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%I #15 Sep 01 2023 04:07:49
%S 1,120,4790016000,1703748471578689536000000,
%T 44045334006101976766560297729172439040000000000,
%U 389438360216723307909581902233109465138002465491175688781168640000000000000000
%N a(n) = Product_{k=1..n} Gamma(6*k).
%F a(n) = A^(35/6) * exp(-35/72) * Gamma(1/3)^(5/3) * 2^(-125/72 + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * Pi^(-25/12 - 5*n/2) * BarnesG(1 + n) * BarnesG(7/6 + n) * BarnesG(4/3 + n) * BarnesG(3/2 + n) * BarnesG(5/3 + n) * BarnesG(11/6 + n), where A = A074962 is the Glaisher-Kinkelin constant.
%F a(n) ~ A^(-1/6) * Gamma(1/3)^(5/3) * 2^(-35/72 + 3*n + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * exp(1/72 - 5*n/2 - 9*n^2/2) * n^(19/72 + 5*n/2 + 3*n^2) * Pi^(-5/6 + n/2), where A = A074962 is the Glaisher-Kinkelin constant.
%t Table[Product[Gamma[6*k], {k, 1, n}], {n, 0, 10}]
%t Table[Product[(6*k-1)!, {k, 1, n}], {n, 0, 10}]
%Y Cf. A000178, A168467, A294319, A294322, A294326.
%Y Cf. A055462, A306635, A306651.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Sep 01 2023
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