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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. 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-François Alcover, Feb 05 2014 *) CROSSREFS Cf. A182919. Sequence in context: A259318 A231744 A262783 * A299858 A308285 A008662 Adjacent sequences:  A182917 A182918 A182919 * A182921 A182922 A182923 KEYWORD nonn,frac AUTHOR Peter Luschny, Mar 11 2011 STATUS approved

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Last modified October 14 07:15 EDT 2019. Contains 327995 sequences. (Running on oeis4.)