A156645 - OEIS

%I #9 Sep 08 2022 08:45:41

%S 1,1,1,1,36,1,1,1225,1225,1,1,41616,1416100,41616,1,1,1413721,

%T 1634261476,1634261476,1413721,1,1,48024900,1885939157025,

%U 64069586905104,1885939157025,48024900,1,1,1631432881,2176372249076025,2511659716192658889,2511659716192658889,2176372249076025,1631432881,1

%N 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 = 2, read by rows.

%H G. C. Greubel, <a href="/A156645/b156645.txt">Rows n = 0..25 of the triangle, flattened</a>

%F 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 = 2.

%F From _G. C. Greubel_, Jul 03 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 = 2.

%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 = 2. (End)

%e Triangle begins as:

%e   1;

%e   1,        1;

%e   1,       36,             1;

%e   1,     1225,          1225,              1;

%e   1,    41616,       1416100,          41616,             1;

%e   1,  1413721,    1634261476,     1634261476,       1413721,        1;

%e   1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1;

%t (* First program *)

%t b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];

%t T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);

%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jul 03 2021 *)

%t (* Second program *)

%t 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 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 03 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,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 03 2021

%o (Sage)

%o 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 def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))

%o flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 03 2021

%Y Cf. A007318 (m=0), A173585 (m=1), this sequence (m=2), A156646 (m=10).

%Y Cf. A123583, A156647.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 12 2009

%E Edited by _G. C. Greubel_, Jul 03 2021