OFFSET
0,5
COMMENTS
Row sums are: 1, 2, 28, 1304, 440080, 510253856, 4298676317632, 124582292154881408, ...
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (1 -q^(2*j))/(1-q) for q = 5.
From G. C. Greubel, Feb 22 2021: (Start)
T(n, k, q) = q-binomial(n, k, q^2), for q = 5.
T(n, k) = T(n-1, k-1) + p^k * T(n-1, k), with p = 25 (as a number triangle). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 26, 1;
1, 651, 651, 1;
1, 16276, 407526, 16276, 1;
1, 406901, 254720026, 254720026, 406901, 1;
1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1;
MATHEMATICA
(* First program *)
c[n_, q_]:= Product[(1 -q^(2*j))/(1-q), {j, 1, n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 5], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
Table[QBinomial[n, k, 5^2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
T[n_, k_, p_]:= T[n, k, p] = If[k==0 || k==n, 1, T[n-1, k-1, p] + p^k*T[n-1, k, q]]; Table[T[n, k, 25], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
PROG
(Sage) flatten([[q_binomial(n, k, 5^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2021
(Magma) q:=5;; [q^(k*(n-k))*GaussianBinomial(n, k, q): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 22 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 22 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 22 2021
STATUS
approved