A155863 Triangle T(n,k) = n*(n^2 - 1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 60, 120, 60, 1, 1, 120, 360, 360, 120, 1, 1, 210, 840, 1260, 840, 210, 1, 1, 336, 1680, 3360, 3360, 1680, 336, 1, 1, 504, 3024, 7560, 10080, 7560, 3024, 504, 1, 1, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 1, 1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1

Offset: 0

Author

Roger L. Bagula, Jan 29 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
  1;
  1,   1;
  1,   6,    1;
  1,  24,   24,     1;
  1,  60,  120,    60,     1;
  1, 120,  360,   360,   120,     1;
  1, 210,  840,  1260,   840,   210,     1;
  1, 336, 1680,  3360,  3360,  1680,   336,     1;
  1, 504, 3024,  7560, 10080,  7560,  3024,   504,    1,
  1, 720, 5040, 15120, 25200, 25200, 15120,  5040,  720,   1;
  1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1;
  ...
		

Links

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

Crossrefs

Cf. A001789, A052771, A155864, A155865.

Programs

Magma

A155863:= func< n,k | k eq 0 or k eq n select 1 else 6*Binomial(n+1, 3)*Binomial(n-2, k-1) >;
[A155863(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

Mathematica

(* First program *)
p[n_, x_]:= p[n, x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n+1), {x, 3}]];
Flatten[Table[CoefficientList[p[n,x], x], {n,0,12}]]
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, 6*Binomial[n+1, 3]*Binomial[n-2, k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2021 *)

Maxima

T(n, k):= ratcoef(expand(x^n + n*(n^2 -1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2), x, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 03 2018 */

Sage

def A155863(n,k): return 1 if (k==0 or k==n) else 6*binomial(n+1, 3)*binomial(n-2, k-1)
flatten([[A155863(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 + x*((d/dx)^3 (x+1)^(n+1)) and T(0, 0) = 1.

From Franck Maminirina Ramaharo, Dec 03 2018: (Start)

T(n, k) = (n-1)*n*(n+1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.

n-th row polynomial is x^n + n*(n^2 - 1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2.

G.f.: 1/(1 - y) + 1/(1 - x*y) + (6*x*y^2)/(1 - y - x*y)^4 - 1.

E.g.f.: exp(y) + exp(x*y) + (3*x*y^2 + (x + x^2)*y^3)*exp((1 + x)*y) - 1. (End)

Sum_{k=0..n} T(n, k) = 2 - [n=0] + 6*A001789(n+1) = 2 - [n=0] + A052771(n+1). - G. C. Greubel, Jun 04 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 03 2018