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%I #14 Jun 27 2023 09:23:37
%S 1,2,18,800,183750,224042112,1475939646720,53195808994099200,
%T 10587785727897772143750,11721562427290210695200000000,
%U 72596493516095364770534596279431168,2527156530619699341247423878706695556300800,496395279097923766533851314609410101501472675840000
%N a(n) = Product_{k=0..n} binomial(n + k, n).
%C Sum_{k=0..n} binomial(n + k, n) = binomial(2*n + 1, n).
%C Product_{k=1..n} binomial(k*n, n) = (n^2)! / (n!)^n.
%F a(n) = (n+1)^n * BarnesG(2*n+2) / (Gamma(n+2)^n * BarnesG(n+2)^2).
%F a(n) ~ A * 2^(2*n^2 + 3*n/2 - 1/12) / (exp(n^2/2 + 1/6) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.
%F a(n) = a(n-1)*2n*(2n-1)!^2/(n!^3*n^(n-1)). - _Chai Wah Wu_, Jun 26 2023
%t Table[Product[Binomial[n+k, n], {k, 0, n}], {n, 0, 13}]
%t Table[(n+1)^n * BarnesG[2*n+2] / (Gamma[n+2]^n * BarnesG[n+2]^2), {n, 0, 13}]
%o (Python)
%o from math import factorial
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A306789(n): return A306789(n-1)*2*n*factorial(2*n-1)**2//factorial(n)**3//n**(n-1) if n else 1 # _Chai Wah Wu_, Jun 26 2023
%Y Cf. A001700, A306760.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Mar 10 2019
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