OFFSET
0,2
COMMENTS
The next term has 104 digits.
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).
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.
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)).
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.
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).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..13
Eric Weisstein's World of Mathematics, Barnes G-Function
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant
FORMULA
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.
MATHEMATICA
Table[Product[(7*k)!, {k, 0, n}], {n, 0, 8}]
PROG
(PARI) {a(n) = prod(k=1, n, (7*k)!)} \\ Seiichi Manyama, Jul 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Apr 17 2016
STATUS
approved