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.

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

Offset

1,5

Comments

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}.

Links

Table of n, a(n) for n=1..32.

Weisstein, Eric W. Beta Function.

Formula

M(i,j)=if[i>=,1/Gamma(i,j),0);i,j<=n IM(i,j)=Inverse(M(i,j))

Example

{1},
{0, 1},
{1, -2, 1},
{6, -13, 8, -1},
{720, -1566, 973, -128, 1},
{3628800, -7893360, 4905486, -646093, 5168, -1}

Mathematica

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}]];

Crossrefs

Sequence in context: A128534 A002562 A218492 * A123968 A282329 A343806

Adjacent sequences: A136453 A136454 A136455 * A136457 A136458 A136459

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Mar 20 2008

Status

approved