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%I #9 Oct 31 2023 10:26:38
%S 1,0,1,1,-2,1,6,-13,8,-1,720,-1566,973,-128,1,3628800,-7893360,
%T 4905486,-646093,5168,-1,1316818944000,-2864346105600,1780110653040,
%U -234459133326,1876009933,-368048,1,52563198423859200000,-114335531944833024000,71056323779613177600,-9358860113257929840
%N 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.
%C Based on:
%C Beta[n,m]=Gamma[n]*Gamma[m]/Gamma[n+m]=Integate[x^n&(1-x)^m,{x,0,1}];
%C f[x,n]=x^n/Gamma[n]
%C g[x,n]=(1-x)^n/Gamma[n]
%C Integral:
%C Matrix[n,m]=Integrate[f[x,n]*g[x,m],{x,0,1}]=1/Gamma[n,m]
%C IM[n]=n*Inverse[Matrix[n,m]]
%C These matrices are made to be like the transorthogonal or simplex coding:
%C -1/(2^n-1)
%C 1/Gamma[n+m] is mostly less than that.
%C These results get really big really fast.
%C The Cornelius-Schultz lower triangular form makes them smaller and the row sums are mostly zero.
%C The row sums are {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}.
%H Weisstein, Eric W. <a href="http://mathworld.wolfram.com/BetaFunction.html">Beta Function</a>.
%F M(i,j)=if[i>=,1/Gamma(i,j),0);i,j<=n IM(i,j)=Inverse(M(i,j))
%e {1},
%e {0, 1},
%e {1, -2, 1},
%e {6, -13, 8, -1},
%e {720, -1566, 973, -128, 1},
%e {3628800, -7893360, 4905486, -646093, 5168, -1}
%t 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}]];
%K uned,tabl,sign
%O 1,5
%A _Roger L. Bagula_, Mar 20 2008
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