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%I #12 Jan 05 2024 14:57:17
%S 1,1,1,1,12,1,1,180,180,1,1,2565,38475,2565,1,1,36936,7895070,7895070,
%T 36936,1,1,530712,1633531536,23277824388,1633531536,530712,1,1,
%U 7628985,337399490610,69234375473172,69234375473172,337399490610,7628985,1,1,109656180,69713779364775,205544107079102610,2959835141939077584,205544107079102610,69713779364775,109656180,1
%N Triangle T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3, read by rows.
%H G. C. Greubel, <a href="/A172376/b172376.txt">Rows n = 1..30 of the triangle, flattened</a>
%F T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3.
%F T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k,q)), where c(n, q) = Product_{j=1..n} f(j, q), and f(n, q) = (-I*sqrt(q))^(n-1)*ChebyshevU(n-1, i*sqrt(q)/2). - _G. C. Greubel_, May 07 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 180, 180, 1;
%e 1, 2565, 38475, 2565, 1;
%e 1, 36936, 7895070, 7895070, 36936, 1;
%e 1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1;
%t f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
%t c[n_, q_]:= Product[f[j, q], {j,n}];
%t T[n_, k_, q_]:= c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]*f[k, q]);
%t Table[T[n, k, 3], {n,12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, May 07 2021 *)
%o (Sage)
%o @CachedFunction
%o def f(n,q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
%o def c(n,q): return product( f(j,q) for j in (1..n) )
%o def T(n,k,q): return c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q))
%o flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, May 07 2021
%Y Cf. A002605, A030195, A172375 (q=2), this sequence (q=3).
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_, Feb 01 2010
%E Edited by _G. C. Greubel_, May 07 2021
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