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 A182919 Numerators of an asymptotic series for the factorial function. 3
 1, 0, 1, -23, 5, 4939, 11839, -1110829, -14470283, 1684880593181, 13113784231, -28792751815367863, -40127106428444687, 97116294357644526719, 15137700541235610329, -17271137929251359193013081753, -622005606550391960056009 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 18 05:11 EDT 2021. Contains 343072 sequences. (Running on oeis4.)