OFFSET
0,1
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 + Sum_{j=0..n} binomial((j+1)*(n-j+1), n+2)*x^j and p(0, x) = 2.
T(n, k) = binomial((k+1)*(n-k+1), n+2) with T(0, 0) = 2, T(n, 0) = T(n, n) = 1. - G. C. Greubel, Jun 09 2021
EXAMPLE
Triangle begins as:
2;
1, 1;
1, 1, 1;
1, 6, 6, 1;
1, 28, 84, 28, 1;
1, 120, 792, 792, 120, 1;
1, 495, 6435, 12870, 6435, 495, 1;
1, 2002, 48620, 167960, 167960, 48620, 2002, 1;
1, 8008, 352716, 1961256, 3268760, 1961256, 352716, 8008, 1;
1, 31824, 2496144, 21474180, 54627300, 54627300, 21474180, 2496144, 31824, 1;
MATHEMATICA
(* First program *)
p[n_, x_]:= If[n==0, 2, 1 +x^n +Sum[Binomial[(j+1)*(n-j+1), n+2]*x^j, {j, 0, n}]];
Table[CoefficientList[p[n, x], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Jun 09 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n==0, 2, If[k==0 || k==n, 1, Binomial[(k+1)*(n-k+1), n + 2]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 09 2021 *)
PROG
(Magma)
A155869:= func< n, k | n eq 0 select 2 else k eq 0 or k eq n select 1 else Binomial((k+1)*(n-k+1), n+2) >;
[A155869(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021
(Sage)
def A155869(n, k): return 2 if (n==0) else 1 if (k==0 or k==n) else binomial((k+1)*(n-k+1), n+2)
flatten([[A155869(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 09 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 29 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 09 2021
STATUS
approved