A382824 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))^2 ).

1, 1, 1, 2, 3, 2, 6, 8, 8, 6, 24, 28, 34, 28, 24, 120, 124, 150, 150, 124, 120, 720, 668, 768, 854, 768, 668, 720, 5040, 4248, 4584, 5204, 5204, 4584, 4248, 5040, 40320, 31176, 31512, 35188, 37556, 35188, 31512, 31176, 40320, 362880, 259488, 246072, 265896, 290380, 290380, 265896, 246072, 259488, 362880

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f.: 1 / ( (1-x) * (1-y) * (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+1,j+1)| * |Stirling1(k+1,j+1)|.

Example

Square array begins:
    1,   1,    2,     6,     24,     120, ...
    1,   3,    8,    28,    124,     668, ...
    2,   8,   34,   150,    768,    4584, ...
    6,  28,  150,   854,   5204,   35188, ...
   24, 124,  768,  5204,  37556,  290380, ...
  120, 668, 4584, 35188, 290380, 2546852, ...

Prog

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

Crossrefs

Main diagonal gives A382827.

Cf. A382823, A382825.

Sequence in context: A209582 A158279 A153984 * A025502 A393409 A110777

Adjacent sequences: A382821 A382822 A382823 * A382825 A382826 A382827

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 05 2025

Status

approved