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A241140 Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation. 2

%I #15 Nov 16 2021 18:44:13

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

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

%U 9,2,3,8,4,4,0,6,2,1,4,0,4,4,6,8,3,0,2,0,1,6,7,3,0,1,6,6,3,3,2,3,3

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

%C The same product where the ratio is replaced by sqrt(2*Pi) evaluates as (2*Pi)^(1/4) = 1.58323...

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 135.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 20.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>

%F Product_{n>=1} (n! / ((sqrt(2*Pi*n)*n^n)/e^n))^((-1)^(n-1)) = A^3/(2^(7/12)*Pi^(1/4)), where A is the Glaisher-Kinkelin constant.

%e 1.057328141001876924952657094184286643131791252628433822095146...

%t RealDigits[Glaisher^3/(2^(7/12)*Pi^(1/4)), 10, 101] // First

%Y Cf. A019727, A074962.

%K nonn,cons,easy

%O 1,3

%A _Jean-François Alcover_, Aug 08 2014

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