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%I #15 Sep 09 2018 08:36:54
%S -1,7,-31,127,-511,1414477,-8191,118518239,-5749691557,91546277357,
%T -23273283019,1982765468311237,-22076500342261,455371239541065869,
%U -925118910976041358111,16555640865486520478399,-1302480594081611886641,904185845619475242495834469891
%N Numerators of an asymptotic series for the Gamma function (odd power series).
%C Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*(y/E)^y*exp(Sum_{k>=1} r(k)/y^(2*k-1)) as x -> oo and r(k) = A277002(k)/A277003(k) (see example 7.1 in the Wang reference).
%C See also theorem 2 and formula (58) in Borwein and Corless. - _Peter Luschny_, Mar 31 2017
%H J. M. Borwein, R. M. Corless, <a href="https://arxiv.org/abs/1703.05349">Gamma and Factorial in the Monthly</a>, arXiv:1703.05349 [math.HO], 2017.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/FactorialFunction">Approximations to 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 a(n) = numerator(b(2*n-1)) with b(n) = Bernoulli(n+1, 1/2)/(n*(n+1)) for n>=1, b(0)=0.
%e The underlying rational sequence b(n) starts:
%e 0, -1/24, 0, 7/2880, 0, -31/40320, 0, 127/215040, 0, -511/608256, ...
%p b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
%p a := n -> numer(b(2*n-1)):
%p seq(a(n), n=1..18);
%t b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
%t a[n_] := Numerator[b[2n-1]];
%t Array[a, 18] (* _Jean-François Alcover_, Sep 09 2018 *)
%Y Cf. A277003 (denominators), A277000/A277001 (even power series).
%K sign,frac
%O 1,2
%A _Peter Luschny_, Sep 26 2016
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