A355607 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).

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0

Offset: 0

Author

Seiichi Manyama, Jul 09 2022

Keywords

• sign
• tabl
• look

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,   6,   0,    0,   0, 0, ...
  0,  20, -12,  24,    0,   0, 0, ...
  0, -90,  40, -60,  120,   0, 0, ...
  0, 594, 180, 240, -360, 720, 0, ...
		

Links

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

Crossrefs

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.

Cf. A292892, A355609, A355619.

Programs

PARI

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = -(n-1)! * 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)/(n-k*j)!.