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%I #29 Mar 27 2024 05:39:16
%S 1,0,1,2,3,7,8,5,5,2,7,2,2,9,1,2,2,4,9,5,3,9,6,0,2,9,6,0,4,9,6,6,8,8,
%T 6,9,2,9,7,8,0,4,4,8,7,5,8,6,9,1,7,7,1,5,0,2,8,2,0,2,2,6,5,9,5,9,2,9,
%U 3,5,4,3,2,4,3,1,0,7,8,0,9,2,3,4,6,6,1,5,9,2,9,7,4,0,3,1,1,5,8,6,8,2
%N Decimal expansion of Product_{n>=1} (1/e * (1/(3*n)+1)^(3*n+1/2)).
%H Mathematics Stackexchange, <a href="https://math.stackexchange.com/questions/4343366">Proof of a closed-form of Product_{n=1..oo} 1/e * (1/(3*n)+1)^(3*n+1/2)</a>, Dec 28, 2021.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%F 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.
%e 1.012378552722912249539602960496688692978044875869177150282...
%p evalf(product(exp(-1)*(1/(3*n)+1)^(3*n+1/2), n = 1..infinity), 104); # _Vaclav Kotesovec_, Aug 16 2015
%t 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_ *)
%t N[Product[1/E*(1/(3*n) + 1)^(3*n + 1/2), {n, 1, Infinity}], 101] (* _Vaclav Kotesovec_, Aug 16 2015 *)
%Y Cf. A074962.
%K nonn,cons
%O 1,4
%A _Eric W. Weisstein_, Nov 02 2004
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