A173583 - OEIS

%I #8 Feb 22 2021 04:21:36

%S 1,1,1,1,26,1,1,651,651,1,1,16276,407526,16276,1,1,406901,254720026,

%T 254720026,406901,1,1,10172526,159200423151,3980255126276,

%U 159200423151,10172526,1,1,254313151,99500274641901,62191645548485651,62191645548485651,99500274641901,254313151,1

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

%C Row sums are: 1, 2, 28, 1304, 440080, 510253856, 4298676317632, 124582292154881408, ...

%H G. C. Greubel, <a href="/A173583/b173583.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F From _G. C. Greubel_, Feb 22 2021: (Start)

%F T(n, k, q) = q-binomial(n, k, q^2), for q = 5.

%F T(n, k) = T(n-1, k-1) + p^k * T(n-1, k), with p = 25 (as a number triangle). (End)

%e Triangle begins as:

%e   1;

%e   1,        1;

%e   1,       26,            1;

%e   1,      651,          651,             1;

%e   1,    16276,       407526,         16276,            1;

%e   1,   406901,    254720026,     254720026,       406901,        1;

%e   1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1;

%t (* First program *)

%t c[n_, q_]:= Product[(1 -q^(2*j))/(1-q), {j,1,n}];

%t T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);

%t Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten

%t (* Second program *)

%t Table[QBinomial[n,k,5^2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 22 2021 *)

%t 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 *)

%o (Sage) flatten([[q_binomial(n, k, 5^2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 22 2021

%o (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

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

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 22 2010

%E Edited by _G. C. Greubel_, Feb 22 2021