A392817 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. 1/(1 - x*(exp(x) - 1)^k).

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 6, 28, 120, 1, 0, 0, 0, 24, 125, 720, 1, 0, 0, 0, 24, 70, 1146, 5040, 1, 0, 0, 0, 0, 180, 900, 8827, 40320, 1, 0, 0, 0, 0, 120, 900, 10514, 94200, 362880, 1, 0, 0, 0, 0, 0, 1440, 3780, 88368, 1007001, 3628800

Offset

0,6

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..495 (rows 0..31, flattened)

Formula

A(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (k*j)! * Stirling2(n-j,k*j)/(n-j)!.

Example

Square array begins:
    1,    1,   1,   1,    1,   1, 1, ...
    1,    0,   0,   0,    0,   0, 0, ...
    2,    2,   0,   0,    0,   0, 0, ...
    6,    3,   6,   0,    0,   0, 0, ...
   24,   28,  24,  24,    0,   0, 0, ...
  120,  125,  70, 180,  120,   0, 0, ...
  720, 1146, 900, 900, 1440, 720, 0, ...

Mathematica

a[n_, k_]:=n! Sum[(k j)! StirlingS2[n-j, k j]/(n-j)!, {j, 0, Floor[n/(k+1)]}];
seq78=Module[{N=10, K=10, A}, A=Table[a[n, k], {n, 0, N}, {k, 0, K}]; Flatten[Table[A[[i+1, s-i+1]], {s, 0, N+K}, {i, Max[0, s-K], Min[s, N]}]][[1;; 78]]];
seq78 (* Vincenzo Librandi, Feb 03 2026 *)

Prog

(PARI) a(n, k) = n!*sum(j=0, n\(k+1), (k*j)!*stirling(n-j, k*j, 2)/(n-j)!);

Crossrefs

Columns k=0..3 give A000142, A052848, A392820, A392821.

Cf. A357868, A392822.

Sequence in context: A295688 A355610 A355609 * A392822 A266994 A383140

Adjacent sequences: A392814 A392815 A392816 * A392818 A392819 A392820

Keyword

nonn,tabl,easy

Author

Seiichi Manyama, Jan 24 2026

Status

approved