A155744 Triangle T(n, k) = (-1)^(n-k)*StirlingS1(n, k) + (-1)^k*StirlingS1(n, n-k) + (-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.

3, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 48, 143, 48, 1, 1, 274, 1835, 1835, 274, 1, 1, 1935, 23649, 51075, 23649, 1935, 1, 1, 15861, 310639, 1195999, 1195999, 310639, 15861, 1, 1, 146188, 4221286, 25753812, 45832899, 25753812, 4221286, 146188, 1, 1, 1491876, 59942994, 535933124, 1510548249, 1510548249, 535933124, 59942994, 1491876, 1

Offset: 0

Author

Roger L. Bagula, Jan 26 2009

Keywords

• nonn
• tabl

Examples

			  3;
  1,      1;
  1,      3,       1;
  1,     11,      11,        1;
  1,     48,     143,       48,        1;
  1,    274,    1835,     1835,      274,        1;
  1,   1935,   23649,    51075,    23649,     1935,       1;
  1,  15861,  310639,  1195999,  1195999,   310639,   15861,      1;
  1, 146188, 4221286, 25753812, 45832899, 25753812, 4221286, 146188, 1;
		

Links

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

Crossrefs

Cf. A000142, A001263, A048994, A342111.

Programs

Magma

A155744:= func< n,k | (-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) + (-1)^k*StirlingFirst(n, n-k) + (-1)^(n-k)*StirlingFirst(n, k) >;
[A155744(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 05 2021

Mathematica

T[n_, k_] = (-1)^(n-k)*StirlingS1[n, k] + (-1)^k*StirlingS1[n, 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 A155744(n,k): return stirling_number1(n, k)*stirling_number1(n, n-k) + stirling_number1(n, k) + stirling_number1(n, n-k)
flatten([[A155744(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

Formula

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

Sum_{k=0..n} T(n, k) = 2*n! + A342111(n). - G. C. Greubel, Jun 05 2021

Extensions

Edited by G. C. Greubel, Jun 05 2021