A155163 - OEIS

%I #27 Sep 01 2018 09:22:10

%S 2,9,3,64,52,4,625,855,195,5,7776,15306,6546,606,6,117649,305571,

%T 201866,38486,1701,7,2097152,6806472,6244680,1950320,194160,4488,8,

%U 43046721,168205743,200503701,90665595,15597315,887949,11367,9

%N 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.

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

%H Muniru A Asiru, <a href="/A155163/b155163.txt">Table of n, a(n) for n = 1..1275</a>

%H Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

%F 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

%F 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

%e [n\k][     0           1          2        3         4       5      6   7]

%e [1]        2;

%e [2]        9,         3;

%e [3]       64,        52,         4;

%e [4]      625,       855,       195,        5;

%e [5]     7776,     15306,      6546,      606,        6;

%e [6]   117649,    305571,    201866,    38486,     1701,      7;

%e [7]  2097152,   6806472,   6244680,  1950320,   194160,   4488,     8;

%e [8] 43046721, 168205743, 200503701, 90665595, 15597315, 887949, 11367, 9;

%p A155163 := proc(n,k)

%p         -(x-1)^(2*n+1)*add(x^(j-n)*(j+1)^n*binomial(j,n),j=0..n+10) ;

%p         coeftayl(%,x=0,k) ;

%p end proc: # _R. J. Mathar_, Feb 13 2013

%t 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}];

%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%o (Maxima)

%o 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 */

%o (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

%Y Cf. A202017.

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 21 2009