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Square array T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n!, read by antidiagonals.
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%I #9 Sep 08 2022 08:45:41

%S 1,1,1,1,-3,2,1,-8,144,6,1,-15,2304,-97200,24,1,-24,14400,-22579200,

%T 914457600,120,1,-35,57600,-857304000,7517247897600,-119833267276800,

%U 720,1,-48,176400,-13548902400,3163657512960000,-85018329720343756800,218719679433615360000,5040

%N Square array T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156647/b156647.txt">Antidiagonal rows n = 0..25, flattened</a>

%F T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n! (square array).

%e Square array begins as:

%e 1, 1, 1, ...;

%e 1, -3, -8, ...;

%e 2, 144, 2304, ...;

%e 6, -97200, -22579200, ...;

%e 24, 914457600, 7517247897600, ...;

%e 120, -119833267276800, -85018329720343756800, ...;

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, -3, 2;

%e 1, -8, 144, 6;

%e 1, -15, 2304, -97200, 24;

%e 1, -24, 14400, -22579200, 914457600, 120;

%e 1, -35, 57600, -857304000, 7517247897600, -119833267276800, 720;

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

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jul 02 2021 *)

%o (Magma)

%o T:= 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(k, n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 02 2021

%o (Sage)

%o def T(n,k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )

%o flatten([[T(k, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 02 2021

%Y Cf. A123583.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 12 2009

%E Edited by _G. C. Greubel_, Jul 02 2021