A272097 - OEIS

A272097

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

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

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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

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