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A182920 Denominators of an asymptotic series for the factorial function.

%I A182920 #12 Feb 16 2025 08:33:13
%S A182920 1,1,144,6480,41472,6531840,1343692800,1881169920,5417769369600,
%T A182920 2011346878464000,5461111524556800,15060965425938432000,
%U A182920 11678040884112261120000,15181453149345939456000,1987390230459832074240000,585336107626182041665536000000
%N A182920 Denominators of an asymptotic series for the factorial function.
%C A182920 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 A182920 Peter Luschny, Approximations to the factorial function, <a href="https://oeis.org/wiki/User:Peter_Luschny/FactorialFunction">Factorial Function</a>.
%H A182920 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 A182920 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation.</a>
%F A182920 Let G = Sum_{k>=0} G[k]/n^k, then n! ~ sqrt(2Pi(n+1/6))*(n/e)^n*G.
%e A182920 G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -23/6480, G_4 = 5/41472.
%p A182920 CoefDenom := f -> denom([1,seq(coeff(convert(series(f,n=infinity,20), polynom),n^(-k)),k=1..16)]): CoefDenom(n!/(n^n/exp(n)*sqrt(2*Pi)*sqrt(n+1/6)));
%t A182920 a[n_] := SeriesCoefficient[ x!/(x^x/Exp[x]*Sqrt[2*Pi]*Sqrt[x+1/6]) /. x -> 1/y, {y, 0, n}]; Table[a[n] // Denominator, {n, 0, 15}] (* _Jean-François Alcover_, Feb 05 2014 *)
%Y A182920 Cf. A182919.
%K A182920 nonn,frac
%O A182920 0,3
%A A182920 _Peter Luschny_, Mar 11 2011