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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
Peter Luschny, Approximations to the factorial function, Factorial Function.
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-François Alcover, Feb 05 2014 *)
CROSSREFS
Cf. A182919.
Sequence in context: A259318 A231744 A262783 * A299858 A308285 A008662
KEYWORD
nonn,frac
AUTHOR
Peter Luschny, Mar 11 2011
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)