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A182913 Denominators of an asymptotic series for the Gamma function (G. Nemes).

%I A182913 #10 Jun 02 2025 03:15:01
%S A182913 1,1,144,12960,207360,2612736,9405849600,18811699200,1083553873920,
%T A182913 4022693756928000,300361133850624000,210853515963138048000,
%U A182913 151814531493459394560000,151814531493459394560000,21861292535058152816640000
%N A182913 Denominators of an asymptotic series for the Gamma function (G. Nemes).
%C A182913 G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.
%D A182913 G. Nemes, More Accurate Approximations for the Gamma Function,
%D A182913 Thai Journal of Mathematics Volume 9(1) (2011), 21-28.
%H A182913 Peter Luschny, <a href="http://www.luschny.de/math/factorial/approx/SimpleCases.html">Approximation Formulas for the Factorial Function.</a>
%F A182913 Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi (x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.
%e A182913 G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.
%p A182913 # See A182912 for G(n).
%p A182913 A182913 := n -> denom(G(n)); seq(A182913(i),i=0..15);
%t A182913 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 A182913 Cf. A001163, A001164, A182912.
%K A182913 nonn,frac
%O A182913 0,3
%A A182913 _Peter Luschny_, Feb 09 2011