A292892 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 * (exp(x) - 1)).

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570

Offset: 0

Author

Seiichi Manyama, Sep 26 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1,  1,  1,  1,   1, ...
   1,  0,  0,  0,   0, ...
   2,  2,  0,  0,   0, ...
   5,  3,  6,  0,   0, ...
  15, 16, 12, 24,   0, ...
  52, 65, 20, 60, 120, ...
		

Links

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

Crossrefs

Columns k=0..3 give A000110, A052506, A240989, A292891.

Rows n=0..1 give A000012, A000007.

Main diagonal gives A000007.

Cf. A292894, A355607.

Programs

PARI

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.

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