A382799 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^2.

1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 14, 4, 0, 0, 12, 40, 40, 12, 0, 0, 48, 144, 260, 144, 48, 0, 0, 240, 648, 1284, 1284, 648, 240, 0, 0, 1440, 3528, 6936, 9588, 6936, 3528, 1440, 0, 0, 10080, 22608, 42744, 65928, 65928, 42744, 22608, 10080, 0, 0, 80640, 166896, 300240, 476808, 581952, 476808, 300240, 166896, 80640, 0

Offset: 0

Author

Seiichi Manyama, Apr 05 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  0,   0,    0,     0,      0, ...
  0,  2,   2,    4,    12,     48, ...
  0,  2,  14,   40,   144,    648, ...
  0,  4,  40,  260,  1284,   6936, ...
  0, 12, 144, 1284,  9588,  65928, ...
  0, 48, 648, 6936, 65928, 581952, ...
		

Crossrefs

Main diagonal gives A382804.

Cf. A379821, A382800.

Cf. A382734, A382801.

Programs

PARI

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

Formula

E.g.f.: 1 / (1 - log(1-x) * log(1-y))^2.

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

A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * |Stirling1(n,j)| * |Stirling1(k,j)|.