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Triangle T(n, k, q) = ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1,q)^2*c(n-k,q)*c(n-k+1, q)), where c(n, q) = Product_{j=1..n} (1-q^j) and q = 4, read by rows.
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%I #8 May 07 2021 09:17:53

%S 1,1,1,1,21,1,1,357,357,1,1,5797,98549,5797,1,1,93093,25698101,

%T 25698101,93093,1,1,1490853,6608951349,107316781429,6608951349,

%U 1490853,1,1,23859109,1693829725237,441691010116213,441691010116213,1693829725237,23859109,1

%N Triangle T(n, k, q) = ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1,q)^2*c(n-k,q)*c(n-k+1, q)), where c(n, q) = Product_{j=1..n} (1-q^j) and q = 4, read by rows.

%H G. C. Greubel, <a href="/A172301/b172301.txt">Rows n = 1..30 of the triangle, flattened</a>

%F T(n, k, q) = ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1,q)^2*c(n-k,q)*c(n-k+1, q)), where c(n, q) = Product_{j=1..n} (1-q^j) and q = 4.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 21, 1;

%e 1, 357, 357, 1;

%e 1, 5797, 98549, 5797, 1;

%e 1, 93093, 25698101, 25698101, 93093, 1;

%e 1, 1490853, 6608951349, 107316781429, 6608951349, 1490853, 1;

%t c[n_, q_]:= QPochhammer[q, q, n];

%t T[n_, k_, q_]:= ((1-q)/(1-q^k))*c[n-1,q]*c[n,q]/(c[k-1,q]^2*c[n-k,q]*c[n-k+1,q]);

%t Table[T[n, k, 4], {n,10}, {k,n}]//Flatten (* modified by _G. C. Greubel_, May 07 2021 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o def c(n,q): return q_pochhammer(n,q,q)

%o def T(n,k,q): return ((1-q)/(1-q^k))*c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q))

%o [[T(n,k,4) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, May 07 2021

%Y Cf. A156916 (q=2), A172300 (q=3), this sequence (q=4), A172302 (q=5).

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_, Jan 31 2010

%E Edited by _G. C. Greubel_, May 07 2021