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

0,4

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.

Links

Table of n, a(n) for n=0..16.

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.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A182920.

Sequence in context: A281923 A040514 A098103 * A040512 A158514 A040511

Adjacent sequences: A182916 A182917 A182918 * A182920 A182921 A182922

Keyword

sign,frac

Author

Peter Luschny, Mar 11 2011

Status

approved