cp's OEIS Frontend

A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

%I A143503 #32 Feb 16 2025 08:33:08
%S A143503 1,-1,1,5,-21,-399,869,39325,-334477,-28717403,59697183,8400372435,
%T A143503 -34429291905,-7199255611995,14631594576045,4251206967062925,
%U A143503 -68787420596367165,-26475975382085110035,53392138323683746235,26275374869163335461975,-105772979046693606062363
%N A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).
%H A143503 F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://arxiv.org/abs/1009.4274">Lehmer's Interesting Series</a>, arXiv:1009.4274 [math-ph], 2010-2011. [Except for the signs, see the unnumbered table on p. 7.]
%H A143503 F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.116">Lehmer's interesting series</a>, Amer. Math. Monthly, 120 (2013), 116-130. [Except for the signs, see Table 4.]
%H A143503 D. H. Lehmer, <a href="https://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Amer. Math. Monthly, 92(7) (1985), 449-457.
%H A143503 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>.
%e A143503 1/sqrt(x^(-1)) - sqrt(x^(-1))/8 + (x^(-1))^(3/2)/128 + (5*(x^(-1))^(5/2))/1024 - (21*(x^(-1))^(7/2))/32768 + ...
%p A143503 H := proc(n) local S, i; S := (x/(exp(x)-1))^(3/2)*exp(x/2);
%p A143503 -pochhammer(1/2,n-1)*coeff(series(S,x,n+2),x,n)*2^(4*n-1-add(i,i= convert(n,base,2))) end:
%p A143503 A143503 := n -> (-1)^irem(n-1,6)*H(n-1);
%p A143503 seq(A143503(n), n=1..16); # _Peter Luschny_, Apr 05 2014
%t A143503 Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* _Vaclav Kotesovec_, Oct 09 2023 *)
%Y A143503 Cf. A061549, A088802 (denominators), A222411, A222412.
%K A143503 sign,frac
%O A143503 1,4
%A A143503 _Eric W. Weisstein_, Aug 20 2008
%E A143503 More terms from _Vaclav Kotesovec_, Oct 09 2023