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a(n) = Product_{k=0..n} (7*k)!.
9

%I #18 Jul 06 2019 04:11:50

%S 1,5040,439378587648000,22448266013011335649028997120000000,

%T 6844214664110424043644485692109939233534721371668480000000000000

%N a(n) = Product_{k=0..n} (7*k)!.

%C The next term has 104 digits.

%C In general, for p > 0, Product_{k=0..n} (p*k)! = p^((n+1)*(n*p+1)/2) * (2*Pi)^((n+1)*(1-p)/2) * Product_{j=1..p} BarnesG(n+1+j/p) / BarnesG(j/p).

%C Equivalently, Product_{k=0..n} (p*k)! = A^(p - 1/p) * exp(1/(12*p) - p/12) * (2*Pi)^((1-p)*n/2) * p^(p*n^2/2 + (p+1)*n/2 - 1/(12*p)) * Product_{j=1..p} (BarnesG(n + 1 + j/p) / Gamma(j/p)^(j/p)), where A = A074962 is the Glaisher-Kinkelin constant.

%C Asymptotics: Product_{k=0..n} (p*k)! ~ exp(p/12 - (p+1)*n/2 - 3*p*n^2/4) * n^(1/4 + p/12 + 1/(12*p) + (p+1)*n/2 + p*n^2/2) * p^(1/2 + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(n/2 - p/4 + 3/4) / (A^p * Product_{j=1..p} BarnesG(j/p)).

%C Equivalently, Product_{k=0..n} (p*k)! ~ A^(-1/p) * exp(1/(12*p) - (p+1)*n/2 - 3*p*n^2/4) * n^(1/4 + p/12 + 1/(12*p) + (p+1)*n/2 + p*n^2/2) * p^(-1/(12*p) + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(n/2 + (p+1)/4) / Product_{j=1..p-1} Gamma(j/p)^(j/p), where A = A074962 is the Glaisher-Kinkelin constant.

%C The general formula for the product of Barnes-G functions is: Product_{j=1..p} BarnesG(j/p) = A^(1/p - p) * exp(p/12 - 1/(12*p)) * p^(1/2 + 1/(12*p)) * (2*Pi)^((1-p)/2) * Product_{j=1..p-1} Gamma(j/p)^(j/p).

%H Seiichi Manyama, <a href="/A271947/b271947.txt">Table of n, a(n) for n = 0..13</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>

%F a(n) ~ A^(-1/7) * exp(1/84 - 4*n - 21*n^2/4) * n^(71/84 + 4*n + 7*n^2/2) * 7^(-1/84 + 4*n + 7*n^2/2) * (2*Pi)^(n/2 + 2) / (Gamma(1/7)^(1/7) * Gamma(2/7)^(2/7) * Gamma(3/7)^(3/7) * Gamma(4/7)^(4/7) * Gamma(5/7)^(5/7) * Gamma(6/7)^(6/7)), where A = A074962 is the Glaisher-Kinkelin constant.

%t Table[Product[(7*k)!,{k,0,n}],{n,0,8}]

%o (PARI) {a(n) = prod(k=1, n, (7*k)!)} \\ _Seiichi Manyama_, Jul 06 2019

%Y Cf. A000178 (p=1), A098694 (p=2), A268504 (p=3), A268505 (p=4), A268506 (p=5), A271946 (p=6).

%Y Partial products of A195391.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Apr 17 2016