login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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

%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