A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0

Offset

0,9

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Formula

For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).

Example

Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  1,   0,   0,   0,   0, ...
  0,  2,   2,   0,   0,   0, ...
  0,  5,   6,   6,   0,   0, ...
  0, 15,  26,  36,  24,   0, ...
  0, 52, 150, 150, 240, 120, ...

Prog

(PARI) T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));

Crossrefs

Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.

Cf. A324162, A357293, A357868.

Sequence in context: A343156 A163577 A132178 * A039655 A357882 A103775

Adjacent sequences: A357866 A357867 A357868 * A357870 A357871 A357872

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 17 2022

Status

approved