A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 20, 0, 1, 0, 8, 9, 64, 90, 0, 1, 0, 10, 12, 132, 300, 594, 0, 1, 0, 12, 15, 224, 630, 2568, 4200, 0, 1, 0, 14, 18, 340, 1080, 6642, 20160, 34544, 0, 1, 0, 16, 21, 480, 1650, 13536, 55440, 193856, 316008, 0

Offset

0,9

Links

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

Formula

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

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, 20,  64, 132,  224,  340, ...
  0, 90, 300, 630, 1080, 1650, ...

Prog

(PARI) T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^j*stirling(n-j, j, 1)/(n-j)!);

Crossrefs

Columns k=0..3 give: A000007, A066166, A053489, A053490.

Main diagonal gives A318615.

Cf. A361652.

Sequence in context: A123641 A217377 A361652 * A362839 A362837 A276193

Adjacent sequences: A362831 A362832 A362833 * A362835 A362836 A362837

Keyword

nonn,tabl

Author

Seiichi Manyama, May 05 2023

Status

approved