A382801 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (1 - log(1-x) * log(1-y))^2 - 1).

1, 1, 1, 2, 7, 2, 6, 20, 20, 6, 24, 72, 130, 72, 24, 120, 324, 642, 642, 324, 120, 720, 1764, 3468, 4794, 3468, 1764, 720, 5040, 11304, 21372, 32964, 32964, 21372, 11304, 5040, 40320, 83448, 150120, 238404, 290976, 238404, 150120, 83448, 40320

Offset: 1

Author

Seiichi Manyama, Apr 05 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,    1,     2,      6,      24,      120, ...
    1,    7,    20,     72,     324,     1764, ...
    2,   20,   130,    642,    3468,    21372, ...
    6,   72,   642,   4794,   32964,   238404, ...
   24,  324,  3468,  32964,  290976,  2524080, ...
  120, 1764, 21372, 238404, 2524080, 26048256, ...
		

Crossrefs

Cf. A382740, A382799.

Programs

PARI

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

Formula

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

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

A(n,k) = (1/2) * A382799(n,k).