login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A182920 Denominators of an asymptotic series for the factorial function. 3
1, 1, 144, 6480, 41472, 6531840, 1343692800, 1881169920, 5417769369600, 2011346878464000, 5461111524556800, 15060965425938432000, 11678040884112261120000, 15181453149345939456000, 1987390230459832074240000, 585336107626182041665536000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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..15.

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

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-Fran├žois Alcover, Feb 05 2014 *)

CROSSREFS

Cf. A182919.

Sequence in context: A259318 A231744 A262783 * A008662 A181014 A035821

Adjacent sequences:  A182917 A182918 A182919 * A182921 A182922 A182923

KEYWORD

nonn,frac

AUTHOR

Peter Luschny, Mar 11 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 20 16:14 EDT 2017. Contains 292276 sequences.