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%I #7 Sep 08 2022 08:45:50
%S 1,1,1,1,16,1,1,225,225,1,1,3136,44100,3136,1,1,43681,8561476,8561476,
%T 43681,1,1,608400,1660970025,23150231104,1660970025,608400,1,1,
%U 8473921,322220846025,62555239000969,62555239000969,322220846025,8473921,1
%N Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.
%H G. C. Greubel, <a href="/A173585/b173585.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} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3.
%F From _G. C. Greubel_, Jul 06 2021: (Start)
%F T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 1.
%F T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 1. (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 16, 1;
%e 1, 225, 225, 1;
%e 1, 3136, 44100, 3136, 1;
%e 1, 43681, 8561476, 8561476, 43681, 1;
%e 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1;
%t (* First program *)
%t f[n_, q_]:= (1/4)*((2+Sqrt[q])^n + (2-Sqrt[q])^n -2);
%t c[n_, q_]:= Product[f[k, q], {k, 2, n, 2}]//Simplify;
%t T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n - k, q]);
%t Table[T[n, k, 3], {n, 0, 10, 2}, {k, 0, n, 2}]//Flatten (* modified by _G. C. Greubel_, Jul 06 2021 *)
%t (* Second program *)
%t t[n_, q_]:= (1/4)*(Round[(2+Sqrt[q])^n + (2-Sqrt[q])^n] -2);
%t c[n_, q_]:= Product[t[2*j, q], {j,n}];
%t T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
%t Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 06 2021 *)
%o (Magma)
%o b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
%o T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
%o [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 06 2021
%o (Sage)
%o @CachedFunction
%o def f(n,q): return (1/4)*( round((2 + sqrt(q))^n + (2 - sqrt(q))^n) - 2 )
%o def c(n,q): return product( f(2*j, q) for j in (1..n))
%o def T(n,k,q): return c(n, q)/(c(k, q)*c(n-k, q))
%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 06 2021
%Y Cf. A022168 (q=0), A022173 (q=1), this sequence (q=3).
%Y Cf. A007318 (m=0), this sequence (m=1), A156645 (m=2), A156646 (m=10).
%Y Cf. A123583, A156647.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 22 2010
%E Edited by _G. C. Greubel_, Jul 06 2021
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