A173583 Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.

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, 1, 254313151, 99500274641901, 62191645548485651, 62191645548485651, 99500274641901, 254313151, 1

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

Cf. A000012 (q=0), A007318 (q=1), A022168 (q=2), A022173 (q=3), A022180 (q=4), A173583 (q=5).

Sequence in context: A225483 A183065 A157630 * A040687 A040688 A040686

Adjacent sequences: A173580 A173581 A173582 * A173584 A173585 A173586

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 22 2010

Extensions

Edited by G. C. Greubel, Feb 22 2021

Status

approved