cp's OEIS Frontend

A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.

%I A156646 #7 Sep 08 2022 08:45:41
%S A156646 1,1,1,1,484,1,1,233289,233289,1,1,112444816,54198633636,112444816,1,
%T A156646 1,54198168025,12591535188240100,12591535188240100,54198168025,1,1,
%U A156646 26123404543236,2925290638056514680225,1409984043580226203632400,2925290638056514680225,26123404543236,1
%N A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.
%H A156646 G. C. Greubel, <a href="/A156646/b156646.txt">Rows n = 0..25 of the triangle, flattened</a>
%F A156646 T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10.
%F A156646 From _G. C. Greubel_, Jul 03 2021: (Start)
%F A156646 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 = 10.
%F A156646 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 = 10. (End)
%e A156646 Triangle begins as:
%e A156646   1;
%e A156646   1,           1;
%e A156646   1,         484,                 1;
%e A156646   1,      233289,            233289,                 1;
%e A156646   1,   112444816,       54198633636,         112444816,           1;
%e A156646   1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1;
%t A156646 (* First program *)
%t A156646 b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
%t A156646 T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
%t A156646 Table[T[n,k,10], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jul 03 2021 *)
%t A156646 (* Second program *)
%t A156646 T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
%t A156646 Table[T[n,k,12], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 03 2021 *)
%o A156646 (Magma)
%o A156646 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 A156646 T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
%o A156646 [T(n,k,10): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 03 2021
%o A156646 (Sage)
%o A156646 def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
%o A156646 def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
%o A156646 flatten([[T(n, k, 10) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 03 2021
%Y A156646 Cf. A007318 (m=0), A173585 (m=1), A156645 (m=2), this sequence (m=10).
%Y A156646 Cf. A123583, A156647.
%K A156646 nonn,tabl
%O A156646 0,5
%A A156646 _Roger L. Bagula_, Feb 12 2009
%E A156646 Edited by _G. C. Greubel_, Jul 03 2021