A340052 a(n) = Product_{1<=i

1, 1, 5, 91, 5661, 1173821, 801125065, 1786768287095, 12964564030176889, 305121026002697122873, 23243604301636717923421133, 5722586073277932639539150258131, 4548248834078776410469611991220703125

Offset: 0

Author

Seiichi Manyama, Dec 29 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A007726, A065072, A127605.

Programs

Mathematica

Table[2^(n*(n-1)) * Product[Product[Sin[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, j-1}], {j, 2, n}], {n, 0, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)

PARI

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

Formula

a(n)^2 = A127605(n)/(2^n * (2*n+1)).

a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/(2*Pi)) / (2^(n/2 + 5/4) * Pi^(3/8) * n^(3/4)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020