%I #15 Feb 25 2015 11:20:42
%S 1,1,2,4,12,36,160,680,4368,24976,219616,1599104,18145600,160805568,
%T 2245960704,23437238656,389201754368,4669754525952,89893314615808,
%U 1218303508421632,26673872401980416,402798557886395392,9883555312758398976
%N E.g.f. sqrt(1-t^2)*exp(t*(1+arctanh(t))).
%F a(n) = a(n-1)+sum(k=1..n/2, (n-1)!/(n-2*k)!*(1/(2*k-1))*a(n-2*k)), a(0)=1.
%F a(n) ~ (n-2)! * (exp(1) + (-1)^n*exp(-1)). - _Vaclav Kotesovec_, Feb 24 2015
%t CoefficientList[Series[Sqrt[1-x^2]*E^(x*(1+Log[(1+x)/(1-x)]/2)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 24 2015 *)
%o (Maxima)
%o a(n):=if n=0 then 1 else a(n-1)+sum((n-1)!/(n-2*k)!*(1/(2*k-1))*a(n-2*k),k,1,n/2);
%o (PARI) default(seriesprecision, 50); Vec(serlaplace(sqrt(1-t^2)*exp(t*(1+atanh(t))) + O(t^50)))
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Feb 23 2015