A355650 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k/k! * (exp(x) - 1)).

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570

Offset: 0

Author

Seiichi Manyama, Jul 12 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, ...
    5,   3,   3,  0,  0, 0, 0, ...
   15,  16,   6,  4,  0, 0, 0, ...
   52,  65,  10, 10,  5, 0, 0, ...
  203, 336, 105, 20, 15, 6, 0, ...
		

Crossrefs

Columns k=0..3 give A000110, A052506, A354000, A354001.

Cf. A145460, A292892, A355610.

Programs

PARI

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(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))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).