A155163
Triangle T(n,k): the coefficient of [x^k] of the series -(x-1)^(2*n+1) *Sum_{j>=0} (j+1)^n *binomial(j,n) * x^(j-n); columns 0<=k
2, 9, 3, 64, 52, 4, 625, 855, 195, 5, 7776, 15306, 6546, 606, 6, 117649, 305571, 201866, 38486, 1701, 7, 2097152, 6806472, 6244680, 1950320, 194160, 4488, 8, 43046721, 168205743, 200503701, 90665595, 15597315, 887949, 11367, 9
Offset: 1
Examples
[n\k][ 0 1 2 3 4 5 6 7] [1] 2; [2] 9, 3; [3] 64, 52, 4; [4] 625, 855, 195, 5; [5] 7776, 15306, 6546, 606, 6; [6] 117649, 305571, 201866, 38486, 1701, 7; [7] 2097152, 6806472, 6244680, 1950320, 194160, 4488, 8; [8] 43046721, 168205743, 200503701, 90665595, 15597315, 887949, 11367, 9;
Links
- Muniru A Asiru, Table of n, a(n) for n = 1..1275
- Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Crossrefs
Cf. A202017.
Programs
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GAP
T := Flat(List([1..50], n->List([1..n], m->Sum([1..n], k->Factorial(k) * (-1)^(n+m+k+1) * Stirling2(n,k) * Binomial(n-k,m-1) * Binomial(n+k,k))))); # Muniru A Asiru, Jan 27 2018
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Maple
A155163 := proc(n,k) -(x-1)^(2*n+1)*add(x^(j-n)*(j+1)^n*binomial(j,n),j=0..n+10) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Feb 13 2013
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Mathematica
Clear[p, x, n, m]; p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[( k + 1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%] -
Maxima
T(n,m):=sum(k!*(-1)^(n+m+k+1)*stirling2(n,k)*binomial(n-k,m-1)*binomial(n+k,k),k,1,n); /* Vladimir Kruchinin, Jan 27 2018 */
Formula
T(n,m) = Sum_{k=1..n} k!*(-1)^(n+m+k+1)*Stirling2(n,k)*C(n-k,m-1)*C(n+k,k). - Vladimir Kruchinin, Jan 27 2018
E.g.f. A(x,y) = E(A(x,y),y), where E(x,y)=(1-y)/(exp(x*(y-1))-y) - e.g.f. Eulerian numbers (A173018). - Vladimir Kruchinin, Aug 31 2018
Comments