A382823 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)) ).

1, 1, 1, 2, 2, 2, 6, 5, 5, 6, 24, 17, 17, 17, 24, 120, 74, 69, 69, 74, 120, 720, 394, 338, 337, 338, 394, 720, 5040, 2484, 1962, 1894, 1894, 1962, 2484, 5040, 40320, 18108, 13228, 12194, 12152, 12194, 13228, 18108, 40320, 362880, 149904, 101812, 89160, 87320, 87320, 89160, 101812, 149904, 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)) ).

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

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

Example

Square array begins:
    1,   1,    2,     6,    24,    120, ...
    1,   2,    5,    17,    74,    394, ...
    2,   5,   17,    69,   338,   1962, ...
    6,  17,   69,   337,  1894,  12194, ...
   24,  74,  338,  1894, 12152,  87320, ...
  120, 394, 1962, 12194, 87320, 696076, ...

Prog

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

Crossrefs

Columns k=0..1 give A000142, A000774.

Main diagonal gives A382826.

Cf. A382824, A382825.

Cf. A099594, A379821.

Sequence in context: A210740 A209820 A145890 * A097091 A094204 A088681

Adjacent sequences: A382820 A382821 A382822 * A382824 A382825 A382826

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 05 2025

Status

approved