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A272097
Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.
2
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, 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, 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
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
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]]
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Apr 20 2016
STATUS
approved