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Denominators of an asymptotic series for the Gamma function (G. Nemes)
3

%I #8 Jan 06 2014 14:55:55

%S 1,1,144,12960,207360,2612736,9405849600,18811699200,1083553873920,

%T 4022693756928000,300361133850624000,210853515963138048000,

%U 151814531493459394560000,151814531493459394560000,21861292535058152816640000

%N Denominators of an asymptotic series for the Gamma function (G. Nemes)

%C G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.

%D G. Nemes, More Accurate Approximations for the Gamma Function,

%D Thai Journal of Mathematics Volume 9(1) (2011), 21-28.

%H Peter Luschny, <a href="http://www.luschny.de/math/factorial/approx/SimpleCases.html">Approximation Formulas for the Factorial Function.</a>

%F Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi (x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.

%e G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.

%p # See A182912 for G(n).

%p A182913 := n -> denom(G(n)); seq(A182913(i),i=0..15);

%t 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 *)

%Y Cf. A001163, A001164, A182912.

%K nonn,frac

%O 0,3

%A _Peter Luschny_, Feb 09 2011