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Numerators of an asymptotic series for the factorial function.
3

%I #14 Feb 28 2023 09:45:55

%S 1,0,1,-23,5,4939,11839,-1110829,-14470283,1684880593181,13113784231,

%T -28792751815367863,-40127106428444687,97116294357644526719,

%U 15137700541235610329,-17271137929251359193013081753,-622005606550391960056009

%N Numerators of an asymptotic series for the factorial function.

%C G_n = A182919(n)/A182920(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="https://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 G = Sum_{k>=0} G[k]/n^k, then n! ~ sqrt(2Pi(n+1/6))*(n/e)^n*G.

%e G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -23/6480, G_4 = 5/41472.

%p CoefNumer := f -> numer([1,seq(coeff(convert(series(f,n=infinity,20), polynom),n^(-k)),k=1..16)]): CoefNumer(n!/(n^n/exp(n)*sqrt(2*Pi)*sqrt(n+1/6)));

%t a[n_] := SeriesCoefficient[ x!/(x^x/Exp[x]*Sqrt[2*Pi]*Sqrt[x+1/6]) /. x -> 1/y, {y, 0, n}]; Table[a[n] // Numerator, {n, 0, 16}] (* _Jean-François Alcover_, Feb 05 2014 *)

%Y Cf. A182920.

%K sign,frac

%O 0,4

%A _Peter Luschny_, Mar 11 2011