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A182913
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Denominators of an asymptotic series for the Gamma function (G. Nemes)
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3
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1, 1, 144, 12960, 207360, 2612736, 9405849600, 18811699200, 1083553873920, 4022693756928000, 300361133850624000, 210853515963138048000, 151814531493459394560000, 151814531493459394560000, 21861292535058152816640000
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OFFSET
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0,3
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COMMENTS
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G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.
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REFERENCES
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G. Nemes, More Accurate Approximations for the Gamma Function,
Thai Journal of Mathematics Volume 9(1) (2011), 21-28.
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LINKS
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Table of n, a(n) for n=0..14.
Peter Luschny, Approximation Formulas for the Factorial Function.
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FORMULA
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Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi (x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.
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EXAMPLE
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G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.
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MAPLE
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# See A182912 for G(n).
A182913 := n -> denom(G(n)); seq(A182913(i), i=0..15);
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MATHEMATICA
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G[n_] := G[n] = Module[{j, J}, J[k_] := J[k] = Module[{j}, If[k == 0, 1, (J[k-1]/k - Sum[J[k-j]*J[j]/(j+1), {j, 1, k-1}])/(1+1/(k+1))]]; Sum[J[2*j]*2^j*6^(j-n)*Gamma[1/2+j]/(Gamma[n-j+1]*Gamma[1/2+j-n]), {j, 0, n}] - Sum[G[j]*(-4)^(j-n)*Gamma[n]/(Gamma[n-j+1]*Gamma[j]), {j, 1, n-1}]]; A182913[n_] := Denominator[G[n]]; Table[A182913[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)
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CROSSREFS
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Cf. A001163, A001164, A182912.
Sequence in context: A231854 A055352 A199406 * A231697 A238932 A230969
Adjacent sequences: A182910 A182911 A182912 * A182914 A182915 A182916
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KEYWORD
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nonn,frac
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AUTHOR
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Peter Luschny, Feb 09 2011
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STATUS
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approved
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