A382734 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^2.

1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 14, 2, 0, 0, 2, 38, 38, 2, 0, 0, 2, 86, 254, 86, 2, 0, 0, 2, 182, 1118, 1118, 182, 2, 0, 0, 2, 374, 4142, 8654, 4142, 374, 2, 0, 0, 2, 758, 14078, 51662, 51662, 14078, 758, 2, 0, 0, 2, 1526, 45614, 267566, 467102, 267566, 45614, 1526, 2, 0

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

E.g.f.: 1 / (exp(x) + exp(y) - exp(x+y))^2.

A(n,k) = A(k,n).

A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * Stirling2(n,j) * Stirling2(k,j).

Example

Square array begins:
  1, 0,   0,    0,     0,      0, ...
  0, 2,   2,    2,     2,      2, ...
  0, 2,  14,   38,    86,    182, ...
  0, 2,  38,  254,  1118,   4142, ...
  0, 2,  86, 1118,  8654,  51662, ...
  0, 2, 182, 4142, 51662, 467102, ...

Prog

(PARI) a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2));

Crossrefs

Main diagonal gives A382737.

Cf. A371761, A382735, A382736.

Cf. A136126, A382740.

Sequence in context: A099766 A194947 A132339 * A333941 A137676 A333755

Adjacent sequences: A382731 A382732 A382733 * A382735 A382736 A382737

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 04 2025

Status

approved