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%I #28 Apr 19 2016 14:04:03
%S 1,4,112,24640,44844800,695273779200,93581069585203200,
%T 110803729631663996928000,1165466869384731418887782400000,
%U 109720873815210197693149787062272000000,93006053830822450607559730484293052399616000000
%N a(n) = Product_{k=0..n} binomial(4*k,k).
%C In general, for p > 1, Product_{k=0..n} binomial(p*k,k) ~ A^(1 + 1/(p*(p-1))) * exp(n/2 - 1/12 - 1/(12*p*(p-1))) * n^(-1/3 - n/2 - 1/(12*p*(p-1))) * (p-1)^(1/(12*(p-1)) - p*n/2 - (p-1)*n^2/2) * p^(-1/(12*p) + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(-1/4 - n/2) * Product_{j=1..p-1} (Gamma(j/(p-1))^(j/(p-1)) / Gamma(j/p)^(j/p)), where A = A074962 is the Glaisher-Kinkelin constant.
%F a(n) ~ A^(13/12) * 2^(9*n/2 + 4*n^2) * exp(n/2 - 13/144) * Gamma(1/4)^(1/2) / (Gamma(1/3)^(1/3) * 3^(11/36 + 2*n + 3*n^2/2) * Pi^(7/12 + n/2) * n^(49/144 + n/2)), where A = A074962 is the Glaisher-Kinkelin constant.
%t Table[Product[Binomial[4*k,k],{k,0,n}],{n,0,10}]
%Y Cf. A007685 (p=2), A268196 (p=3).
%Y Cf. A000178, A098694, A268504, A268505, A268506, A271946, A271947.
%Y Cf. A005810, A165975.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Apr 17 2016
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