A182919 Numerators of an asymptotic series for the factorial function.

1, 0, 1, -23, 5, 4939, 11839, -1110829, -14470283, 1684880593181, 13113784231, -28792751815367863, -40127106428444687, 97116294357644526719, 15137700541235610329, -17271137929251359193013081753, -622005606550391960056009

Offset: 0

Author

Peter Luschny, Mar 11 2011

Keywords

• sign
• 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. A182920.

Programs

Maple

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)));

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] // Numerator, {n, 0, 16}] (* 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.