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a(n) = sqrt( Product_{1<=j, k<=n-1} (4*sin(j*Pi/n)^2 + 4*cos(k*Pi/n)^2) ).
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%I #16 Mar 18 2023 05:51:48

%S 1,1,2,16,384,30976,7741440,6369316864,16435095011328,

%T 138915523039657984,3696387867279360000000,

%U 321533678904455375050768384,88192375153215003517412966400000,78996127242669742603293261855977373696,223311937686075869460797609709638544686841856

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

%F a(n) ~ c * (sqrt(2) - 1)^n * exp(2*G*n^2/Pi), where c = sqrt(Pi) / Gamma(3/4)^2 if n is even and c = 2^(1/4) if n is odd, G is Catalan's constant A006752. - _Vaclav Kotesovec_, Mar 18 2023

%t Table[Sqrt[Product[Product[(4*Sin[j*Pi/n]^2 + 4*Cos[k*Pi/n]^2), {j, 1, n - 1}], {k, 1, n - 1}]], {n, 0, 15}] // Round (* _Vaclav Kotesovec_, Mar 18 2023 *)

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

%o {a(n) = round(sqrt(prod(j=1, n-1, prod(k=1, n-1, 4*sin(j*Pi/n)^2+4*cos(k*Pi/n)^2))))}

%Y Main diagonal of A340561.

%Y Cf. A127606, A340562.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 11 2021