A340295 a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).

1, 13, 18281, 2732887529, 43384923739812577, 73125714588602035608260981, 13085551252412040683513520733767180041, 248596840858215958581954513797323868183183928594833

Offset: 0

Author

Seiichi Manyama, Jan 03 2021

Keywords

• nonn

Comments

a(n)/A001570(n+1) is an integer.

Links

• Wikipedia, Chebyshev polynomials
• Wikipedia, Resultant

Crossrefs

Cf. A001570, A334089, A340166, A340291, A340292.

Programs

Mathematica

Table[Resultant[ChebyshevT[4*n+2, x/2], ChebyshevT[4*n+2, I*x/2], x]^(1/4) / 2^n, {n, 0, 10}] (* Vaclav Kotesovec, Jan 04 2021 *)

PARI

default(realprecision, 120);
{a(n) = round(4^(2*n^2)*prod(j=1, n, prod(k=1, n, 1-(sin(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))}

PARI

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

Formula

a(n) = A334089(2*n+1).

a(n) ~ exp(2*G*(2*n+1)^2/Pi) / 2^(3*n + 7/8), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 04 2021