OFFSET
1,4
LINKS
Mathematics Stackexchange, Proof of a closed-form of Product_{n=1..oo} 1/e * (1/(3*n)+1)^(3*n+1/2), Dec 28, 2021.
Eric Weisstein's World of Mathematics, Infinite Product
FORMULA
3^(13/24) * exp(1 + (2*Pi^2 - 3*PolyGamma(1, 1/3))/(12*sqrt(3)*Pi)) * sqrt(Gamma(1/3)/(2*Pi)) / A^4, where A = A074962 is the Glaisher-Kinkelin constant.
EXAMPLE
1.012378552722912249539602960496688692978044875869177150282...
MAPLE
evalf(product(exp(-1)*(1/(3*n)+1)^(3*n+1/2), n = 1..infinity), 104); # Vaclav Kotesovec, Aug 16 2015
MATHEMATICA
RealDigits[(3^(13/24)*E^(1 + (2*Pi^2 - 3*PolyGamma[1, 1/3])/(12*Sqrt[3]*Pi)) * Sqrt[Gamma[1/3]/(2*Pi)])/Glaisher^4, 10, 100][[1]] (* Vaclav Kotesovec, Aug 16 2015 after Eric W. Weisstein *)
N[Product[1/E*(1/(3*n) + 1)^(3*n + 1/2), {n, 1, Infinity}], 101] (* Vaclav Kotesovec, Aug 16 2015 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Nov 02 2004
STATUS
approved