cp's OEIS Frontend

A172375 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 = 2, read by rows.

%I A172375 #11 Jan 05 2024 15:01:19
%S A172375 1,1,1,1,6,1,1,48,48,1,1,352,2816,352,1,1,2640,154880,154880,2640,1,1,
%T A172375 19680,8659200,63500800,8659200,19680,1,1,146944,481976320,
%U A172375 26508697600,26508697600,481976320,146944,1,1,1096704,26859012096,11012194959360,82591462195200,11012194959360,26859012096,1096704,1
%N A172375 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 = 2, read by rows.
%H A172375 G. C. Greubel, <a href="/A172375/b172375.txt">Rows n = 1..30 of the triangle, flattened</a>
%F A172375 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 = 2.
%F A172375 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 A172375 Triangle begins as:
%e A172375   1;
%e A172375   1,      1;
%e A172375   1,      6,         1;
%e A172375   1,     48,        48,           1;
%e A172375   1,    352,      2816,         352,           1;
%e A172375   1,   2640,    154880,      154880,        2640,         1;
%e A172375   1,  19680,   8659200,    63500800,     8659200,     19680,      1;
%e A172375   1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1;
%t A172375 f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
%t A172375 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172375 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 A172375 Table[T[n, k, 2], {n,12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, May 07 2021 *)
%o A172375 (Sage)
%o A172375 @CachedFunction
%o A172375 def f(n,q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
%o A172375 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172375 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 A172375 flatten([[T(n,k,2) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, May 07 2021
%Y A172375 Cf. A002605, A030195, this sequence (q=2), A172376 (q=3).
%K A172375 nonn,tabl
%O A172375 1,5
%A A172375 _Roger L. Bagula_, Feb 01 2010
%E A172375 Edited by _G. C. Greubel_, May 07 2021