%I #7 Dec 21 2017 17:38:18

%S 0,1,-4,48,-1186,50060,-3226206,294835184,-36270477034,5779302944436,

%T -1157856177719830,284876691727454552,-84442374415240892898,

%U 29680054107768128647388,-12205478262363331593956686,5805823539844285054558025280,-3163004294186696659107788567386

%N Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

%H Vaclav Kotesovec, <a href="/A296838/b296838.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).

%F a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - _Vaclav Kotesovec_, Dec 21 2017

%e log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

%t nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

%Y Cf. A001469, A003707, A009379, A009399, A110501, A296837.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Dec 21 2017