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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 29 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 04 2021
STATUS
approved