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A340139 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i sin(i*Pi/(2*n))^2 * sin(j*Pi/(2*n))^2).

%I A340139 #22 Feb 28 2023 23:47:18
%S A340139 1,1,13,1904,3016365,50771120400,8993476465721657,
%T A340139 16670531837245286776832,322175275214070402711647486361,
%U A340139 64754609334534873770923002355900227840
%N A340139 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i<j<=n-1} (1 - sin(i*Pi/(2*n))^2 * sin(j*Pi/(2*n))^2).
%H A340139 Seiichi Manyama, <a href="/A340139/b340139.txt">Table of n, a(n) for n = 1..45</a>
%F A340139 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i<j<=n-1} (1 - cos(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
%F A340139 a(n) ~ sqrt(Gamma(1/4)) * exp(4*G*n^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(3*n - 9/4) * (1 + sqrt(2))^n), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Jan 05 2021
%t A340139 Table[4^((n-2)*(n-1)) * Product[Product[1 - Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, j-1}], {j, 2, n-1}], {n, 1, 12}] // Round (* _Vaclav Kotesovec_, Dec 31 2020 *)
%o A340139 (PARI) default(realprecision, 120);
%o A340139 {a(n) = round(4^((n-2)*(n-1))*prod(j=2, n-1, prod(i=1, j-1, 1-(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}
%Y A340139 Cf. A007725.
%K A340139 nonn
%O A340139 1,3
%A A340139 _Seiichi Manyama_, Dec 29 2020