A202995 Triangle read by rows, based on expansion of (x^2*exp(x)/(exp(x)-1))^m = x^m + sum(n>m T(n,m)*m!/((n-m)!*n!)*x^n).

1, 1, 1, 1, 3, 1, 0, 10, 6, 1, -4, 30, 40, 10, 1, 0, 36, 270, 110, 15, 1, 120, -420, 1596, 1260, 245, 21, 1, 0, -2400, 5040, 14056, 4200, 476, 28, 1, -12096, 30240, -46080, 136080, 72576, 11340, 840, 36, 1, 0, 423360, -756000, 795600, 1197000, 276192, 26460, 1380, 45, 1

Offset

1,5

Comments

Triangle T(n,m)*m!/((n-m)!*n!)=

1. Riordan Array (1,x^2*exp(x)/(exp(x)-1)) without first column.

2. Riordan Array (x*exp(x)/(exp(x)-1),x^2*exp(x)/(exp(x)-1)) numbering triangle (0,0).

Links

Table of n, a(n) for n=1..55.

Formula

T(n,m):=(n!*(n-m)!*sum(j=0..m, (j!*binomial(m,j)*sum(k=0..n-2*m+j, (k!*stirling1(k+j,j)*stirling2(n-2*m+j,k))/((n-2*m+j)!*(k+j)!)))))/m!.

Example

1,
1, 1,
1, 3, 1,
0, 10, 6, 1,
-4, 30, 40, 10, 1,
0, 36, 270, 110, 15, 1,
120, -420, 1596, 1260, 245, 21, 1

Prog

(Maxima) T(n, m):=(n!*(n-m)!*sum((j!*binomial(m, j)*sum((k!*stirling1(k+j, j)*stirling2(n-2*m+j, k))/((n-2*m+j)!*(k+j)!), k, 0, n-2*m+j)), j, 0, m))/m!;

Crossrefs

Sequence in context: A261765 A079669 A143398 * A191578 A288385 A245667

Adjacent sequences: A202992 A202993 A202994 * A202996 A202997 A202998

Keyword

sign,tabl,changed

Author

Vladimir Kruchinin, Dec 27 2011

Status

approved