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%I #6 Nov 23 2023 10:36:06
%S 1,6,540,907200,31434480000,23788231346880000,
%T 408042767492495815680000,162838835029822082951032012800000,
%U 1541352909587869227178909850805190656000000,351233376660297011570511252132131832794456064000000000,1949695346852822356399298814748829537555898997004605685760000000000
%N a(n) = Product_{k=0..n} (3*k)! / k!^3.
%F a(n) = Product_{k=0..n} binomial(3*k,k) * binomial(2*k,k).
%F a(n) = A268504(n) / A000178(n)^3.
%F a(n) = A268504(n) / A061719(n).
%F a(n) = A007685(n) * A268196(n).
%F a(n) ~ A^(8/3) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36) * exp(n - 2/9) / (n^(n + 13/18) * (2*Pi)^(n + 7/6)), where A is the Glaisher-Kinkelin constant A074962.
%t Table[Product[(3*k)!/k!^3, {k, 0, n}], {n, 0, 10}]
%t Table[Product[Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]
%Y Cf. A000178, A006480, A268504, A061719, A268196.
%Y Cf. A007685, A367568, A367569, A367570, A367571.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Nov 23 2023