A382825 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-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).

1, 1, 1, 2, 4, 2, 6, 11, 11, 6, 24, 39, 55, 39, 24, 120, 174, 255, 255, 174, 120, 720, 942, 1338, 1623, 1338, 942, 720, 5040, 6012, 8106, 10434, 10434, 8106, 6012, 5040, 40320, 44244, 56292, 72762, 82116, 72762, 56292, 44244, 40320, 362880, 369072, 442860, 560988, 668580, 668580, 560988, 442860, 369072, 362880

Offset: 0

Author

Seiichi Manyama, Apr 06 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,   1,    2,     6,     24,     120, ...
    1,   4,   11,    39,    174,     942, ...
    2,  11,   55,   255,   1338,    8106, ...
    6,  39,  255,  1623,  10434,   72762, ...
   24, 174, 1338, 10434,  82116,  668580, ...
  120, 942, 8106, 72762, 668580, 6302028, ...
		

Crossrefs

Main diagonal gives A382828.

Cf. A382823, A382824.

Cf. A382673, A382800.

Programs

PARI

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

Formula

E.g.f.: 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).

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

A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * |Stirling1(n+1,j+1)| * |Stirling1(k+1,j+1)|.