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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
1, 0, 1, -1, 1, 1, -48, 28, 25, -1, 233280, -91368, -60993, 2305, 1, 222953472000, -65503641600, -33198846720, 985867696, 446161, -1, -69132994560000000000, 16249035196800000000, 6593300559405000000, -157196644177875000, -59060479175425, 144069601, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

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.

The row sums are:

1, 1, 1, 4, 83225, 125237297536, -46447914508307980823,

-7674877258831704425297015668284,

826995023940325227060455484795500305924239025,

78707026870454795917065346205715405165749494298851023916447744,

-8564339911597252330069530473576149288702875987041232549818479682054967832279248559, -1336289257117044917306250532620084989502533702227664929930431988712778911134146754043951495782103516647100

REFERENCES

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

LINKS

Roger L. Bagula, Mar 20 2008, Table of n, a(n) for n = 1..78

FORMULA

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

EXAMPLE

{1},

{0, 1},

{-1, 1, 1},

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

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

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

MATHEMATICA

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}]]

CROSSREFS

Sequence in context: A128380 A298451 A094658 * A085517 A299525 A033694

Adjacent sequences:  A136452 A136453 A136454 * A136456 A136457 A136458

KEYWORD

uned,tabl,sign

AUTHOR

Roger L. Bagula, Mar 20 2008

STATUS

approved

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Last modified September 18 19:30 EDT 2021. Contains 347534 sequences. (Running on oeis4.)