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<n.

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

1,1

Comments

Row sums are A001813: 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400.

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.

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

Example

[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;

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

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[%]

Prog

(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 */
(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

Crossrefs

Cf. A202017.

Sequence in context: A162615 A275716 A295197 * A309929 A339799 A161934

Adjacent sequences: A155160 A155161 A155162 * A155164 A155165 A155166

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Jan 21 2009

Status

approved