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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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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
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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 *) -
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
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)
Extensions
Edited by G. C. Greubel, Jun 04 2021
Comments