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
KEYWORD
AUTHOR
Roger L. Bagula, Jan 29 2009
EXTENSIONS
Edited and name clarified by Franck Maminirina Ramaharo, Dec 03 2018
STATUS
approved