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A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

%I A144618 #19 Sep 23 2016 12:58:48
%S A144618 1,24,1152,414720,39813120,6688604160,4815794995200,115579079884800,
%T A144618 22191183337881600,263631258054033408000,88580102706155225088000,
%U A144618 27636992044320430227456000,39797268543821419527536640000
%N A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
%C A144618 From _Peter Luschny_, Feb 24 2011 (Start):
%C A144618 G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.
%C A144618 The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie). (End)
%C A144618 Also denominators of polynomials mentioned in A144617.
%C A144618 Also denominators of polynomials mentioned in A144622.
%H A144618 Chris Kormanyos, <a href="/A144618/b144618.txt">Table denominators of u_k for k=0..121</a>
%H A144618 Dirk Laurie, <a href="http://dip.sun.ac.za/~laurie/papers/computing_gamma.pdf">Old and new ways of computing the gamma function</a>, page 14, 2005.
%H A144618 Peter Luschny, <a href="http://www.luschny.de/math/factorial/approx/SimpleCases.html">Approximation Formulas for the Factorial Function.</a>
%H A144618 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 A144618 z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2)  Sum_{n>=0} G_n / (z+1/2)^n.
%F A144618 - _Peter Luschny_, Feb 24 2011
%e A144618 G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
%p A144618 G := proc(n) option remember; local j,R;
%p A144618 R := seq(2*j,j=1..iquo(n+1,2));
%p A144618 `if`(n=0,1,add(bernoulli(j,1/2)*G(n-j+1)/(n*j),j=R)) end:
%p A144618 A144618 := n -> denom(G(n)); seq(A144618(i),i=0..12);
%p A144618 # _Peter Luschny_, Feb 24 2011
%t A144618 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] // Denominator, {n, 0, 12}] (* _Jean-François Alcover_, Jul 26 2013, after Maple *)
%Y A144618 Cf. A001163, A001164, A182935, A144617, A144622.
%K A144618 nonn,frac
%O A144618 0,2
%A A144618 _N. J. A. Sloane_, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
%E A144618 Added more terms up to polynomial number u_12, v_12 for the denominators of u_k, v_k. Christopher Kormanyos (ckormanyos(AT)yahoo.com), Jan 31 2009
%E A144618 Typo in definition corrected Aug 05 2010 by _N. J. A. Sloane_
%E A144618 A-number in definition corrected - _R. J. Mathar_, Aug 05 2010
%E A144618 Edited and new definition by Peter Luschny, Feb 24 2011