A389641 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..floor(n/k)} n!/((j!)^k*(n - j*k)!) for k > 0 and A(n, 0) = 1.

1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 8, 3, 1, 1, 1, 16, 7, 1, 1, 1, 1, 32, 19, 7, 1, 1, 1, 1, 64, 51, 25, 1, 1, 1, 1, 1, 128, 141, 61, 25, 1, 1, 1, 1, 1, 256, 393, 211, 121, 1, 1, 1, 1, 1, 1, 512, 1107, 841, 361, 121, 1, 1, 1, 1, 1, 1, 1024, 3139, 2857, 841, 721, 1, 1, 1, 1, 1, 1

Offset

0,5

Links

Table of n, a(n) for n=0..77.

Formula

For the transposed form C(k, n) = A(n, k) and k >= 1 one has

C(k, n) = hypergeom(s(n, k), t(k), (-1)^k*k^k), where s(n, k) = [(j - n)/k, j = 0..k-1] and t(k) = [1, j = 1...k-1].

A(n, n) = n! + 1 for n > 0, cf. A038507.

A(n, k) = 1 for k > n.

Example

Array starts:
  [0]  1,   1,    1,    1,     1,    1,      1,      1,      1,      1, ...
  [1]  1,   2,    1,    1,     1,    1,      1,      1,      1,      1, ...
  [2]  1,   4,    3,    1,     1,    1,      1,      1,      1,      1, ...
  [3]  1,   8,    7,    7,     1,    1,      1,      1,      1,      1, ...
  [4]  1,  16,   19,   25,    25,    1,      1,      1,      1,      1, ...
  [5]  1,  32,   51,   61,   121,   121,     1,      1,      1,      1, ...
  [6]  1,  64,  141,  211,   361,   721,   721,      1,      1,      1, ...
  [7]  1, 128,  393,  841,   841,  2521,  5041,   5041,      1,      1, ...
  [8]  1, 256, 1107, 2857,  4201,  6721, 20161,  40321,  40321,      1, ...
  [9]  1, 512, 3139, 9745, 25705, 15121, 60481, 181441, 362881, 362881, ...

Maple

A := (n, k) -> ifelse(k = 0 or k > n, 1, add(n!/((j!)^k*(n - j*k)!), j = 0..floor(n/k))):
for n from 0 to 9 do seq(A(n, k), k = 0..9) od;
# Alternative: computing the columns:
s := (n, m) -> [seq((j - n)/m, j = 0..m-1)]: t := m -> [seq(1, j = 1...m-1)]:
C := (k, n) -> hypergeom(s(n, k), t(k), (-1)^k*k^k):
for k from 1 to 6 do seq(simplify(C(k, j)), j = 0..9) od;

Prog

(PARI) T(n, k) = if ((k==0) || (k>n), 1, sum(j=0, n\k, n!/((j!)^k*(n - j*k)!))); \\ Michel Marcus, Oct 22 2025

Crossrefs

Columns: A000012, A002426, A344560, A361637, A361703.

Cf. A038507.

Sequence in context: A205553 A178411 A257598 * A294580 A294587 A064645

Adjacent sequences: A389638 A389639 A389640 * A389642 A389643 A389644

Keyword

nonn,tabl

Author

Peter Luschny, Oct 22 2025

Status

approved