A379821 Array read by ascending antidiagonals: A(n, k) = (-1)^(n + k) * Sum_{j=0..k} (j!)^2 * Stirling1(n, j) * Stirling1(k, j).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 5, 2, 0, 0, 6, 14, 14, 6, 0, 0, 24, 50, 76, 50, 24, 0, 0, 120, 224, 360, 360, 224, 120, 0, 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0, 0, 5040, 7776, 11628, 15664, 15664, 11628, 7776, 5040, 0

Offset

0,12

Links

Table of n, a(n) for n=0..54.

Example

Array begins:
  [0] 1,   0,    0,     0,      0,       0,        0,        0, ...
  [1] 0,   1,    1,     2,      6,      24,      120,      720, ...
  [2] 0,   1,    5,    14,     50,     224,     1216,     7776, ...
  [3] 0,   2,   14,    76,    360,    1908,    11628,    81072, ...
  [4] 0,   6,   50,   360,   2392,   15664,   110336,   856080, ...
  [5] 0,  24,  224,  1908,  15664,  126676,  1046780,  9169920, ...
  [6] 0, 120, 1216, 11628, 110336, 1046780, 10057204, 99846144, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 0,   0;
  [2] 0,   1,    0;
  [3] 0,   1,    1,    0;
  [4] 0,   2,    5,    2,    0;
  [5] 0,   6,   14,   14,    6,    0;
  [6] 0,  24,   50,   76,   50,   24,    0;
  [7] 0, 120,  224,  360,  360,  224,  120,   0;
  [8] 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0;

Maple

A := (n, k) -> local j; (-1)^(n + k)*add((j!)^2*Stirling1(n, j)*Stirling1(k, j), j = 0..k):
seq(lprint(seq(A(n, k), k = 0..7)), n = 0..8);

Crossrefs

The corresponding array with Stirling2 numbers is A371761.

Sequence in context: A355676 A159985 A259667 * A321216 A193083 A146103

Adjacent sequences: A379818 A379819 A379820 * A379822 A379823 A379824

Keyword

nonn,tabl

Author

Peter Luschny, Jan 03 2025

Status

approved