A272097 - OEIS

%I #14 Jan 16 2017 12:26:30

%S 1,0,0,2,6,8,7,9,1,3,2,4,1,5,2,7,9,4,1,5,8,4,3,4,5,5,4,6,4,3,4,5,2,0,

%T 9,6,1,8,1,8,1,0,4,0,3,1,9,2,3,6,7,8,8,8,3,7,2,8,6,6,5,6,7,3,8,0,6,4,

%U 7,7,8,5,0,6,2,1,1,1,0,0,7,3,8,5,3,8,1,0,9,5,8,8,6,6,7,8,2,6,3,5,8,8,0,1,9

%N Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.

%C 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.

%C 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.

%H G. C. Greubel, <a href="/A272097/b272097.txt">Table of n, a(n) for n = 1..5000</a>

%F Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n)))).

%F Equals exp(1/12) * (2*Pi)^(1/4) * Gamma(1/12) / (12 * A^2), where A = A074962 is the Glaisher-Kinkelin constant.

%e 1.00268791324152794158434554643452096181810403192367888372866567380647785...

%t Product[n!/(n^n/E^n*Sqrt[2*Pi*n]*(1 + 1/(12*n))), {n, 1, Infinity}]

%t RealDigits[E^(1/12)*(2*Pi)^(1/4)*Gamma[13/12]/Glaisher^2, 10, 120][[1]]

%Y Cf. A000142, A001163, A001164, A074962, A203138, A203140, A213080, A241140.

%K nonn,cons

%O 1,4

%A _Vaclav Kotesovec_, Apr 20 2016