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%I #28 Feb 28 2023 23:48:20
%S 1,2,14,50,722,9922,401998,19681538,2415542018,400448833106,
%T 152849502772958,83804387156528018,100644292294423977842,
%U 180483873668860889130642,686161117968330536875295134,4001215836806010384390623471618
%N a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4*sin((2*j-1)*Pi/n)^2 + 4*sin((2*k-1)*Pi/(n+1))^2) )^(1/4).
%C Number of perfect matchings in the graph C_n X C_{n+1} for n > 0.
%H S. N. Perepechko, <a href="http://www.jip.ru/2016/333-361-2016.pdf">The number of perfect matchings on C_m X C_n graphs</a>, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectMatching.html">Perfect Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%F a(n) ~ 2^(3/4) * exp(G*n*(n+1)/Pi), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Feb 14 2021
%t Table[Product[4*Sin[(2*j - 1)*Pi/n]^2 + 4*Sin[(2*k - 1)*Pi/(n+1)]^2, {k, 1, n+1}, {j, 1, n}]^(1/4), {n, 0, 15}] // Round (* _Vaclav Kotesovec_, Feb 14 2021 *)
%o (PARI) default(realprecision, 120);
%o a(n) = round(prod(j=1, n, prod(k=1, n+1, 4*sin((2*j-1)*Pi/n)^2+4*sin((2*k-1)*Pi/(n+1))^2))^(1/4));
%Y Cf. A162484, A220864, A230033, A231087, A231485, A232804, A253678, A281583, A281679, A308761, A309018, A335586.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Feb 13 2021
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