OFFSET
1,4
COMMENTS
Product_{k=1..n} (k! / (sqrt(2*Pi*k) * k^k * exp(-k))) ~ c * n^(1/12), where c = exp(1/12)*(2*Pi)^(1/4) / A^2 = A213080 = 1.04633506677050318098095065697776..., where A = A074962 is the Glaisher-Kinkelin constant.
Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n) + 1/(288*n^2)))) = exp(1/12) * (2*Pi)^(1/4) * abs(Gamma(25/24 + i/24))^2 / A^2 = 0.997305599490607358564533726617761207426462854447669845..., where A = A074962 is the Glaisher-Kinkelin constant and i is the imaginary unit.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
FORMULA
Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n)))).
Equals exp(1/12) * (2*Pi)^(1/4) * Gamma(1/12) / (12 * A^2), where A = A074962 is the Glaisher-Kinkelin constant.
EXAMPLE
1.00268791324152794158434554643452096181810403192367888372866567380647785...
MATHEMATICA
Product[n!/(n^n/E^n*Sqrt[2*Pi*n]*(1 + 1/(12*n))), {n, 1, Infinity}]
RealDigits[E^(1/12)*(2*Pi)^(1/4)*Gamma[13/12]/Glaisher^2, 10, 120][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Apr 20 2016
STATUS
approved