A355610 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k/k!).

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 3, 20, 120, 1, 0, 0, 0, 6, 90, 720, 1, 0, 0, 0, 4, 20, 594, 5040, 1, 0, 0, 0, 0, 10, 180, 4200, 40320, 1, 0, 0, 0, 0, 5, 40, 1134, 34544, 362880, 1, 0, 0, 0, 0, 0, 15, 210, 7980, 316008, 3628800, 1, 0, 0, 0, 0, 0, 6, 70, 1904, 71280, 3207240, 39916800

Offset: 0

Author

Seiichi Manyama, Jul 09 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,   1,   1,  1,  1, 1, 1, ...
    1,   0,   0,  0,  0, 0, 0, ...
    2,   2,   0,  0,  0, 0, 0, ...
    6,   3,   3,  0,  0, 0, 0, ...
   24,  20,   6,  4,  0, 0, 0, ...
  120,  90,  20, 10,  5, 0, 0, ...
  720, 594, 180, 40, 15, 6, 0, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=0..4 give A000142, A066166, A351492, A351493, A355507.

Cf. A355609, A355619.

Programs

PARI

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

Formula

T(0,k) = 1 and T(n,k) = (n-1)!/k! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).