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A136455 Characteristic polynomials of the inverse beta function based matrices as a triangle of integer coefficients: n*IM(i,j) = n*Inverse(1/Gamma(i,j)); i,j>=n. 1

%I

%S 1,0,1,-1,1,1,-48,28,25,-1,233280,-91368,-60993,2305,1,222953472000,

%T -65503641600,-33198846720,985867696,446161,-1,-69132994560000000000,

%U 16249035196800000000,6593300559405000000,-157196644177875000,-59060479175425,144069601,1

%N Characteristic polynomials of the inverse beta function based matrices as a triangle of integer coefficients: n*IM(i,j) = n*Inverse(1/Gamma(i,j)); 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 The row sums are:

%C 1, 1, 1, 4, 83225, 125237297536, -46447914508307980823,

%C -7674877258831704425297015668284,

%C 826995023940325227060455484795500305924239025,

%C 78707026870454795917065346205715405165749494298851023916447744,

%C -8564339911597252330069530473576149288702875987041232549818479682054967832279248559, -1336289257117044917306250532620084989502533702227664929930431988712778911134146754043951495782103516647100

%D Weisstein, Eric W. "Beta Function." http : // mathworld.wolfram.com/BetaFunction.html

%H Roger L. Bagula, Mar 20 2008, <a href="/A136455/b136455.txt">Table of n, a(n) for n = 1..78</a>

%F M(i,j)=1/Gamma[i+j]; i,j<=n IM(i,j)=Inverse(M(i,j))

%e {1},

%e {0, 1},

%e {-1, 1, 1},

%e {-48, 28, 25, -1},

%e {233280, -91368, -60993, 2305, 1},

%e {222953472000, -65503641600, -33198846720, 985867696, 446161, -1}

%t M[w_] := Table[Table[1/Gamma[n + m], {n, 0, w}], {m, 0, w}] IM[w_] := Inverse[M[w]] Join[{1, x}, Table[CharacteristicPolynomial[n*IM[n], x], {n, 1, 10}]] a = Join[{{1}, {0,1}}, Table[CoefficientList[CharacteristicPolynomial[n*IM[n], x], x], {n, 1, 10}]]; Flatten[a] Join[{1, 1}, Table[Apply[Plus, CoefficientList[CharacteristicPolynomial[n*IM[n], x], x]], {n, 1, 10}]]

%K uned,tabl,sign

%O 1,7

%A _Roger L. Bagula_, Mar 20 2008

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Last modified October 26 22:19 EDT 2021. Contains 348269 sequences. (Running on oeis4.)