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%I #23 Jan 05 2021 03:27:25
%S 1,12,17745,2958176256,54090331699622625,107181043200192494332800000,
%T 22868509031094388112997259982567521313,
%U 523389340935243821042846225254323436248483571433472
%N a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 - sin(i*Pi/(2*n))^2 * sin(j*Pi/(2*n))^2).
%H Seiichi Manyama, <a href="/A340166/b340166.txt">Table of n, a(n) for n = 1..30</a>
%H D. E. Knuth, <a href="https://arxiv.org/abs/math/9501234">Aztec Diamonds, Checkerboard Graphs, and Spanning Trees</a>, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.
%F a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 - cos(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
%F a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 - sin(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
%F a(n) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 2^(6*n - 2)), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Jan 05 2021
%t Table[4^(2*(n-1)^2) * Product[Product[1 - Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, n-1}], {j, 1, n-1}], {n, 1, 10}] // Round (* _Vaclav Kotesovec_, Dec 31 2020 *)
%o (PARI) default(realprecision, 120);
%o {a(n) = round(4^(2*(n-1)^2)*prod(i=1, n-1, prod(j=1, n-1, 1-(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}
%Y Cf. A007725, A340139, A340167.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Dec 30 2020
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