A155742 Triangle T(n, k) = (-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.

1, 0, 0, 0, 1, 0, 0, 6, 6, 0, 0, 36, 121, 36, 0, 0, 240, 1750, 1750, 240, 0, 0, 1800, 23290, 50625, 23290, 1800, 0, 0, 15120, 308700, 1193640, 1193640, 308700, 15120, 0, 0, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 0, 0, 1451520, 59832864, 535810464, 1510458516, 1510458516, 535810464, 59832864, 1451520, 0

Offset: 0

Author

Roger L. Bagula, Jan 26 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  0,      0;
  0,      1,       0;
  0,      6,       6,        0;
  0,     36,     121,       36,        0;
  0,    240,    1750,     1750,      240,        0;
  0,   1800,   23290,    50625,    23290,     1800,       0;
  0,  15120,  308700,  1193640,  1193640,   308700,   15120,      0;
  0, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 0;
		

Links

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

Crossrefs

Cf. A048994, A342111 (row sums).

Programs

Magma

A155742:= func< n,k | (-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
[A155742(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 05 2021

Mathematica

T[n_, k_]:= (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 05 2021 *)

Sage

def A155742(n,k): return stirling_number1(n,k)*stirling_number1(n, n-k)
flatten([[A155742(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

Formula

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

Sum_{k=0..n} T(n, k) = A342111(n). - G. C. Greubel, Jun 05 2021

Extensions

Edited by G. C. Greubel, Jun 05 2021