A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 0

Offset

0,9

Links

Table of n, a(n) for n=0..65.

Formula

E.g.f. of column k: exp(x * (exp(k * x) - 1)).

G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).

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

Example

Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  0,   0,    0,    0,     0, ...
  0,  2,   4,    6,    8,    10, ...
  0,  3,  12,   27,   48,    75, ...
  0, 16,  80,  216,  448,   800, ...
  0, 65, 560, 2025, 5120, 10625, ...

Prog

(PARI) T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);

Crossrefs

Columns k=0..3 give: A000007, A052506, A351736, A351737.

Main diagonal gives A356806.

Cf. A362652.

Sequence in context: A217377 A361652 A362834 * A362837 A276193 A357400

Adjacent sequences: A362836 A362837 A362838 * A362840 A362841 A362842

Keyword

nonn,tabl

Author

Seiichi Manyama, May 05 2023

Status

approved