Search: id:A136456
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%I A136456
%S A136456 1,0,1,1,2,1,6,13,8,1,720,1566,973,128,1,3628800,7893360,4905486,646093,
               5168,1,
%T A136456 1316818944000,2864346105600,1780110653040,234459133326,1876009933,368048,
               1,
%U A136456 52563198423859200000,114335531944833024000,71056323779613177600,9358860113257929840
%V A136456 1,0,1,1,-2,1,6,-13,8,-1,720,-1566,973,-128,1,3628800,-7893360,4905486,
               -646093,5168,-1,
%W A136456 1316818944000,-2864346105600,1780110653040,-234459133326,1876009933,-368048,
               1,
%X A136456 52563198423859200000,-114335531944833024000,71056323779613177600,-9358860113257929840
%N A136456 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 A136456 Based on:
%C A136456 Beta[n,m]=Gamma[n]*Gamma[m]/Gamma[n+m]=Integate[x^n&(1-x)^m,{x,0,1}];
%C A136456 f[x,n]=x^n/Gamma[n]
%C A136456 g[x,n]=(1-x)^n/Gamma[n]
%C A136456 Integral:
%C A136456 Matrix[n,m]=Integrate[f[x,n]*g[x,m],{x,0,1}]=1/Gamma[n,m]
%C A136456 IM[n]=n*Inverse[Matrix[n,m]]
%C A136456 These matrices are made to be like the transorthogonal or simplex coding 
               :
%C A136456 -1/(2^n-1)
%C A136456 1/Gamma[n+m] is mostly less than that.
%C A136456 These results get really big really fast.
%C A136456 The Cornelius -Schultz lower triangular form makes them smaller and the
%C A136456 row sums are mostly zero.
%C A136456 The row sums are:
%C A136456 {1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
%D A136456 Weisstein, Eric W. "Beta Function." http : // mathworld.wolfram.com/BetaFunction.html
%F A136456 M(i,j)=if[i>=,1/Gamma(i,j),0);i,j<=n IM(i,j)=Inverse(M(i,j))
%e A136456 {1},
%e A136456 {0, 1},
%e A136456 {1, -2, 1},
%e A136456 {6, -13, 8, -1},
%e A136456 {720, -1566, 973, -128, 1},
%e A136456 {3628800, -7893360, 4905486, -646093, 5168, -1}
%t A136456 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}]];
%Y A136456 Sequence in context: A081064 A128534 A002562 this_sequence A123968 A068797 
               A049951
%Y A136456 Adjacent sequences: A136453 A136454 A136455 this_sequence A136457 A136458 
               A136459
%K A136456 uned,tabl,sign
%O A136456 1,5
%A A136456 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 20 2008

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