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Numerators of an asymptotic series for the factorial function (S. Wehmeier).
3

%I #12 Sep 23 2016 12:40:23

%S 1,1,-31,-139,9871,324179,-8225671,-69685339,1674981058019,

%T 24279707153761,-25107254122618741,-1539511255447387949,

%U 181685125700898353671,368622611536873334281,-538867967772767903436298889

%N Numerators of an asymptotic series for the factorial function (S. Wehmeier).

%C W_n = A182916(n)/A182917(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function. It is a generalization of Gosper's approximation.

%H Peter Luschny, Approximations to the factorial function, <a href="http://oeis.org/wiki/User:Peter_Luschny/FactorialFunction">Factorial Function</a>.

%H W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation</a>.

%F Let A = Sum_{k>=0} W[k]/n^k, then n! ~ sqrt(2Pi*(n+A))*(n/e)^n.

%e W_0 = 1/6, W_1 = 1/72, W_2 = -31/6480, W_3 = -139/155520, W_4 = 9871/6531840.

%Y Cf. A182917.

%K sign,frac

%O 0,3

%A _Peter Luschny_, Mar 09 2011