%I #8 Feb 28 2014 04:32:40
%S 1,1,3,10,53,376,3607,38032,498409,7122304,121691051,2182921984,
%T 45592175389,987527547904,24479592884671,620921169012736,
%U 17795726532904913,517636848366223360,16851227968120051027,552890360903850459136,20150074601540899828741
%N E.g.f. satisfies: A(x) = exp( x*(A(x) + 1/A(-x))/2 ).
%C Compare to the e.g.f. G(x) of A058014, which satisfies both: G(x) = exp(x*(G(x) + 1/G(x))/2) and G(x) = exp(x*(G(x) + G(-x))/2); A058014 counts labeled trees such that the degrees of all nodes, excluding the first, are odd.
%H Vaclav Kotesovec, <a href="/A199202/b199202.txt">Table of n, a(n) for n = 0..320</a>
%F E.g.f.: A(x) = exp(x*B(x)) where B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2 is the e.g.f. of A198198.
%F E.g.f. satisfies: log(x) = x*log(y)/(x*y^2 - 2*y*log(y)) + log(2*log(y) - x*y), where y = A(x). - _Vaclav Kotesovec_, Feb 28 2014
%F a(n) ~ c * n! * d^n / n^(3/2), where d = 1.9126860724609002014... (see A198198), and c = 1.84843299011729... if n is even, and c = 1.808309580980992... if n is odd. - _Vaclav Kotesovec_, Feb 28 2014
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! +.. .
%e Let B(x) = log(A(x))/x = (A(x) + 1/A(-x))/2 then B(x) begins:
%e B(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 216*x^5/5! + 1561*x^6/6! + 19328*x^7/7! +...+ A198198(n)*x^n/n! +...
%e such that B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2.
%o (PARI) {a(n)=local(A=1+x*O(x^n)); for(n=0, n, A=exp(x*(A+1/subst(A, x, -x))/2+x*O(x^n))); n!*polcoeff(A, n)}
%Y Cf. A198198, A058014.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 03 2011