A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).

1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730

Offset: 0

Author

Seiichi Manyama, Feb 20 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
      1,   1,  1,  1, 1, 1, ...
      1,   0,  0,  0, 0, 0, ...
      4,   1,  0,  0, 0, 0, ...
     21,   3,  1,  0, 0, 0, ...
    148,  12,  4,  1, 0, 0, ...
   1305,  70, 10,  5, 1, 0, ...
  13806, 465, 40, 15, 6, 1, ...
		

Crossrefs

Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.

Cf. A143398, A351761.

Programs

PARI

T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));

PARI

T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

Formula

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

T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022