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

Offset: 1

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

• nonn
• cons

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.

Examples

			1.00268791324152794158434554643452096181810403192367888372866567380647785...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

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

Programs

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]]

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.