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%I #8 Jan 06 2014 14:55:34
%S 1,0,1,-1,-257,-53,5741173,37529,-710165119,-3376971533,
%T 360182239526821,104939254406053,-508096766056991140541,
%U -70637580369737593,289375690552473442964467,796424971106808496421869
%N Numerators 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, 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 G := proc(n) option remember; local j,J;
%p J := proc(k) option remember; local j; `if`(k=0,1,
%p (J(k-1)/k-add((J(k-j)*J(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
%p add(J(2*j)*2^j*6^(j-n)*GAMMA(1/2+j)/(GAMMA(n-j+1)*GAMMA(1/2+j-n)),j=0..n)-add(G(j)*(-4)^(j-n)*(GAMMA(n)/(GAMMA(n-j+1)*GAMMA(j))),j=1..n-1) end:
%p A182912 := n -> numer(G(n)); seq(A182912(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}]]; A182912[n_] := Numerator[G[n]]; Table[A182912[i], {i, 0, 15}] (* _Jean-François Alcover_, Jan 06 2014, translated from Maple *)
%Y Cf. A001163, A001164, A182913.
%K sign,frac
%O 0,5
%A _Peter Luschny_, Feb 09 2011
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