A182920 Denominators of an asymptotic series for the factorial function.

1, 1, 144, 6480, 41472, 6531840, 1343692800, 1881169920, 5417769369600, 2011346878464000, 5461111524556800, 15060965425938432000, 11678040884112261120000, 15181453149345939456000, 1987390230459832074240000, 585336107626182041665536000000

Offset: 0

Author

Peter Luschny, Mar 11 2011

Keywords

• nonn
• frac

Comments

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.

Examples

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

Links

• Peter Luschny, Approximations to the factorial function, Factorial Function.
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
• Eric Weisstein's World of Mathematics, Stirling's Approximation.

Crossrefs

Cf. A182919.

Programs

Maple

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)));

Mathematica

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 *)

Formula

Let G = Sum_{k>=0} G[k]/n^k, then n! ~ sqrt(2Pi(n+1/6))*(n/e)^n*G.