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A136456
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Characteristic polynomials of the Inverse Beta function based matrices as a triangle of Integer coefficients: (lower triangular form: Cornelius-Schultz form) n*IM(i,j)=Inverse(if[i>=,1/Gamma(i,j),0));i.j>=n.
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0
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1, 0, 1, 1, -2, 1, 6, -13, 8, -1, 720, -1566, 973, -128, 1, 3628800, -7893360, 4905486, -646093, 5168, -1, 1316818944000, -2864346105600, 1780110653040, -234459133326, 1876009933, -368048, 1, 52563198423859200000, -114335531944833024000, 71056323779613177600, -9358860113257929840
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OFFSET
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1,5
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COMMENTS
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Based on:
Beta[n,m]=Gamma[n]*Gamma[m]/Gamma[n+m]=Integate[x^n&(1-x)^m,{x,0,1}];
f[x,n]=x^n/Gamma[n]
g[x,n]=(1-x)^n/Gamma[n]
Integral:
Matrix[n,m]=Integrate[f[x,n]*g[x,m],{x,0,1}]=1/Gamma[n,m]
IM[n]=n*Inverse[Matrix[n,m]]
These matrices are made to be like the transorthogonal or simplex coding:
-1/(2^n-1)
1/Gamma[n+m] is mostly less than that.
These results get really big really fast.
The Cornelius-Schultz lower triangular form makes them smaller and the row sums are mostly zero.
The row sums are {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}.
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LINKS
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FORMULA
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M(i,j)=if[i>=,1/Gamma(i,j),0);i,j<=n IM(i,j)=Inverse(M(i,j))
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EXAMPLE
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{1},
{0, 1},
{1, -2, 1},
{6, -13, 8, -1},
{720, -1566, 973, -128, 1},
{3628800, -7893360, 4905486, -646093, 5168, -1}
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MATHEMATICA
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M[w_] := Table[Table[If[n - m == 0 && n == 0 && m == 0, 1, If[n >= m, 1/Gamma[n + m], 0]], {n, 0, w}], {m, 0, w}]; TableForm[Table[M[w], {w, 0, 5}]; ] TableForm[Table[Inverse[M[w]], {w, 0, 5}]]; IM[w_] := Inverse[M[w]]; Join[{1, x}, Table[CharacteristicPolynomial[IM[n], x], {n, 1, 10}]]; a = Join[{{1}, {0, 1}}, Table[CoefficientList[CharacteristicPolynomial[IM[ n], x], x], {n, 1, 10}]]; Flatten[a] Join[{1, 1}, Table[Apply[Plus, CoefficientList[ CharacteristicPolynomial[IM[n], x], x]], {n, 1, 10}]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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