A340476 - OEIS

A340476

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).

1, 1, 1, 1, 4, 1, 1, 19, 11, 1, 1, 91, 176, 29, 1, 1, 436, 2911, 1471, 76, 1, 1, 2089, 48301, 79808, 11989, 199, 1, 1, 10009, 801701, 4375897, 2091817, 97021, 521, 1, 1, 47956, 13307111, 240378643, 372713728, 53924597, 783511, 1364, 1

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..44.

Wikipedia, Chebyshev polynomials

Wikipedia, Resultant

FORMULA

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).

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, ...

1, 4, 19, 91, 436, ...

1, 11, 176, 2911, 48301, ...

1, 29, 1471, 79808, 4375897, ...

1, 76, 11989, 2091817, 372713728, ...

PROG

(PARI) default(realprecision, 120);

{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)))}

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

CROSSREFS