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

0,5

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 + 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

Example

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;
  ...

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 *)

Prog

(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 */
(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
(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

Crossrefs

Cf. A001789, A052771, A155864, A155865.

Sequence in context: A174527 A156139 A309280 * A173882 A174045 A169660

Adjacent sequences: A155860 A155861 A155862 * A155864 A155865 A155866

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 29 2009

Extensions

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

Status

approved