A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1

Offset: 0

Author

Seiichi Manyama, Jul 03 2020

Keywords

• sign
• tabl
• look

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...
		

Links

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

Crossrefs

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.

Main diagonal gives A334241.

Cf. A292861, A335975.

Programs

Mathematica

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

Formula

T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

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