A355619 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, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 3, 20, 0, 1, 0, 0, 0, -6, -90, 0, 1, 0, 0, 0, 4, 20, 594, 0, 1, 0, 0, 0, 0, -10, 0, -4200, 0, 1, 0, 0, 0, 0, 5, 40, -126, 34544, 0, 1, 0, 0, 0, 0, 0, -15, -210, 1260, -316008, 0, 1, 0, 0, 0, 0, 0, 6, 70, 1904, -4320, 3207240, 0

Offset: 0

Author

Seiichi Manyama, Jul 10 2022

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   1,  1,   1,   1, 1, 1, ...
  1,   0,  0,   0,   0, 0, 0, ...
  0,   2,  0,   0,   0, 0, 0, ...
  0,  -3,  3,   0,   0, 0, 0, ...
  0,  20, -6,   4,   0, 0, 0, ...
  0, -90, 20, -10,   5, 0, 0, ...
  0, 594,  0,  40, -15, 6, 0, ...
		

Links

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

Crossrefs

Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.

Cf. A355607, A355610.

Programs

PARI

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