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A182919
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Numerators of an asymptotic series for the factorial function.
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3
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1, 0, 1, -23, 5, 4939, 11839, -1110829, -14470283, 1684880593181, 13113784231, -28792751815367863, -40127106428444687, 97116294357644526719, 15137700541235610329, -17271137929251359193013081753, -622005606550391960056009
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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Let G = Sum_{k>=0} G[k]/n^k, then n! ~ sqrt(2Pi(n+1/6))*(n/e)^n*G.
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EXAMPLE
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G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -23/6480, G_4 = 5/41472.
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MAPLE
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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)));
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A182920.
Sequence in context: A281923 A040514 A098103 * A040512 A158514 A040511
Adjacent sequences: A182916 A182917 A182918 * A182920 A182921 A182922
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KEYWORD
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sign,frac
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AUTHOR
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Peter Luschny, Mar 11 2011
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STATUS
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approved
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