A292894 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 * (1 - exp(x))).

1, 1, -1, 1, 0, 0, 1, 0, -2, 1, 1, 0, 0, -3, 1, 1, 0, 0, -6, 8, -2, 1, 0, 0, 0, -12, 55, -9, 1, 0, 0, 0, -24, -20, 84, -9, 1, 0, 0, 0, 0, -60, 330, -637, 50, 1, 0, 0, 0, 0, -120, -120, 2478, -4992, 267, 1, 0, 0, 0, 0, 0, -360, -210, 11704, -10593, 413, 1, 0, 0, 0, 0, 0, -720, -840, 19824, -15192, 92060, -2180

Offset: 0

Author

Seiichi Manyama, Sep 26 2017

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,  1,   1,   1,    1, ...
  -1,  0,   0,   0,    0, ...
   0, -2,   0,   0,    0, ...
   1, -3,  -6,   0,    0, ...
   1,  8, -12, -24,    0, ...
  -2, 55, -20, -60, -120, ...
		

Links

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

Crossrefs

Columns k=0..2 give A000587, A292893, A292951.

Rows n=0..1 give A000012, (-1)*A000007.

Main diagonal gives A000007.

Cf. A292892, A355607.

Programs

PARI

T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*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))} (-1)^j * 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)