A155868 Triangle T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1, read by rows.

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

Offset

0,8

Comments

Row sums are:

{1, 2, 3, 14, 195, 3982, 100807, 3034922, 105994835, 4215106730, 188097696347,...}

Links

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

Formula

T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + (-1)^n*Sum_{j=0..n} StirlingS1(n, j)*StirlingS1(n, n-j)*x^k and p(0, x) = 1.

From G. C. Greubel, Jun 04 2021: (Start)

T(n, k) = (-1)^n*StirlingS1(n, j)*StirlingS1(n, n-j), with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2 + A342111(n) - 2*[n==0]. (End)

Example

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

Mathematica

(* First program *)
p[n_, x_]:= If[n==0, 1, 1 +x^n +(-1)^n*Sum[StirlingS1[n, j]*StirlingS1[n, n-j]*x^j, {j, 0, n}]];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Jun 04 2021 *)
(* Second program *)
T[n_, k_]:=  If[k==0 || k==n, 1, (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)

Prog

(Magma)
A155868:= func< n, k | k eq 0 or k eq n select 1 else (-1)^n*StirlingFirst(n, k)* StirlingFirst(n, n-k) >;
[A155868(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021
(Sage)
def A155868(n, k): return 1 if (k==0 or k==n) else stirling_number1(n, k)*stirling_number1(n, n-k)
flatten([[A155868(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Crossrefs

Cf. A048994, A342111.

Sequence in context: A046606 A172350 A205457 * A322622 A176565 A176567

Adjacent sequences: A155865 A155866 A155867 * A155869 A155870 A155871

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 29 2009

Extensions

Edited by G. C. Greubel, Jun 04 2021

Status

approved