A357882 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|/j!.

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 6, 0, 1, 0, 0, 6, 24, 0, 1, 0, 0, 6, 34, 120, 0, 1, 0, 0, 0, 36, 220, 720, 0, 1, 0, 0, 0, 24, 210, 1688, 5040, 0, 1, 0, 0, 0, 0, 240, 1710, 14868, 40320, 0, 1, 0, 0, 0, 0, 120, 2040, 17304, 147684, 362880, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 194712, 1631376, 3628800, 0

Offset: 0

Author

Seiichi Manyama, Oct 18 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   2,   2,   0,   0,   0, ...
  0,   6,   6,   6,   0,   0, ...
  0,  24,  34,  36,  24,   0, ...
  0, 120, 220, 210, 240, 120, ...
		

Crossrefs

Columns k=0-5 give: A000007, A000142, (-1)^n * A009199(n), A353344, A353358, A353404.

Cf. A357119, A357869, A357881.

Programs

PARI

T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1))/j!);

PARI

T(n, k) = if(k==0, 0^n, n!*polcoef(exp((-log(1-x+x*O(x^n)))^k), n));

Formula

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

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(j,k)| * T(n-j,k).