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

0,1

Links

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

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

Example

  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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A000142, A001263, A048994, A342111.

Sequence in context: A226203 A327791 A051997 * A086869 A348953 A095345

Adjacent sequences: A155741 A155742 A155743 * A155745 A155746 A155747

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 26 2009

Extensions

Edited by G. C. Greubel, Jun 05 2021

Status

approved