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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*cos(b*Pi/(2*k+1))^2).
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%I #21 Jan 09 2021 07:35:11

%S 1,1,1,1,4,1,1,19,11,1,1,91,176,29,1,1,436,2911,1471,76,1,1,2089,

%T 48301,79808,11989,199,1,1,10009,801701,4375897,2091817,97021,521,1,1,

%U 47956,13307111,240378643,372713728,53924597,783511,1364,1

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*cos(b*Pi/(2*k+1))^2).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Resultant">Resultant</a>

%F T(n,k) = 2^k * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*k}(x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

%e Square array begins:

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

%e 1, 4, 19, 91, 436, ...

%e 1, 11, 176, 2911, 48301, ...

%e 1, 29, 1471, 79808, 4375897, ...

%e 1, 76, 11989, 2091817, 372713728, ...

%o (PARI) default(realprecision, 120);

%o {T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))}

%o (PARI) {T(n, k) = sqrtint(4^k*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*k, 2, x/2)))}

%Y Column k=0..1 give A000012, A002878.

%Y Row n=0..7 give A000012, A004253(n+1), A003729, A028478, A028480, A028482, A028484, A028486.

%Y Main diagonal gives A127606.

%Y Cf. A187617, A340475.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, Jan 09 2021