, 2751, 2752, 2752, 10395, 27720, 37170, 39088, 39207, 39208, 39208, 135135, 405405, 595980, 652960, 659785, 660031, 660032, 660032, 2027025, 6756750, 10855845, 12492480, 12795475, 12818410, 12818911, 12818912, 12818912] | -| TriRevAcc | [1, 1, 1, 4, 4, 3, 26, 26, 25, 15, 236, 236, 235, 210, 105, 2752, 2752, 2751, 2695, 2205, 945, 39208, 39208, 39207, 39088, 37170, 27720, 10395, 660032, 660032, 660031, 659785, 652960, 595980, 405405, 135135, 12818912, 12818912, 12818911, 12818410, 12795475, 12492480, 10855845, 6756750, 2027025] | -| TriAccRev | [1, 0, 1, 0, 1, 4, 0, 1, 11, 26, 0, 1, 26, 131, 236, 0, 1, 57, 547, 1807, 2752, 0, 1, 120, 2038, 11488, 28813, 39208, 0, 1, 247, 7072, 64052, 254627, 524897, 660032, 0, 1, 502, 23437, 326432, 1963067, 6062162, 10791887, 12818912] | -| TriDiag | [1, 1, 3, 0, 15, 1, 105, 10, 0, 945, 105, 1, 10395, 1260, 25, 0, 135135, 17325, 490, 1] | - -WardSetRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912] | -| EvenSum | [1, 1, 3, 16, 130, 1436, 19964, 332536, 6429616] | -| OddSum | [0, 0, 1, 10, 106, 1316, 19244, 327496, 6389296] | -| AltSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| AccSum | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | -| AccRevSum | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | -| RevAccSum | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | -| DiagSum | [1, 1, 3, 16, 115, 1051, 11680, 152951] | - -WardSetRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 3, 30, 525, 52920, 242099550, 45099954900, 367310208815775225] | -| RowGcd | [1, 1, 3, 5, 5, 7, 7, 1, 1] | -| RowMax | [1, 1, 3, 15, 105, 1260, 17325, 270270, 4729725] | -| Middle | [1, 1, 1, 10, 25, 490, 1918, 56980, 302995] | -| Central | [1, 1, 25, 1918, 302995] | -| LeftSide | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | -| RightSide | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| PosHalf | [1, 2, 14, 162, 2622, 54546, 1386702, 41660226, 1444071006] | -| NegHalf | [1, -2, 10, -82, 938, -13778, 247210, -5240338, 128149802] | -| Bin | [1, 1, 5, 48, 679, 12710, 296246, 8267448, 268799091] | -| InvBin | [1, -1, 1, 12, -169, 1010, 11614, -484736, 7251635] | -| TransSqrs | [0, 0, 1, 14, 214, 3740, 74316, 1660776, 41295200] | -| TransNat0 | [0, 0, 1, 12, 158, 2412, 42460, 850896, 19167488] | -| TransNat1 | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | - -WardSetRev Diagonals as rows +WardSet:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 0], [3, 1, 0], [15, 10, 1, 0], [105, 105, 25, 1, 0], [945, 1260, 490, 56, 1, 0], [10395, 17325, 9450, 1918, 119, 1, 0], [135135, 270270, 190575, 56980, 6825, 246, 1, 0], [2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 0]] | +| RevTabl | [[1], [0, 1], [0, 1, 3], [0, 1, 10, 15], [0, 1, 25, 105, 105], [0, 1, 56, 490, 1260, 945], [0, 1, 119, 1918, 9450, 17325, 10395], [0, 1, 246, 6825, 56980, 190575, 270270, 135135], [0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025]] | +| AntiDiag | [[1], [1], [3, 0], [15, 1], [105, 10, 0], [945, 105, 1], [10395, 1260, 25, 0], [135135, 17325, 490, 1], [2027025, 270270, 9450, 56, 0]] | +| AccTabl | [[1], [1, 1], [3, 4, 4], [15, 25, 26, 26], [105, 210, 235, 236, 236], [945, 2205, 2695, 2751, 2752, 2752], [10395, 27720, 37170, 39088, 39207, 39208, 39208], [135135, 405405, 595980, 652960, 659785, 660031, 660032, 660032], [2027025, 6756750, 10855845, 12492480, 12795475, 12818410, 12818911, 12818912, 12818912]] | +| RevAccTabl | [[1], [1, 1], [4, 4, 3], [26, 26, 25, 15], [236, 236, 235, 210, 105], [2752, 2752, 2751, 2695, 2205, 945], [39208, 39208, 39207, 39088, 37170, 27720, 10395], [660032, 660032, 660031, 659785, 652960, 595980, 405405, 135135], [12818912, 12818912, 12818911, 12818410, 12795475, 12492480, 10855845, 6756750, 2027025]] | +| AccRevTabl | [[1], [0, 1], [0, 1, 4], [0, 1, 11, 26], [0, 1, 26, 131, 236], [0, 1, 57, 547, 1807, 2752], [0, 1, 120, 2038, 11488, 28813, 39208], [0, 1, 247, 7072, 64052, 254627, 524897, 660032], [0, 1, 502, 23437, 326432, 1963067, 6062162, 10791887, 12818912]] | + +WardSet:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912] | +| EvenSum | [1, 1, 3, 16, 130, 1436, 19964, 332536, 6429616] | +| OddSum | [0, 0, 1, 10, 106, 1316, 19244, 327496, 6389296] | +| AltSum | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| AccSum | [1, 2, 11, 92, 1022, 14100, 231996, 4429360, 96202720] | +| AccRevSum | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | +| AntiDiagSum | [1, 1, 3, 16, 115, 1051, 11680, 152951, 2306801] | + +WardSet:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 30, 525, 52920, 242099550, 45099954900, 367310208815775225] | +| RowGcd | [1, 1, 3, 5, 5, 7, 7, 1, 1] | +| RowMax | [1, 1, 3, 15, 105, 1260, 17325, 270270, 4729725] | +| ColMiddle | [1, 1, 1, 10, 25, 490, 1918, 56980, 302995] | +| ColECenter | [1, 1, 25, 1918, 302995] | +| ColOCenter | [1, 10, 490, 56980] | +| ColLeft | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | +| ColRight | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| TransSqrs | [0, 0, 1, 14, 214, 3740, 74316, 1660776, 41295200] | +| TransNat0 | [0, 0, 1, 12, 158, 2412, 42460, 850896, 19167488] | +| TransNat1 | [1, 1, 5, 38, 394, 5164, 81668, 1510928, 31986400] | + +WardSet:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 0, 0, 0, 0, 0, 0, 0, 0]| @@ -175,7 +167,7 @@ WardSetRev Diagonals as rows | DiagRow7 | [135135, 4729725, 94594500, 1422280860, 17892864990, 199124936010, 2026763158420, 19282395272140, 174073797222325]| | DiagRow8 | [2027025, 91891800, 2343240900, 44346982680, 694740296250, 9540421090200, 118885395048420, 1375295856374440, 15006064187108995]| -WardSetRev Diagonals as columns +WardSet:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | @@ -188,7 +180,7 @@ WardSetRev Diagonals as columns | DiagCol7 | [0, 1, 1012, 235092, 30950920, 3049616570, 254752658160, 19282395272140, 1375295856374440] | | DiagCol8 | [0, 1, 2035, 731731, 134779645, 17539336815, 1861763348445, 174073797222325, 15006064187108995] | -WardSetRev Polynomial values as rows +WardSet:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ WardSetRev Polynomial values as rows | PolyRow7 | [135135, 660032, 2010951, 4812912, 9915335, 18423360, 31729887, 51548336, 79946127] | | PolyRow8 | [2027025, 12818912, 46589967, 127780416, 294396725, 601682400, 1126164627, 1970081792, 3266196921] | -WardSetRev Polynomial values as columns +WardSet:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025] | diff --git a/data/md/Worpitzky.md b/data/md/Worpitzky.md index 3fa2e80..cf61e0a 100644 --- a/data/md/Worpitzky.md +++ b/data/md/Worpitzky.md @@ -14,45 +14,41 @@ Worpitzky Triangle view | Row7 | [1, 127, 1932, 10206, 25200, 31920, 20160, 5040] | | Row8 | [1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320] | -Worpitzky Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1, 15, 50, 60, 24, 1, 31, 180, 390, 360, 120, 1, 63, 602, 2100, 3360, 2520, 720, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320] | -| TriRev | [1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1] | -| TriAcc | [1, 1, 2, 1, 4, 6, 1, 8, 20, 26, 1, 16, 66, 126, 150, 1, 32, 212, 602, 962, 1082, 1, 64, 666, 2766, 6126, 8646, 9366, 1, 128, 2060, 12266, 37466, 69386, 89546, 94586, 1, 256, 6306, 52926, 219750, 537270, 869910, 1051350, 1091670] | -| TriRevAcc | [1, 2, 1, 6, 4, 1, 26, 20, 8, 1, 150, 126, 66, 16, 1, 1082, 962, 602, 212, 32, 1, 9366, 8646, 6126, 2766, 666, 64, 1, 94586, 89546, 69386, 37466, 12266, 2060, 128, 1, 1091670, 1051350, 869910, 537270, 219750, 52926, 6306, 256, 1] | -| TriAccRev | [1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670] | -| TriDiag | [1, 1, 1, 1, 1, 3, 1, 7, 2, 1, 15, 12, 1, 31, 50, 6, 1, 63, 180, 60] | +Worpitzky Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [1, 3, 2], [1, 7, 12, 6], [1, 15, 50, 60, 24], [1, 31, 180, 390, 360, 120], [1, 63, 602, 2100, 3360, 2520, 720], [1, 127, 1932, 10206, 25200, 31920, 20160, 5040], [1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320]] | +| RevTabl | [[1], [1, 1], [2, 3, 1], [6, 12, 7, 1], [24, 60, 50, 15, 1], [120, 360, 390, 180, 31, 1], [720, 2520, 3360, 2100, 602, 63, 1], [5040, 20160, 31920, 25200, 10206, 1932, 127, 1], [40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1]] | +| AntiDiag | [[1], [1], [1, 1], [1, 3], [1, 7, 2], [1, 15, 12], [1, 31, 50, 6], [1, 63, 180, 60], [1, 127, 602, 390, 24]] | +| AccTabl | [[1], [1, 2], [1, 4, 6], [1, 8, 20, 26], [1, 16, 66, 126, 150], [1, 32, 212, 602, 962, 1082], [1, 64, 666, 2766, 6126, 8646, 9366], [1, 128, 2060, 12266, 37466, 69386, 89546, 94586], [1, 256, 6306, 52926, 219750, 537270, 869910, 1051350, 1091670]] | +| RevAccTabl | [[1], [2, 1], [6, 4, 1], [26, 20, 8, 1], [150, 126, 66, 16, 1], [1082, 962, 602, 212, 32, 1], [9366, 8646, 6126, 2766, 666, 64, 1], [94586, 89546, 69386, 37466, 12266, 2060, 128, 1], [1091670, 1051350, 869910, 537270, 219750, 52926, 6306, 256, 1]] | +| AccRevTabl | [[1], [1, 2], [2, 5, 6], [6, 18, 25, 26], [24, 84, 134, 149, 150], [120, 480, 870, 1050, 1081, 1082], [720, 3240, 6600, 8700, 9302, 9365, 9366], [5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586], [40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670]] | Worpitzky Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670] | -| EvenSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| OddSum | [0, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | -| AccRevSum | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | -| RevAccSum | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | -| DiagSum | [1, 1, 2, 4, 10, 28, 88, 304] | +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670] | +| EvenSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| OddSum | [0, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | +| AccRevSum | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | +| AntiDiagSum | [1, 1, 2, 4, 10, 28, 88, 304, 1144] | Worpitzky Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 84, 600, 145080, 2167200, 453138235200, 319959556963200] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 12, 60, 390, 3360, 31920, 332640] | -| Middle | [1, 1, 3, 7, 50, 180, 2100, 10206, 166824] | -| Central | [1, 3, 50, 2100, 166824] | -| LeftSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RightSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| PosHalf | [1, 3, 12, 66, 480, 4368, 47712, 608016, 8855040] | -| NegHalf | [1, -1, 0, 2, 0, -16, 0, 272, 0] | -| Bin | [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721] | -| InvBin | [1, 0, -3, -10, 25, 574, 2653, -30234, -644079] | -| TransSqrs | [0, 1, 11, 109, 1139, 13021, 164051, 2273629, 34497299] | -| TransNat0 | [0, 1, 7, 49, 391, 3601, 37927, 451249, 5995591] | -| TransNat1 | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 84, 600, 145080, 2167200, 453138235200, 319959556963200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 12, 60, 390, 3360, 31920, 332640] | +| ColMiddle | [1, 1, 3, 7, 50, 180, 2100, 10206, 166824] | +| ColECenter | [1, 3, 50, 2100, 166824] | +| ColOCenter | [1, 7, 180, 10206] | +| ColLeft | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| ColRight | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| TransSqrs | [0, 1, 11, 109, 1139, 13021, 164051, 2273629, 34497299] | +| TransNat0 | [0, 1, 7, 49, 391, 3601, 37927, 451249, 5995591] | +| TransNat1 | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | Worpitzky Diagonals as rows | DiagRow | Seq | @@ -106,10 +102,10 @@ Worpitzky Polynomial values as columns | PolyCol7 | [1, 8, 120, 2696, 80760, 3024008, 135878520, 7123041416, 426748573560] | | PolyCol8 | [1, 9, 153, 3897, 132345, 5618169, 286195833, 17008993017, 1155276253305] | -# WorpitzkyRev -True +# Worpitzky:Rev +[] -WorpitzkyRev Triangle view +Worpitzky:Rev Triangle view | Row | Seq | | :--- | :--- | | Row0 | [1] | @@ -122,47 +118,43 @@ WorpitzkyRev Triangle view | Row7 | [5040, 20160, 31920, 25200, 10206, 1932, 127, 1] | | Row8 | [40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1] | -WorpitzkyRev Flattened triangles -| Flat | Seq | -| :--- | :--- | -| Triangle | [1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1] | -| TriRev | [1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1, 15, 50, 60, 24, 1, 31, 180, 390, 360, 120, 1, 63, 602, 2100, 3360, 2520, 720, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320] | -| TriAcc | [1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670] | -| TriRevAcc | [1, 2, 1, 6, 5, 2, 26, 25, 18, 6, 150, 149, 134, 84, 24, 1082, 1081, 1050, 870, 480, 120, 9366, 9365, 9302, 8700, 6600, 3240, 720, 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040, 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320] | -| TriAccRev | [1, 1, 2, 1, 4, 6, 1, 8, 20, 26, 1, 16, 66, 126, 150, 1, 32, 212, 602, 962, 1082, 1, 64, 666, 2766, 6126, 8646, 9366, 1, 128, 2060, 12266, 37466, 69386, 89546, 94586, 1, 256, 6306, 52926, 219750, 537270, 869910, 1051350, 1091670] | -| TriDiag | [1, 1, 2, 1, 6, 3, 24, 12, 1, 120, 60, 7, 720, 360, 50, 1, 5040, 2520, 390, 15] | - -WorpitzkyRev Row sums -| Sum | Seq | -| :--- | :--- | -| Sum | [1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670] | -| EvenSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| OddSum | [0, 1, 3, 13, 75, 541, 4683, 47293, 545835] | -| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | -| AccSum | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | -| AccRevSum | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | -| RevAccSum | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | -| DiagSum | [1, 1, 3, 9, 37, 187, 1131, 7965] | - -WorpitzkyRev Transforms -| Trans | Seq | -| :--- | :--- | -| RowLcm | [1, 1, 6, 84, 600, 145080, 2167200, 453138235200, 319959556963200] | -| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| RowMax | [1, 1, 3, 12, 60, 390, 3360, 31920, 332640] | -| Middle | [1, 1, 3, 12, 50, 390, 2100, 25200, 166824] | -| Central | [1, 3, 50, 2100, 166824] | -| LeftSide | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | -| RightSide | [1, 1, 1, 1, 1, 1, 1, 1, 1] | -| PosHalf | [1, 3, 15, 111, 1095, 13503, 199815, 3449631, 68062695] | -| NegHalf | [1, -1, 3, -13, 75, -541, 4683, -47293, 545835] | -| Bin | [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721] | -| InvBin | [1, 0, -3, 10, 25, -574, 2653, 30234, -644079] | -| TransSqrs | [0, 1, 7, 49, 411, 4061, 46103, 590857, 8434723] | -| TransNat0 | [0, 1, 5, 29, 209, 1809, 18269, 210853, 2737769] | -| TransNat1 | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | - -WorpitzkyRev Diagonals as rows +Worpitzky:Rev Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [1, 1], [2, 3, 1], [6, 12, 7, 1], [24, 60, 50, 15, 1], [120, 360, 390, 180, 31, 1], [720, 2520, 3360, 2100, 602, 63, 1], [5040, 20160, 31920, 25200, 10206, 1932, 127, 1], [40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1]] | +| RevTabl | [[1], [1, 1], [1, 3, 2], [1, 7, 12, 6], [1, 15, 50, 60, 24], [1, 31, 180, 390, 360, 120], [1, 63, 602, 2100, 3360, 2520, 720], [1, 127, 1932, 10206, 25200, 31920, 20160, 5040], [1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320]] | +| AntiDiag | [[1], [1], [2, 1], [6, 3], [24, 12, 1], [120, 60, 7], [720, 360, 50, 1], [5040, 2520, 390, 15], [40320, 20160, 3360, 180, 1]] | +| AccTabl | [[1], [1, 2], [2, 5, 6], [6, 18, 25, 26], [24, 84, 134, 149, 150], [120, 480, 870, 1050, 1081, 1082], [720, 3240, 6600, 8700, 9302, 9365, 9366], [5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586], [40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670]] | +| RevAccTabl | [[1], [2, 1], [6, 5, 2], [26, 25, 18, 6], [150, 149, 134, 84, 24], [1082, 1081, 1050, 870, 480, 120], [9366, 9365, 9302, 8700, 6600, 3240, 720], [94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040], [1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320]] | +| AccRevTabl | [[1], [1, 2], [1, 4, 6], [1, 8, 20, 26], [1, 16, 66, 126, 150], [1, 32, 212, 602, 962, 1082], [1, 64, 666, 2766, 6126, 8646, 9366], [1, 128, 2060, 12266, 37466, 69386, 89546, 94586], [1, 256, 6306, 52926, 219750, 537270, 869910, 1051350, 1091670]] | + +Worpitzky:Rev Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670] | +| EvenSum | [1, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| OddSum | [0, 1, 3, 13, 75, 541, 4683, 47293, 545835] | +| AltSum | [1, 0, 0, 0, 0, 0, 0, 0, 0] | +| AccSum | [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261] | +| AccRevSum | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | +| AntiDiagSum | [1, 1, 3, 9, 37, 187, 1131, 7965, 64021] | + +Worpitzky:Rev Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 6, 84, 600, 145080, 2167200, 453138235200, 319959556963200] | +| RowGcd | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| RowMax | [1, 1, 3, 12, 60, 390, 3360, 31920, 332640] | +| ColMiddle | [1, 1, 3, 12, 50, 390, 2100, 25200, 166824] | +| ColECenter | [1, 3, 50, 2100, 166824] | +| ColOCenter | [1, 12, 390, 25200] | +| ColLeft | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 7, 49, 411, 4061, 46103, 590857, 8434723] | +| TransNat0 | [0, 1, 5, 29, 209, 1809, 18269, 210853, 2737769] | +| TransNat1 | [1, 3, 11, 55, 359, 2891, 27635, 305439, 3829439] | + +Worpitzky:Rev Diagonals as rows | DiagRow | Seq | | :--- | :--- | | DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| @@ -175,7 +167,7 @@ WorpitzkyRev Diagonals as rows | DiagRow7 | [5040, 181440, 3780000, 59875200, 801496080, 9574044480, 105398092800, 1091804313600, 10794490827120]| | DiagRow8 | [40320, 1814400, 46569600, 898128000, 14495120640, 207048441600, 2706620716800, 33094020960000, 384202115256960]| -WorpitzkyRev Diagonals as columns +Worpitzky:Rev Diagonals as columns | DiagCol | Seq | | :--- | :--- | | DiagCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -188,7 +180,7 @@ WorpitzkyRev Diagonals as columns | DiagCol7 | [1, 255, 18660, 874500, 33105600, 1118557440, 35517081600, 1091804313600, 33094020960000] | | DiagCol8 | [1, 511, 57002, 3669006, 180204024, 7612364760, 294293759760, 10794490827120, 384202115256960] | -WorpitzkyRev Polynomial values as rows +Worpitzky:Rev Polynomial values as rows | PolyRow | Seq | | :--- | :--- | | PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | @@ -201,7 +193,7 @@ WorpitzkyRev Polynomial values as rows | PolyRow7 | [5040, 94586, 608016, 2424132, 7336880, 18532590, 41173776, 83094536, 155612592] | | PolyRow8 | [40320, 1091670, 8855040, 41967540, 145879680, 413749770, 1015741440, 2238150600, 4533828480] | -WorpitzkyRev Polynomial values as columns +Worpitzky:Rev Polynomial values as columns | PolyCol | Seq | | :--- | :--- | | PolyCol0 | [1, 1, 2, 6, 24, 120, 720, 5040, 40320] | @@ -214,3 +206,107 @@ WorpitzkyRev Polynomial values as columns | PolyCol7 | [1, 8, 72, 776, 10440, 174728, 3525192, 83094536, 2238150600] | | PolyCol8 | [1, 9, 90, 1062, 15480, 279864, 6103440, 155612592, 4533828480] | +# Worpitzky:Rev:Inv +[] + +Worpitzky:Rev:Inv Triangle view +| Row | Seq | +| :--- | :--- | +| Row0 | [1] | +| Row1 | [-1, 1] | +| Row2 | [1, -3, 1] | +| Row3 | [-1, 9, -7, 1] | +| Row4 | [1, -45, 55, -15, 1] | +| Row5 | [-1, 585, -835, 285, -31, 1] | +| Row6 | [1, -21105, 30835, -11025, 1351, -63, 1] | +| Row7 | [-1, 1858185, -2719675, 977445, -121891, 6069, -127, 1] | +| Row8 | [1, -367958745, 538607755, -193649085, 24187051, -1213065, 26335, -255, 1] | + +Worpitzky:Rev:Inv Triangles +| Flat | Seq | +| :--- | :--- | +| Tabl | [[1], [-1, 1], [1, -3, 1], [-1, 9, -7, 1], [1, -45, 55, -15, 1], [-1, 585, -835, 285, -31, 1], [1, -21105, 30835, -11025, 1351, -63, 1], [-1, 1858185, -2719675, 977445, -121891, 6069, -127, 1], [1, -367958745, 538607755, -193649085, 24187051, -1213065, 26335, -255, 1]] | +| RevTabl | [[1], [1, -1], [1, -3, 1], [1, -7, 9, -1], [1, -15, 55, -45, 1], [1, -31, 285, -835, 585, -1], [1, -63, 1351, -11025, 30835, -21105, 1], [1, -127, 6069, -121891, 977445, -2719675, 1858185, -1], [1, -255, 26335, -1213065, 24187051, -193649085, 538607755, -367958745, 1]] | +| AntiDiag | [[1], [-1], [1, 1], [-1, -3], [1, 9, 1], [-1, -45, -7], [1, 585, 55, 1], [-1, -21105, -835, -15], [1, 1858185, 30835, 285, 1]] | +| AccTabl | [[1], [-1, 0], [1, -2, -1], [-1, 8, 1, 2], [1, -44, 11, -4, -3], [-1, 584, -251, 34, 3, 4], [1, -21104, 9731, -1294, 57, -6, -5], [-1, 1858184, -861491, 115954, -5937, 132, 5, 6], [1, -367958744, 170649011, -23000074, 1186977, -26088, 247, -8, -7]] | +| RevAccTabl | [[1], [0, -1], [-1, -2, 1], [2, 1, 8, -1], [-3, -4, 11, -44, 1], [4, 3, 34, -251, 584, -1], [-5, -6, 57, -1294, 9731, -21104, 1], [6, 5, 132, -5937, 115954, -861491, 1858184, -1], [-7, -8, 247, -26088, 1186977, -23000074, 170649011, -367958744, 1]] | +| AccRevTabl | [[1], [1, 0], [1, -2, -1], [1, -6, 3, 2], [1, -14, 41, -4, -3], [1, -30, 255, -580, 5, 4], [1, -62, 1289, -9736, 21099, -6, -5], [1, -126, 5943, -115948, 861497, -1858178, 7, 6], [1, -254, 26081, -1186984, 23000067, -170649018, 367958737, -8, -7]] | + +Worpitzky:Rev:Inv Row sums +| Trait | Seq | +| :--- | :--- | +| RowSum | [1, 0, -1, 2, -3, 4, -5, 6, -7] | +| EvenSum | [1, -1, 2, -8, 57, -867, 32188, -2841694, 562821143] | +| OddSum | [0, 1, -3, 10, -60, 871, -32193, 2841700, -562821150] | +| AltSum | [1, -2, 5, -18, 117, -1738, 64381, -5683394, 1125642293] | +| AccSum | [1, -1, -2, 10, -39, 373, -12620, 1106852, -219148685] | +| AccRevSum | [1, 1, -2, 0, 21, -345, 12580, -1106798, 219148615] | +| AntiDiagSum | [1, -1, 2, -4, 11, -53, 642, -21956, 1889307] | + +Worpitzky:Rev:Inv Transforms +| Trans | Seq | +| :--- | :--- | +| RowLcm | [1, 1, 3, 63, 495, 57542355, 125599126275, 16844027146592997737925, 4147431491472964184928637039792321755] | +| RowGcd | [1, 1, 3, 1, 5, 1, 7, 1, 1] | +| RowMax | [1, 1, 3, 9, 55, 835, 30835, 2719675, 538607755] | +| ColMiddle | [1, -1, -3, 9, 55, -835, -11025, 977445, 24187051] | +| ColECenter | [1, -3, 55, -11025, 24187051] | +| ColOCenter | [-1, 9, -835, 977445] | +| ColLeft | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| ColRight | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| TransSqrs | [0, 1, 1, -10, 56, -661, 23087, -2026564, 401232330] | +| TransNat0 | [0, 1, -1, -2, 24, -349, 12585, -1106804, 219148622] | +| TransNat1 | [1, 1, -2, 0, 21, -345, 12580, -1106798, 219148615] | + +Worpitzky:Rev:Inv Diagonals as rows +| DiagRow | Seq | +| :--- | :--- | +| DiagRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1]| +| DiagRow1 | [-1, -3, -7, -15, -31, -63, -127, -255, -511]| +| DiagRow2 | [1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751]| +| DiagRow3 | [-1, -45, -835, -11025, -121891, -1213065, -11291995, -100551825, -868631731]| +| DiagRow4 | [1, 585, 30835, 977445, 24187051, 518499765, 10155703195, 187411686165, 3320927260651]| +| DiagRow5 | [-1, -21105, -2719675, -193649085, -10332652771, -466233951645, -18927199821475, -716488744816845, -25859622364266691]| +| DiagRow6 | [1, 1858185, 538607755, 82717588485, 9290808830611, 868912857645525, 72359887737458035, 5579198891656835445, 408101537756309768371]| +| DiagRow7 | [-1, -367958745, -230061795355, -74376706042485, -17315107997964931, -3321907873772452965, -563456403054687663235, -88047674103739207102245, -13006300970771965124279011]| +| DiagRow8 | [1, 157169540745, 206863086384955, 138614447472278085, 66196730510741081251, 25867235277155609502165, 8892141435741268108524835, 2806102018204396870180667445, 834961274440513999132373870851]| + +Worpitzky:Rev:Inv Diagonals as columns +| DiagCol | Seq | +| :--- | :--- | +| DiagCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| DiagCol1 | [1, -3, 9, -45, 585, -21105, 1858185, -367958745, 157169540745] | +| DiagCol2 | [1, -7, 55, -835, 30835, -2719675, 538607755, -230061795355, 206863086384955] | +| DiagCol3 | [1, -15, 285, -11025, 977445, -193649085, 82717588485, -74376706042485, 138614447472278085] | +| DiagCol4 | [1, -31, 1351, -121891, 24187051, -10332652771, 9290808830611, -17315107997964931, 66196730510741081251] | +| DiagCol5 | [1, -63, 6069, -1213065, 518499765, -466233951645, 868912857645525, -3321907873772452965, 25867235277155609502165] | +| DiagCol6 | [1, -127, 26335, -11291995, 10155703195, -18927199821475, 72359887737458035, -563456403054687663235, 8892141435741268108524835] | +| DiagCol7 | [1, -255, 111645, -100551825, 187411686165, -716488744816845, 5579198891656835445, -88047674103739207102245, 2806102018204396870180667445] | +| DiagCol8 | [1, -511, 465751, -868631731, 3320927260651, -25859622364266691, 408101537756309768371, -13006300970771965124279011, 834961274440513999132373870851] | + +Worpitzky:Rev:Inv Polynomial values as rows +| PolyRow | Seq | +| :--- | :--- | +| PolyRow0 | [1, 1, 1, 1, 1, 1, 1, 1, 1] | +| PolyRow1 | [-1, 0, 1, 2, 3, 4, 5, 6, 7] | +| PolyRow2 | [1, -1, -1, 1, 5, 11, 19, 29, 41] | +| PolyRow3 | [-1, 2, -3, -10, -13, -6, 17, 62, 135] | +| PolyRow4 | [1, -3, 27, 37, -3, -99, -233, -363, -423] | +| PolyRow5 | [-1, 4, -355, -334, 307, 1424, 2609, 3310, 2951] | +| PolyRow6 | [1, -5, 12595, 11377, -11219, -49649, -90305, -115835, -108743] | +| PolyRow7 | [-1, 6, -1106819, -1000306, 981171, 4357174, 7945361, 10230366, 9666439] | +| PolyRow8 | [1, -7, 219148643, 198091693, -194190067, -862703599, -1573469297, -2026393627, -1915092487] | + +Worpitzky:Rev:Inv Polynomial values as columns +| PolyCol | Seq | +| :--- | :--- | +| PolyCol0 | [1, -1, 1, -1, 1, -1, 1, -1, 1] | +| PolyCol1 | [1, 0, -1, 2, -3, 4, -5, 6, -7] | +| PolyCol2 | [1, 1, -1, -3, 27, -355, 12595, -1106819, 219148643] | +| PolyCol3 | [1, 2, 1, -10, 37, -334, 11377, -1000306, 198091693] | +| PolyCol4 | [1, 3, 5, -13, -3, 307, -11219, 981171, -194190067] | +| PolyCol5 | [1, 4, 11, -6, -99, 1424, -49649, 4357174, -862703599] | +| PolyCol6 | [1, 5, 19, 17, -233, 2609, -90305, 7945361, -1573469297] | +| PolyCol7 | [1, 6, 29, 62, -363, 3310, -115835, 10230366, -2026393627] | +| PolyCol8 | [1, 7, 41, 135, -423, 2951, -108743, 9666439, -1915092487] | + diff --git a/src/Abel.py b/src/Abel.py index e6cee60..6e66098 100644 --- a/src/Abel.py +++ b/src/Abel.py @@ -28,7 +28,7 @@ def abel(n: int) -> list[int]: @set_attributes( abel, "Abel", - ['A061356', 'A137452', 'A139526'], + ['A137452', 'A061356', 'A139526'], True) def Abel(n: int, k: int) -> int: return abel(n)[k] diff --git a/src/Bessel.py b/src/Bessel.py index 4f070f8..d6161d9 100644 --- a/src/Bessel.py +++ b/src/Bessel.py @@ -29,7 +29,7 @@ def bessel(n: int) -> list[int]: @set_attributes( bessel, "Bessel", - ['A001497', 'A001498', 'A122850', 'A132062'], + ['A132062', 'A001497', 'A001498', 'A122850'], True) def Bessel(n: int, k: int) -> int: return bessel(n)[k] diff --git a/src/CatalanAer.py b/src/CatalanAer.py index e1701f5..4a97fbe 100644 --- a/src/CatalanAer.py +++ b/src/CatalanAer.py @@ -33,7 +33,7 @@ def r(k: int) -> int: @set_attributes( catalanaer, "CatalanAer", - ['A052173', 'A053121', 'A112554', 'A322378'], + ['A053121', 'A052173', 'A112554', 'A322378'], True) def CatalanAer(n: int, k: int) -> int: return catalanaer(n)[k] diff --git a/src/CentralCyc.py b/src/CentralCyc.py index b76d838..556ddaa 100644 --- a/src/CentralCyc.py +++ b/src/CentralCyc.py @@ -30,7 +30,7 @@ def centralcycle(n: int) -> list[int]: @set_attributes( centralcycle, "CentralCycle", - ['A111999', 'A259456', 'A269940'], + ['A269940', 'A111999', 'A259456'], False) def CentralCycle(n: int, k: int) -> int: return centralcycle(n)[k] diff --git a/src/CentralSet.py b/src/CentralSet.py index 54665e3..943b2a3 100644 --- a/src/CentralSet.py +++ b/src/CentralSet.py @@ -30,7 +30,7 @@ def centralset(n: int) -> list[int]: @set_attributes( centralset, "CentralSet", - ['A008957', 'A036969', 'A269945'], + ['A269945', 'A008957', 'A036969'], True) def CentralSet(n: int, k: int) -> int: return centralset(n)[k] diff --git a/src/ChebyshevT.py b/src/ChebyshevT.py index 99f2ee9..d0dbf8d 100644 --- a/src/ChebyshevT.py +++ b/src/ChebyshevT.py @@ -33,7 +33,7 @@ def chebyshevt(n: int) -> list[int]: @set_attributes( chebyshevt, "ChebyshevT", - ['A039991', 'A053120', 'A081265'], + ['A053120', 'A039991', 'A081265'], True) def ChebyshevT(n: int, k: int) -> int: return chebyshevt(n)[k] diff --git a/src/Euler.py b/src/Euler.py index ee11cfa..69e501d 100644 --- a/src/Euler.py +++ b/src/Euler.py @@ -32,7 +32,7 @@ def euler(n: int) -> list[int]: @set_attributes( euler, "Euler", - ['A109449', 'A247453'], + ['A247453', 'A109449'], True) def Euler(n: int, k: int) -> int: return euler(n)[k] diff --git a/src/Eulerian.py b/src/Eulerian.py index b1591fd..f3db276 100644 --- a/src/Eulerian.py +++ b/src/Eulerian.py @@ -29,7 +29,7 @@ def eulerian(n: int) -> list[int]: @set_attributes( eulerian, "Eulerian", - ['A008292', 'A123125', 'A173018'], + ['A173018', 'A008292', 'A123125'], False) def Eulerian(n: int, k: int) -> int: return eulerian(n)[k] diff --git a/src/Eulerian2.py b/src/Eulerian2.py index 4aafc98..8d1ab2e 100644 --- a/src/Eulerian2.py +++ b/src/Eulerian2.py @@ -30,7 +30,7 @@ def eulerian2(n: int) -> list[int]: @set_attributes( eulerian2, "Eulerian2", - ['A008517', 'A112007', 'A163936', 'A340556'], + ['A340556', 'A008517', 'A112007', 'A163936'], False) def Eulerian2(n: int, k: int) -> int: return eulerian2(n)[k] diff --git a/src/Fibonacci.py b/src/Fibonacci.py index e9f5f6f..bf72f42 100644 --- a/src/Fibonacci.py +++ b/src/Fibonacci.py @@ -33,7 +33,7 @@ def fibonacci(n: int) -> list[int]: @set_attributes( fibonacci, "Fibonacci", - ['A105809', 'A228074', 'A354267'], + ['A354267', 'A105809', 'A228074'], False) def Fibonacci(n: int, k: int) -> int: return fibonacci(n)[k] diff --git a/src/Fubini.py b/src/Fubini.py index 915f9da..88dba74 100644 --- a/src/Fubini.py +++ b/src/Fubini.py @@ -32,7 +32,7 @@ def r(k: int) -> int: @set_attributes( fubini, "Fubini", - ['A019538', 'A090582', 'A131689', 'A278075'], + ['A131689', 'A019538', 'A090582', 'A278075'], False) def Fubini(n: int, k: int) -> int: return fubini(n)[k] diff --git a/src/FussCatalan.py b/src/FussCatalan.py index 7a50cb8..4f665e0 100644 --- a/src/FussCatalan.py +++ b/src/FussCatalan.py @@ -28,7 +28,7 @@ def fusscatalan(n: int) -> list[int]: @set_attributes( fusscatalan, "FussCatalan", - ['A030237', 'A054445', 'A355173'], + ['A355173', 'A030237', 'A054445'], False) def FussCatalan(n: int, k: int) -> int: return fusscatalan(n)[k] diff --git a/src/Harmonic.py b/src/Harmonic.py index 74530d4..2d02f4c 100644 --- a/src/Harmonic.py +++ b/src/Harmonic.py @@ -33,7 +33,7 @@ def harmonic(n: int) -> list[int]: @set_attributes( harmonic, "Harmonic", - ['A109822', 'A358694'], + ['A358694', 'A109822'], True) def Harmonic(n: int, k: int) -> int: return harmonic(n)[k] diff --git a/src/HermiteE.py b/src/HermiteE.py index 18927fc..1ee84a5 100644 --- a/src/HermiteE.py +++ b/src/HermiteE.py @@ -26,7 +26,7 @@ def hermitee(n: int) -> list[int]: @set_attributes( hermitee, "HermiteE", - ['A066325', 'A073278', 'A099174'], + ['A099174', 'A066325', 'A073278'], True) def HermiteE(n: int, k: int) -> int: return hermitee(n)[k] diff --git a/src/LabeledGraphs.py b/src/LabeledGraphs.py new file mode 100644 index 0000000..09f65b6 --- /dev/null +++ b/src/LabeledGraphs.py @@ -0,0 +1,41 @@ +from functools import cache +from Binomial import Binomial +from _tabltypes import set_attributes + +"""Labeled graphs. + +[0] 1; +[1] 0, 1; +[2] 0, 1, 1; +[3] 0, 2, 2, 4; +[4] 0, 8, 6, 12, 38; +[5] 0, 64, 32, 48, 152, 728; +[6] 0, 1024, 320, 320, 760, 3640, 26704; +[7] 0, 32768, 6144, 3840, 6080, 21840, 160224, 1866256; +""" + + +@cache +def labeledgraphs(n: int) -> list[int]: + + if n == 0: return [1] + + s = [2 ** (((k - n + 1) * (k - n)) // 2) * Binomial(n - 1, k - 1) * labeledgraphs(k)[k] for k in range(1, n)] + b = 2 ** (((n - 1) * n) // 2) - sum(s) + + return [0] + s + [b] + + +@set_attributes( + labeledgraphs, + "LabeledGraphs", + ['A360603'], + True) +def LabeledGraphs(n: int, k: int) -> int: + return labeledgraphs(n)[k] + + +if __name__ == "__main__": + from _tabltest import TablTest + + TablTest(LabeledGraphs) \ No newline at end of file diff --git a/src/Lah.py b/src/Lah.py index d025aaf..7ca995b 100644 --- a/src/Lah.py +++ b/src/Lah.py @@ -32,7 +32,7 @@ def lah(n: int) -> list[int]: @set_attributes( lah, "Lah", - ['A008297', 'A066667', 'A089231', 'A105278', 'A111596', 'A271703'], + ['A271703', 'A008297', 'A066667', 'A089231', 'A105278', 'A111596'], True) def Lah(n: int, k: int) -> int: return lah(n)[k] diff --git a/src/Lehmer.py b/src/Lehmer.py index 64d2b47..ed2a6dd 100644 --- a/src/Lehmer.py +++ b/src/Lehmer.py @@ -32,7 +32,7 @@ def lehmer(n: int) -> list[int]: @set_attributes( lehmer, "Lehmer", - ['A039621', 'A354794'], + ['A354794', 'A039621'], True) def Lehmer(n: int, k: int) -> int: return lehmer(n)[k] diff --git a/src/MotzkinGF.py b/src/MotzkinGF.py index d8d7f65..c6dd309 100644 --- a/src/MotzkinGF.py +++ b/src/MotzkinGF.py @@ -33,7 +33,7 @@ def r(k: int) -> int: @set_attributes( motzkingf, "MotzkinGF", - ['A026300', 'A064189', 'A009766'], + ['A064189', 'A026300', 'A009766'], True) def MotzkinGF(n: int, k: int) -> int: return motzkingf(n)[k] diff --git a/src/Narayana.py b/src/Narayana.py index dcb349a..df9e04f 100644 --- a/src/Narayana.py +++ b/src/Narayana.py @@ -37,7 +37,7 @@ def narayana(n: int) -> list[int]: @set_attributes( narayana, "Narayana", - ['A001263', 'A090181', 'A131198'], + ['A090181', 'A001263', 'A131198'], True) def Narayana(n: int, k: int) -> int: return narayana(n)[k] diff --git a/src/OrderedCyc.py b/src/OrderedCyc.py index 2bd817e..beb1c2c 100644 --- a/src/OrderedCyc.py +++ b/src/OrderedCyc.py @@ -31,7 +31,7 @@ def orderedcycle(n: int) -> list[int]: @set_attributes( orderedcycle, "OrderedCycle", - ['A048594', 'A075181', 'A225479'], + ['A225479', 'A048594', 'A075181'], False) def OrderedCycle(n: int, k: int) -> int: return orderedcycle(n)[k] diff --git a/src/Ordinals.py b/src/Ordinals.py index 2b93a73..eb6e1dc 100644 --- a/src/Ordinals.py +++ b/src/Ordinals.py @@ -24,7 +24,7 @@ def ordinals(n: int) -> list[int]: @set_attributes( ordinals, "Ordinals", - ['A002260', 'A002262', 'A004736', 'A025581'], + ['A002262', 'A002260', 'A004736', 'A025581'], False) def Ordinals(n: int, k: int) -> int: return ordinals(n)[k] diff --git a/src/Partition.py b/src/Partition.py index 4c6b82f..78eaf03 100644 --- a/src/Partition.py +++ b/src/Partition.py @@ -33,7 +33,7 @@ def partnumexact(n: int) -> list[int]: @set_attributes( partnumexact, "Partition", - ['A008284', 'A058398', 'A072233'], + ['A072233', 'A008284', 'A058398'], True) def PartnumExact(n: int, k: int) -> int: return partnumexact(n)[k] diff --git a/src/PartitionMax.py b/src/PartitionMax.py index 41211df..93b16a5 100644 --- a/src/PartitionMax.py +++ b/src/PartitionMax.py @@ -3,7 +3,7 @@ from _tabltypes import set_attributes from Partition import partnumexact -"""Partition numbers at most. +"""Euler's table, partition numbers at most. [0] 1 [1] 0, 1 @@ -26,7 +26,7 @@ def partnummax(n: int) -> list[int]: @set_attributes( partnummax, "PartitionMax", - ['A008284', 'A058398', 'A072233'], + ['A026820'], False) def PartnumMax(n: int, k: int) -> int: return partnummax(n)[k] diff --git a/src/Polygonal.py b/src/Polygonal.py index 872d280..95fb084 100644 --- a/src/Polygonal.py +++ b/src/Polygonal.py @@ -44,7 +44,7 @@ def polygonal(n: int) -> list[int]: @set_attributes( polygonal, "Polygonal", - ['A057145', 'A134394', 'A139600', 'A139601'], + ['A139600', 'A057145', 'A134394', 'A139601'], False) def Polygonal(n: int, k: int) -> int: return polygonal(n)[k] diff --git a/src/PowLaguerre.py b/src/PowLaguerre.py index 114a2fe..fc8c643 100644 --- a/src/PowLaguerre.py +++ b/src/PowLaguerre.py @@ -31,7 +31,7 @@ def powlaguerre(n: int) -> list[int]: @set_attributes( powlaguerre, "PowLaguerre", - ['A021012', 'A196347'], + ['A196347', 'A021012'], False) def PowLaguerre(n: int, k: int) -> int: return powlaguerre(n)[k] diff --git a/src/Schroeder.py b/src/Schroeder.py index 68823e4..3b24b02 100644 --- a/src/Schroeder.py +++ b/src/Schroeder.py @@ -32,7 +32,7 @@ def schroeder(n: int) -> list[int]: @set_attributes( schroeder, "Schroeder", - ['A033877', 'A080245', 'A080247', 'A122538', 'A106579'], + ['A122538', 'A033877', 'A080245', 'A080247', 'A106579'], True) def Schroeder(n: int, k: int) -> int: return schroeder(n)[k] diff --git a/src/SchroederL.py b/src/SchroederL.py index 81876df..e2d38c0 100644 --- a/src/SchroederL.py +++ b/src/SchroederL.py @@ -33,7 +33,7 @@ def schroederl(n: int) -> list[int]: @set_attributes( schroederl, - "schroederL", + "SchroederL", ['A172094'], True) def SchroederL(n: int, k: int) -> int: diff --git a/src/SchroederP.py b/src/SchroederP.py index efa4bdf..5f9cffc 100644 --- a/src/SchroederP.py +++ b/src/SchroederP.py @@ -33,7 +33,7 @@ def schroederpaths(n: int) -> list[int]: @set_attributes( schroederpaths, "SchroederP", - ['A063007', 'A104684'], + ['A104684', 'A063007'], True) def SchroederPaths(n: int, k: int) -> int: return schroederpaths(n)[k] diff --git a/src/StirlingCyc.py b/src/StirlingCyc.py index 807a653..cabc180 100644 --- a/src/StirlingCyc.py +++ b/src/StirlingCyc.py @@ -32,7 +32,7 @@ def stirlingcycle(n: int) -> list[int]: @set_attributes( stirlingcycle, "StirlingCyc", - ['A008275', 'A008276', 'A048994', 'A054654', 'A094638', 'A130534', 'A132393'], + ['A132393', 'A008275', 'A008276', 'A048994', 'A054654', 'A094638', 'A130534'], True) def StirlingCycle(n: int, k: int) -> int: return stirlingcycle(n)[k] diff --git a/src/StirlingSet.py b/src/StirlingSet.py index 575b906..cfe7bb6 100644 --- a/src/StirlingSet.py +++ b/src/StirlingSet.py @@ -30,7 +30,7 @@ def stirlingset(n: int) -> list[int]: @set_attributes( stirlingset, "StirlingSet", - ['A008277', 'A008278', 'A048993', 'A080417', 'A106800', 'A151511', 'A151512', 'A154959', 'A213735'], + ['A048993', 'A008277', 'A008278', 'A080417', 'A106800', 'A151511', 'A151512', 'A154959', 'A213735'], True) def StirlingSet(n: int, k: int) -> int: return stirlingset(n)[k] diff --git a/src/SymPoly.py b/src/SymPoly.py index ac7b3c8..3e33b36 100644 --- a/src/SymPoly.py +++ b/src/SymPoly.py @@ -31,7 +31,7 @@ def sympoly(n: int) -> list[int]: @set_attributes( sympoly, "SymPoly", - ['A093905', 'A105954', 'A165674', 'A165675'], + [ 'A165675', 'A093905', 'A105954', 'A165674'], True) def Sympoly(n: int, k: int) -> int: return sympoly(n)[k] diff --git a/src/WardSet.py b/src/WardSet.py index a33c86a..9e3d162 100644 --- a/src/WardSet.py +++ b/src/WardSet.py @@ -32,7 +32,7 @@ def wardset(n: int) -> list[int]: @set_attributes( wardset, "WardSet", - ['A134991', 'A269939'], + ['A269939', 'A134991'], False) def WardSet(n: int, k: int) -> int: return wardset(n)[k] diff --git a/src/_tabldata.py b/src/_tabldata.py index b09736d..a749db8 100644 --- a/src/_tabldata.py +++ b/src/_tabldata.py @@ -4,6 +4,18 @@ # #@ +def fnv(data) -> int: + """ + FNV-1a hash algorithm. + """ + assert isinstance(data, bytes) + + hval = 0xCBF29CE484222325 + for byte in data: + hval = hval ^ byte + hval = (hval * 0x100000001B3) % 0x10000000000000000 + return hval + def get_compressed() -> None: oeisstripped = "https://oeis.org/stripped.gz" @@ -32,11 +44,17 @@ def oeisabsdata() -> None: def GetOEISdata() -> None: get_compressed() oeisabsdata() + print("OEIS data updated!") if __name__ == "__main__": - #get_compressed() - #oeisabsdata() GetOEISdata() print("Done") + + ''' + data = b"1 1 1 3 30 630 3780 207900 8108100 56756700 1929727800 " + print(hex(fnv(data))) + data = b"[1 1 6 84 600 145080 2167200 453138235200 319959556963200 " + print(hex(fnv(data))) + ''' diff --git a/src/_tablmake.py b/src/_tablmake.py index 9a27605..3d42fe4 100644 --- a/src/_tablmake.py +++ b/src/_tablmake.py @@ -44,6 +44,7 @@ "Harmonic.py", "HermiteE.py", "HermiteH.py", + "LabeledGraphs.py", "Laguerre.py", "Lah.py", "Lehmer.py", @@ -65,6 +66,7 @@ "RisingFact.py", "Schroeder.py", "SchroederP.py", + "SchroederL.py", "Seidel.py", "SeidelBoust.py", "Sierpinski.py", @@ -122,6 +124,7 @@ Harmonic, HermiteE, HermiteH, + LabeledGraphs, Laguerre, Lah, Lehmer, @@ -142,6 +145,7 @@ Rencontres, RisingFactorial, Schroeder, + SchroederL, SchroederPaths, Seidel, SeidelBoust, diff --git a/src/_tabltabls.py b/src/_tabltabls.py index 356f0d9..f0a069d 100644 --- a/src/_tabltabls.py +++ b/src/_tabltabls.py @@ -127,6 +127,15 @@ def RegisterTablsTrait(f: Callable[[tabl], trow]) -> None: from Abel import Abel from StirlingSet import StirlingSet - - PrintTabls(Abel, 4) - PrintTabls(StirlingSet, 6, False) + from LabeledGraphs import LabeledGraphs + + #PrintTabls(Abel, 4) + #PrintTabls(StirlingSet, 6, False) + #PrintTabls(LabeledGraphs, 6) + + T = LabeledGraphs.tab(9) + print(T) + print(AccTabl(T)) + print(FlatAccTabl(T)) + #print(InvTabl(T)) + #print(AntiDiagTabl(T)) diff --git a/src/_tabltraitcard.py b/src/_tabltraitcard.py index 7241dbd..77befb1 100644 --- a/src/_tabltraitcard.py +++ b/src/_tabltraitcard.py @@ -1,11 +1,13 @@ from typing import Callable from _tablpaths import GetDataPath, GetCsvPath, GetMdPath from _tabltypes import rgen, tgen, tabl, trow +from _tabltypes import inversion_wrapper, reversion_wrapper, revinv_wrapper, invrev_wrapper from _tablpoly import PolyRow0, PolyRow1, PolyRow2, PolyRow3, PolyCol0, PolyCol1, PolyCol2, PolyCol3, PolyDiag, PosHalf, NegHalf, PolyTabl from _tablsums import RowSum, EvenSum, OddSum, AltSum, AccSum, AccRevSum, AntiDiagSum from _tabltabls import FlatTabl, FlatAccTabl, FlatRevAccTabl, FlatAccRevTabl, FlatAntiDiagTabl, FlatRevTabl, FlatInvTabl, FlatInvRevTabl, FlatRevInvTabl, FlatDiffxTabl from _tabltransforms import FlatBinTabl, FlatInvBinTabl, BinConv, InvBinConv, RowLcm, RowGcd, RowMax, DiagRow0, DiagRow1, DiagRow2, DiagRow3, DiagCol0, DiagCol1, DiagCol2, DiagCol3, TransSqrs, TransNat0, TransNat1, ColMiddle, ColECentral, ColOCentral, ColLeft, ColRight + """ Pretty printing of triangles trait cards. @@ -106,7 +108,7 @@ def RegisterTrait2(f: Callable[[rgen, int], trow]) -> None: def register() -> None: - RegisterTrait(FlatTabl) + RegisterTrait(FlatTabl) # must always come first! RegisterTrait(FlatRevTabl) RegisterTrait(FlatInvTabl) RegisterTrait(FlatRevInvTabl) @@ -176,6 +178,7 @@ def PrintTraits(g: tgen, size: int, trianglename = g.id T = g.tab(size) gen = g.gen + anum = g.sim[0] # Note that the similars are ordered! if markdown: @@ -190,12 +193,15 @@ def PrintTraits(g: tgen, size: int, name = traitname[4:] if traitname.startswith("Flat") else traitname tt = trait(T) if withanum: - anum = '' if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: continue + if anum == '': # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + # print(traitname) + if onlythefound: continue seqstr = SeqToFixlenString(tt, 70, ' ') print(f'| {trianglename} | {anum:7} | {name:<12} | {seqstr} |') + anum = '' else: seqstr = SeqToFixlenString(tt, 70, ' ') print(f'| {trianglename} | {name:<12} | {seqstr} |') @@ -212,12 +218,15 @@ def PrintTraits(g: tgen, size: int, for traitname, trait in TRAIT2.items(): tt = trait(gen, size) if withanum: - anum = '' if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: continue + if anum == '': # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + # print(traitname) + if onlythefound: continue seqstr = SeqToFixlenString(tt, 70, ' ') print(f'| {trianglename} | {anum:7} | {traitname:<12} | {seqstr} |') + anum = '' else: seqstr = SeqToFixlenString(tt, 70, ' ') print(f'| {trianglename} | {traitname:<12} | {seqstr} |') @@ -228,6 +237,7 @@ def PrintTraits(g: tgen, size: int, seqstr = SeqToFixlenString(trait(gen, size), 70, ' ') print(f'{trianglename}:{traitname:<14} {seqstr}') + def SaveTraitsToFile(g: tgen, size: int, withanum: bool = False, markdown: bool = True, @@ -236,6 +246,8 @@ def SaveTraitsToFile(g: tgen, size: int, trianglename = g.id T = g.tab(size) gen = g.gen + anum = g.sim[0] # Note that the similars are ordered! + print(anum) if markdown: filepath = (GetMdPath() / f"{trianglename}.md").resolve() @@ -266,12 +278,16 @@ def SaveTraitsToFile(g: tgen, size: int, name = traitname[4:] if traitname.startswith("Flat") else traitname tt = trait(T) if withanum: - anum = '' if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: continue + if anum == '': # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: continue + print(anum) seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'| {trianglename} | {anum:7} | {name:<12} | {seqstr} |\n') + anum = '' else: seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'| {trianglename} | {name:<12} | {seqstr} |\n') @@ -283,12 +299,15 @@ def SaveTraitsToFile(g: tgen, size: int, tt = trait(T) if withanum: - anum = '' if tt == [] else GetAnumber(tt) - if anum == "": - print(traitname) - if onlythefound: continue - seqstr = SeqToFixlenString(tt, 70, ' ') + if anum == '': # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: continue + seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'{trianglename},{anum},{name},{seqstr}\n') + anum = '' else: seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'{trianglename},{name},{seqstr}\n') @@ -298,12 +317,15 @@ def SaveTraitsToFile(g: tgen, size: int, for traitname, trait in TRAIT2.items(): tt = trait(gen, size) if withanum: - anum = '' if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: continue + if anum == '': # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: continue seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'| {trianglename} | {anum:7} | {traitname:<12} | {seqstr} |\n') + anum = '' else: seqstr = SeqToFixlenString(tt, 70, ' ') target.write(f'| {trianglename} | {traitname:<12} | {seqstr} |\n') @@ -324,8 +346,6 @@ def SaveTraitsToFile(g: tgen, size: int, target.write(f'{trianglename},{traitname},{seqstr}\n') -from _tabltypes import inversion_wrapper, reversion_wrapper, revinv_wrapper, invrev_wrapper - def PrintExtendedTraits(T: tgen, size: int, withanum: bool = False, markdown: bool = True) -> None: tim: int = size + size // 2 @@ -454,19 +474,31 @@ def SaveAllExtendedTraitsToCSV() -> None: from Binomial import Binomial from CatalanSqr import CatalanSqr from PowLaguerre import PowLaguerre + from LabeledGraphs import LabeledGraphs + + register() + + # SaveAllTraitsToCSV() # SaveExtendedTraitsToCSV(StirlingSet, 20) # SaveAllExtendedTraitsToCSV() - # SaveAllFoundTraitsToCSV() - register() - SaveAllTraitsToCSV() - - # SaveTraitsToFile(PowLaguerre, 20, - # withanum = True, - # markdown = False, - # onlythefound = False) + + #SaveTraitsToFile(LabGraphs, 20, + # withanum = True, + # markdown = False, + # onlythefound = False) + + #PrintTraits(Lah, 20, + # withanum = True, + # markdown = True, + # onlythefound = False) + + SaveTraitsToFile(Lah, 20, + withanum = True, + markdown = False, + onlythefound = False) # SEQ = StirlingSet #PrintTraits(SEQ, 12, withanum = False, markdown = False) diff --git a/src/tabl.py b/src/tabl.py index 3e1c88d..6ca6fe3 100644 --- a/src/tabl.py +++ b/src/tabl.py @@ -1090,7 +1090,7 @@ def abel(n: int) -> list[int]: return [b[k - 1] * n ** (n - k) if k > 0 else 0 for k in range(n + 1)] -@set_attributes(abel, "Abel", ["A061356", "A137452", "A139526"], True) +@set_attributes(abel, "Abel", ["A137452", "A061356", "A139526"], True) def Abel(n: int, k: int) -> int: return abel(n)[k] @@ -1139,7 +1139,7 @@ def bessel(n: int) -> list[int]: return row -@set_attributes(bessel, "Bessel", ["A001497", "A001498", "A122850", "A132062"], True) +@set_attributes(bessel, "Bessel", ["A132062", "A001497", "A001498", "A122850"], True) def Bessel(n: int, k: int) -> int: return bessel(n)[k] @@ -1230,7 +1230,7 @@ def r(k: int) -> int: @set_attributes( - catalanaer, "CatalanAer", ["A052173", "A053121", "A112554", "A322378"], True + catalanaer, "CatalanAer", ["A053121", "A052173", "A112554", "A322378"], True ) def CatalanAer(n: int, k: int) -> int: return catalanaer(n)[k] @@ -1266,7 +1266,7 @@ def centralcycle(n: int) -> list[int]: return row -@set_attributes(centralcycle, "CentralCycle", ["A111999", "A259456", "A269940"], False) +@set_attributes(centralcycle, "CentralCycle", ["A269940", "A111999", "A259456"], False) def CentralCycle(n: int, k: int) -> int: return centralcycle(n)[k] @@ -1283,7 +1283,7 @@ def centralset(n: int) -> list[int]: return row -@set_attributes(centralset, "CentralSet", ["A008957", "A036969", "A269945"], True) +@set_attributes(centralset, "CentralSet", ["A269945", "A008957", "A036969"], True) def CentralSet(n: int, k: int) -> int: return centralset(n)[k] @@ -1322,7 +1322,7 @@ def chebyshevt(n: int) -> list[int]: return row -@set_attributes(chebyshevt, "ChebyshevT", ["A039991", "A053120", "A081265"], True) +@set_attributes(chebyshevt, "ChebyshevT", ["A053120", "A039991", "A081265"], True) def ChebyshevT(n: int, k: int) -> int: return chebyshevt(n)[k] @@ -1416,7 +1416,7 @@ def euler(n: int) -> list[int]: return row -@set_attributes(euler, "Euler", ["A109449", "A247453"], True) +@set_attributes(euler, "Euler", ["A247453", "A109449"], True) def Euler(n: int, k: int) -> int: return euler(n)[k] @@ -1435,7 +1435,7 @@ def eulerian(n: int) -> list[int]: return row -@set_attributes(eulerian, "Eulerian", ["A008292", "A123125", "A173018"], False) +@set_attributes(eulerian, "Eulerian", ["A173018", "A008292", "A123125"], False) def Eulerian(n: int, k: int) -> int: return eulerian(n)[k] @@ -1453,7 +1453,7 @@ def eulerian2(n: int) -> list[int]: @set_attributes( - eulerian2, "Eulerian2", ["A008517", "A112007", "A163936", "A340556"], False + eulerian2, "Eulerian2", ["A340556", "A008517", "A112007", "A163936"], False ) def Eulerian2(n: int, k: int) -> int: return eulerian2(n)[k] @@ -1548,7 +1548,7 @@ def fibonacci(n: int) -> list[int]: return row -@set_attributes(fibonacci, "Fibonacci", ["A105809", "A228074", "A354267"], False) +@set_attributes(fibonacci, "Fibonacci", ["A354267", "A105809", "A228074"], False) def Fibonacci(n: int, k: int) -> int: return fibonacci(n)[k] @@ -1567,7 +1567,7 @@ def r(k: int) -> int: return row -@set_attributes(fubini, "Fubini", ["A019538", "A090582", "A131689", "A278075"], False) +@set_attributes(fubini, "Fubini", ["A131689", "A019538", "A090582", "A278075"], False) def Fubini(n: int, k: int) -> int: return fubini(n)[k] @@ -1582,7 +1582,7 @@ def fusscatalan(n: int) -> list[int]: return list(accumulate(row)) -@set_attributes(fusscatalan, "FussCatalan", ["A030237", "A054445", "A355173"], False) +@set_attributes(fusscatalan, "FussCatalan", ["A355173", "A030237", "A054445"], False) def FussCatalan(n: int, k: int) -> int: return fusscatalan(n)[k] @@ -1637,7 +1637,7 @@ def harmonic(n: int) -> list[int]: return row -@set_attributes(harmonic, "Harmonic", ["A109822", "A358694"], True) +@set_attributes(harmonic, "Harmonic", ["A358694", "A109822"], True) def Harmonic(n: int, k: int) -> int: return harmonic(n)[k] @@ -1651,7 +1651,7 @@ def hermitee(n: int) -> list[int]: return row -@set_attributes(hermitee, "HermiteE", ["A066325", "A073278", "A099174"], True) +@set_attributes(hermitee, "HermiteE", ["A099174", "A066325", "A073278"], True) def HermiteE(n: int, k: int) -> int: return hermitee(n)[k] @@ -1670,6 +1670,26 @@ def HermiteH(n: int, k: int) -> int: return hermiteh(n)[k] +@cache +def labeledgraphs(n: int) -> list[int]: + if n == 0: + return [1] + s = [ + 2 ** (((k - n + 1) * (k - n)) // 2) + * Binomial(n - 1, k - 1) + * labeledgraphs(k)[k] + for k in range(1, n) + ] + b = 2 ** (((n - 1) * n) // 2) - sum(s) + + return [0] + s + [b] + + +@set_attributes(labeledgraphs, "LabeledGraphs", ["A360603"], True) +def LabeledGraphs(n: int, k: int) -> int: + return labeledgraphs(n)[k] + + @cache def laguerre(n: int) -> list[int]: if n == 0: @@ -1697,7 +1717,7 @@ def lah(n: int) -> list[int]: @set_attributes( - lah, "Lah", ["A008297", "A066667", "A089231", "A105278", "A111596", "A271703"], True + lah, "Lah", ["A271703", "A008297", "A066667", "A089231", "A105278", "A111596"], True ) def Lah(n: int, k: int) -> int: return lah(n)[k] @@ -1717,7 +1737,7 @@ def lehmer(n: int) -> list[int]: return [t(k - 1, n - k, n - k) if n != k else 1 for k in range(n + 1)] -@set_attributes(lehmer, "Lehmer", ["A039621", "A354794"], True) +@set_attributes(lehmer, "Lehmer", ["A354794", "A039621"], True) def Lehmer(n: int, k: int) -> int: return lehmer(n)[k] @@ -1806,7 +1826,7 @@ def r(k: int) -> int: return row -@set_attributes(motzkingf, "MotzkinGF", ["A026300", "A064189", "A009766"], True) +@set_attributes(motzkingf, "MotzkinGF", ["A064189", "A026300", "A009766"], True) def MotzkinGF(n: int, k: int) -> int: return motzkingf(n)[k] @@ -1825,7 +1845,7 @@ def narayana(n: int) -> list[int]: return row -@set_attributes(narayana, "Narayana", ["A001263", "A090181", "A131198"], True) +@set_attributes(narayana, "Narayana", ["A090181", "A001263", "A131198"], True) def Narayana(n: int, k: int) -> int: return narayana(n)[k] @@ -1865,7 +1885,7 @@ def ordinals(n: int) -> list[int]: @set_attributes( - ordinals, "Ordinals", ["A002260", "A002262", "A004736", "A025581"], False + ordinals, "Ordinals", ["A002262", "A002260", "A004736", "A025581"], False ) def Ordinals(n: int, k: int) -> int: return ordinals(n)[k] @@ -1884,7 +1904,7 @@ def orderedcycle(n: int) -> list[int]: return row -@set_attributes(orderedcycle, "OrderedCycle", ["A048594", "A075181", "A225479"], False) +@set_attributes(orderedcycle, "OrderedCycle", ["A225479", "A048594", "A075181"], False) def OrderedCycle(n: int, k: int) -> int: return orderedcycle(n)[k] @@ -1903,7 +1923,7 @@ def partnumexact(n: int) -> list[int]: return [part(n, k) for k in range(n + 1)] -@set_attributes(partnumexact, "Partition", ["A008284", "A058398", "A072233"], True) +@set_attributes(partnumexact, "Partition", ["A072233", "A008284", "A058398"], True) def PartnumExact(n: int, k: int) -> int: return partnumexact(n)[k] @@ -1913,7 +1933,7 @@ def partnummax(n: int) -> list[int]: return list(accumulate(partnumexact(n))) -@set_attributes(partnummax, "PartitionMax", ["A008284", "A058398", "A072233"], False) +@set_attributes(partnummax, "PartitionMax", ["A026820"], False) def PartnumMax(n: int, k: int) -> int: return partnummax(n)[k] @@ -1933,7 +1953,7 @@ def polygonal(n: int) -> list[int]: @set_attributes( - polygonal, "Polygonal", ["A057145", "A134394", "A139600", "A139601"], False + polygonal, "Polygonal", ["A139600", "A057145", "A134394", "A139601"], False ) def Polygonal(n: int, k: int) -> int: return polygonal(n)[k] @@ -1950,7 +1970,7 @@ def powlaguerre(n: int) -> list[int]: return row -@set_attributes(powlaguerre, "PowLaguerre", ["A021012", "A196347"], False) +@set_attributes(powlaguerre, "PowLaguerre", ["A196347", "A021012"], False) def PowLaguerre(n: int, k: int) -> int: return powlaguerre(n)[k] @@ -2009,7 +2029,7 @@ def schroeder(n: int) -> list[int]: @set_attributes( schroeder, "Schroeder", - ["A033877", "A080245", "A080247", "A122538", "A106579"], + ["A122538", "A033877", "A080245", "A080247", "A106579"], True, ) def Schroeder(n: int, k: int) -> int: @@ -2027,11 +2047,30 @@ def schroederpaths(n: int) -> list[int]: return row -@set_attributes(schroederpaths, "SchroederP", ["A063007", "A104684"], True) +@set_attributes(schroederpaths, "SchroederP", ["A104684", "A063007"], True) def SchroederPaths(n: int, k: int) -> int: return schroederpaths(n)[k] +@cache +def schroederl(n: int) -> list[int]: + if n == 0: + return [1] + if n == 1: + return [1, 1] + arow: list[int] = schroederl(n - 1) + [0] + row: list[int] = schroederl(n - 1) + [1] + row[0] = row[0] + 2 * row[1] + for k in range(1, n): + row[k] = arow[k - 1] + 3 * arow[k] + 2 * arow[k + 1] + return row + + +@set_attributes(schroederl, "SchroederL", ["A172094"], True) +def SchroederL(n: int, k: int) -> int: + return schroederl(n)[k] + + @cache def seidel(n: int) -> list[int]: if n == 0: @@ -2089,7 +2128,7 @@ def stirlingcycle(n: int) -> list[int]: @set_attributes( stirlingcycle, "StirlingCyc", - ["A008275", "A008276", "A048994", "A054654", "A094638", "A130534", "A132393"], + ["A132393", "A008275", "A008276", "A048994", "A054654", "A094638", "A130534"], True, ) def StirlingCycle(n: int, k: int) -> int: @@ -2149,9 +2188,9 @@ def stirlingset(n: int) -> list[int]: stirlingset, "StirlingSet", [ + "A048993", "A008277", "A008278", - "A048993", "A080417", "A106800", "A151511", @@ -2229,7 +2268,7 @@ def sympoly(n: int) -> list[int]: return row -@set_attributes(sympoly, "SymPoly", ["A093905", "A105954", "A165674", "A165675"], True) +@set_attributes(sympoly, "SymPoly", ["A165675", "A093905", "A105954", "A165674"], True) def Sympoly(n: int, k: int) -> int: return sympoly(n)[k] @@ -2261,7 +2300,7 @@ def wardset(n: int) -> list[int]: return row -@set_attributes(wardset, "WardSet", ["A134991", "A269939"], False) +@set_attributes(wardset, "WardSet", ["A269939", "A134991"], False) def WardSet(n: int, k: int) -> int: return wardset(n)[k] @@ -2320,6 +2359,7 @@ def Worpitzky(n: int, k: int) -> int: Harmonic, HermiteE, HermiteH, + LabeledGraphs, Laguerre, Lah, Lehmer, @@ -2340,6 +2380,7 @@ def Worpitzky(n: int, k: int) -> int: Rencontres, RisingFactorial, Schroeder, + SchroederL, SchroederPaths, Seidel, SeidelBoust, @@ -2641,6 +2682,18 @@ def AllCsvToHtml(csvpath: Path = GetCsvPath(), outpath: Path = GetHtmlPath()) -> CsvToHtml(fun, csvpath, outpath) +def fnv(data) -> int: + """ + FNV-1a hash algorithm. + """ + assert isinstance(data, bytes) + hval = 0xCBF29CE484222325 + for byte in data: + hval = hval ^ byte + hval = (hval * 0x100000001B3) % 0x10000000000000000 + return hval + + def get_compressed() -> None: oeisstripped = "https://oeis.org/stripped.gz" r = requests.get(oeisstripped, stream=True) @@ -2746,7 +2799,7 @@ def RegisterTrait2(f: Callable[[rgen, int], trow]) -> None: def register() -> None: - RegisterTrait(FlatTabl) + RegisterTrait(FlatTabl) # must always come first! RegisterTrait(FlatRevTabl) RegisterTrait(FlatInvTabl) RegisterTrait(FlatRevInvTabl) @@ -2811,6 +2864,7 @@ def PrintTraits( trianglename = g.id T = g.tab(size) gen = g.gen + anum = g.sim[0] # Note that the similars are ordered! if markdown: if withanum: print("| Triangle | Anum | Trait | Sequence |") @@ -2822,13 +2876,16 @@ def PrintTraits( name = traitname[4:] if traitname.startswith("Flat") else traitname tt = trait(T) if withanum: - anum = "" if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: - continue + if anum == "": # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + # print(traitname) + if onlythefound: + continue seqstr = SeqToFixlenString(tt, 70, " ") print(f"| {trianglename} | {anum:7} | {name:<12} | {seqstr} |") + anum = "" else: seqstr = SeqToFixlenString(tt, 70, " ") print(f"| {trianglename} | {name:<12} | {seqstr} |") @@ -2841,13 +2898,16 @@ def PrintTraits( for traitname, trait in TRAIT2.items(): tt = trait(gen, size) if withanum: - anum = "" if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: - continue + if anum == "": # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + # print(traitname) + if onlythefound: + continue seqstr = SeqToFixlenString(tt, 70, " ") print(f"| {trianglename} | {anum:7} | {traitname:<12} | {seqstr} |") + anum = "" else: seqstr = SeqToFixlenString(tt, 70, " ") print(f"| {trianglename} | {traitname:<12} | {seqstr} |") @@ -2867,6 +2927,8 @@ def SaveTraitsToFile( trianglename = g.id T = g.tab(size) gen = g.gen + anum = g.sim[0] # Note that the similars are ordered! + print(anum) if markdown: filepath = (GetMdPath() / f"{trianglename}.md").resolve() else: @@ -2889,15 +2951,19 @@ def SaveTraitsToFile( name = traitname[4:] if traitname.startswith("Flat") else traitname tt = trait(T) if withanum: - anum = "" if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: - continue + if anum == "": # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: + continue + print(anum) seqstr = SeqToFixlenString(tt, 70, " ") target.write( f"| {trianglename} | {anum:7} | {name:<12} | {seqstr} |\n" ) + anum = "" else: seqstr = SeqToFixlenString(tt, 70, " ") target.write(f"| {trianglename} | {name:<12} | {seqstr} |\n") @@ -2907,13 +2973,16 @@ def SaveTraitsToFile( tt = trait(T) if withanum: - anum = "" if tt == [] else GetAnumber(tt) - if anum == "": - print(traitname) - if onlythefound: - continue + if anum == "": # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: + continue seqstr = SeqToFixlenString(tt, 70, " ") target.write(f"{trianglename},{anum},{name},{seqstr}\n") + anum = "" else: seqstr = SeqToFixlenString(tt, 70, " ") target.write(f"{trianglename},{name},{seqstr}\n") @@ -2921,15 +2990,18 @@ def SaveTraitsToFile( for traitname, trait in TRAIT2.items(): tt = trait(gen, size) if withanum: - anum = "" if tt == [] else GetAnumber(tt) - if anum != "": - print(traitname) - if onlythefound: - continue + if anum == "": # which is always the case exept in the first loop + if tt != []: + anum = GetAnumber(tt) + if anum == "": + print(traitname) + if onlythefound: + continue seqstr = SeqToFixlenString(tt, 70, " ") target.write( f"| {trianglename} | {anum:7} | {traitname:<12} | {seqstr} |\n" ) + anum = "" else: seqstr = SeqToFixlenString(tt, 70, " ") target.write(f"| {trianglename} | {traitname:<12} | {seqstr} |\n") diff --git a/src/tablupdate.py b/src/tablupdate.py new file mode 100644 index 0000000..b679f29 --- /dev/null +++ b/src/tablupdate.py @@ -0,0 +1,34 @@ +from pathlib import Path +path = Path(__file__).parent +mkpath = (path / "src/_tablmake.py").resolve() + +'''Make sure to reference tabl.py in its current state.''' +exec(open(mkpath).read()) + +from tabl import GetOEISdata, SaveAllTraitsToCSV, AllCsvToHtml, SaveExtendedTables, CrossReferences + +''' +DO NOT USE this function if you only want to update tabl.py. +If you want to update table.py after you have added a new +triangle implementation just run "src/_tablmake.py". +As a normal user this is all you want and it is quick. + +On the other hand this function updates all data related to the +defined integer triangles (including downloading the latest +compressed database from the OEIS), updates the corresponding +csv files and the corresponding html files as well as the +crossreferences. This will take a considerably amount of time. +Normally this is done only when preparing a new release. +''' +def update() -> None: + GetOEISdata() + SaveAllTraitsToCSV() + AllCsvToHtml() + SaveExtendedTables() + CrossReferences() + + +if __name__ == "__main__": + + '''Update project.''' + update()