A155825 Triangle T(n, k) = (-1)^n * n! * StirlingS1(n, k)*StirlingS1(n, n-k)/binomial(n, k), read by rows.

1, 0, 0, 0, 1, 0, 0, 12, 12, 0, 0, 216, 484, 216, 0, 0, 5760, 21000, 21000, 5760, 0, 0, 216000, 1117920, 1822500, 1117920, 216000, 0, 0, 10886400, 74088000, 171884160, 171884160, 74088000, 10886400, 0, 0, 711244800, 6059370240, 18531878400, 26391951936, 18531878400, 6059370240, 711244800, 0

Offset

0,8

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = (-1)^n * n! * StirlingS1(n, k)*StirlingS1(n, n-k)/binomial(n, k).

Example

Triangle begins as:
  1;
  0,        0;
  0,        1,        0;
  0,       12,       12,         0;
  0,      216,      484,       216,         0;
  0,     5760,    21000,     21000,      5760,        0;
  0,   216000,  1117920,   1822500,   1117920,   216000,        0;
  0, 10886400, 74088000, 171884160, 171884160, 74088000, 10886400, 0;

Mathematica

T[n_, k_]:= (-1)^n*n!*StirlingS1[n, k]StirlingS1[n, n-k]/Binomial[n, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma) [(-1)^n*Factorial(n)*StirlingFirst(n, k)*StirlingFirst(n, n-k)/Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 03 2021
(Sage) flatten([[factorial(n)*stirling_number1(n, k)*stirling_number1(n, n-k)/binomial(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 03 2021

Crossrefs

Cf. A048994.

Sequence in context: A254717 A195748 A038337 * A125509 A281251 A247511

Adjacent sequences: A155822 A155823 A155824 * A155826 A155827 A155828

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 28 2009

Extensions

Edited by G. C. Greubel, Jun 03 2021

Status

approved