A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 16, 0, 1, 0, 8, 9, 56, 65, 0, 1, 0, 10, 12, 120, 250, 336, 0, 1, 0, 12, 15, 208, 555, 1812, 1897, 0, 1, 0, 14, 18, 320, 980, 5148, 12614, 11824, 0, 1, 0, 16, 21, 456, 1525, 11064, 39711, 101040, 80145, 0

Offset

0,9

Links

Table of n, a(n) for n=0..65.

Formula

E.g.f. of column k: exp(k * x * (exp(x) - 1)).

Example

Square array begins:
  1,  1,   1,   1,   1,    1, ...
  0,  0,   0,   0,   0,    0, ...
  0,  2,   4,   6,   8,   10, ...
  0,  3,   6,   9,  12,   15, ...
  0, 16,  56, 120, 208,  320, ...
  0, 65, 250, 555, 980, 1525, ...

Prog

(PARI) T(n, k) = n!*sum(j=0, n\2, k^j*stirling(n-j, j, 2)/(n-j)!);

Crossrefs

Columns k=0..3 give: A000007, A052506, A351733, A351734.

Main diagonal gives (-1)^n * A290158(n).

Cf. A362834.

Sequence in context: A134317 A123641 A217377 * A362834 A362839 A362837

Adjacent sequences: A361649 A361650 A361651 * A361653 A361654 A361655

Keyword

nonn,tabl

Author

Seiichi Manyama, May 05 2023

Status

approved