A340052 - OEIS

A340052

a(n) = Product_{1<=i<j<=n} (4*sin(i*Pi/(2*n+1))^2 + 4*sin(j*Pi/(2*n+1))^2).

%I #47 Feb 28 2023 23:47:14

%S 1,1,5,91,5661,1173821,801125065,1786768287095,12964564030176889,

%T 305121026002697122873,23243604301636717923421133,

%U 5722586073277932639539150258131,4548248834078776410469611991220703125

%N a(n) = Product_{1<=i<j<=n} (4*sin(i*Pi/(2*n+1))^2 + 4*sin(j*Pi/(2*n+1))^2).

%H Seiichi Manyama, <a href="/A340052/b340052.txt">Table of n, a(n) for n = 0..50</a>

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

%F 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

%t 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 *)

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

%o {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)))}

%Y Cf. A007726, A065072, A127605.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 29 2020