%I #37 Apr 24 2022 20:44:00

%S 1,-1,-1,5,7,-19,-869,715,2339,-200821,-12863,2117,7106149,-64604977,

%T -131301607,7629931291,174053933,-19449462373,-46949081169401,

%U 355455588729389,10635113572583999,-6511303438681407901,-349640201588122693,9112944418860287

%N Numerators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

%H Alois P. Heinz, <a href="/A222411/b222411.txt">Table of n, a(n) for n = 0..300</a>

%H David Broadhurst, <a href="/A241885/a241885.txt">Relations between A241885/A242225, A222411/A222412, and A350194/A350154.</a>

%H 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.

%H 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.

%H 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.

%F Theorem: A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - _David Broadhurst_, Apr 23 2022 (see Link).

%e The first few fractions are 1, -1/4, -1/32, 5/384, 7/10240, -19/40960, -869/61931520, 715/49545216, ... = A222411/A222412. - _Petros Hadjicostas_, May 14 2020

%p gf:= (x/(exp(x)-1))^(3/2)*exp(x/2):

%p a:= n-> numer(coeff(series(gf, x, n+3), x, n)):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Mar 02 2013

%t Series[(x/(Exp[x]-1))^(3/2)*Exp[x/2], {x, 0, 25}] // CoefficientList[#, x]& // Numerator (* _Jean-François Alcover_, Mar 18 2014 *)

%Y Cf. A222412 (denominators).

%Y Cf. also A241885/A242225, A350194/A350154.

%K sign,frac

%O 0,4

%A _N. J. A. Sloane_, Feb 28 2013