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%I #13 Oct 09 2023 04:40:36
%S 1,-1,1,1003,-4027,-5128423,168359651,68168266699,-587283555451,
%T -221322134443186643,3253248645450176257,52946591945344238676937,
%U -3276995262387193162157789,-6120218676760621380031990351
%N Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
%C G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.
%C The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie).
%H Dirk Laurie, <a href="https://web.archive.org/web/20150911031428/http://dip.sun.ac.za/~laurie/papers/computing_gamma.pdf">Old and new ways of computing the gamma function</a>, page 14, 2005.
%H Peter Luschny, <a href="http://www.luschny.de/math/factorial/approx/SimpleCases.html">Approximation Formulas for the Factorial Function.</a>
%H W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).
%F z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
%e G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
%p G := proc(n) option remember; local j,R;
%p R := seq(2*j,j=1..iquo(n+1,2));
%p `if`(n=0,1,add(bernoulli(j,1/2)*G(n-j+1)/(n*j),j=R)) end:
%p A182935 := n -> numer(G(n)); seq(A182935(i),i=0..15);
%t a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Numerator, {n, 0, 15}] (* _Jean-François Alcover_, Jul 26 2013, after Maple *)
%Y Cf. A001163, A001164, A144618.
%K sign,frac
%O 0,4
%A Peter Luschny, Feb 24 2011
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