OFFSET
0,8
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
FORMULA
T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1)*(2*j-1)/6 for n > 2 otherwise 1.
From G. C. Greubel, Feb 11 2021: (Start)
T(n, k) = ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k) with T(n, 0) = T(n, n) =1.
Sum_{k=0..n} T(n, k) = (n*(n-1)*(2*n-1)/6)*HypergeometricPFQ[{1-n, 3/2-n, 2-n}, {3/2, 2}, -1] + 2 - [n=0] (n*(n-1)*(2*n-1)/6)*A196148[n-2] + 2 - [n=0]. (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 5, 5, 1;
1, 14, 70, 14, 1;
1, 30, 420, 420, 30, 1;
1, 55, 1650, 4620, 1650, 55, 1;
1, 91, 5005, 30030, 30030, 5005, 91, 1;
1, 140, 12740, 140140, 300300, 140140, 12740, 140, 1;
1, 204, 28560, 519792, 2042040, 2042040, 519792, 28560, 204, 1;
1, 285, 58140, 1627920, 10581480, 19399380, 10581480, 1627920, 58140, 285, 1;
MATHEMATICA
(* First program *)
c[n_]:= If[n<2, 1, Product[j*(j-1)*(2*j-1)/6, {j, 2, n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, ((n-k)/6)*Binomial[n-1, k-1]*Binomial[2*n, 2*k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)
PROG
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else ((n-k)/6)*binomial(n-1, k-1)*binomial(2*n, 2*k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else ((n-k)/6)*Binomial(n-1, k-1)*Binomial(2*n, 2*k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 08 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 11 2021
STATUS
approved