%I #14 Feb 26 2014 07:51:38
%S 1,1,1,4,25,216,1561,19328,225681,3790720,54595921,1141489152,
%T 20620975849,519551642624,11074307974185,325796967940096,
%U 8068953248603809,271784019897188352,7647972455738262817,290140183695205990400,9154902981606486123321
%N E.g.f. satisfies: A(x) = ( exp(x*A(x)) + exp(x*A(-x)) )/2.
%H Vaclav Kotesovec, <a href="/A198198/b198198.txt">Table of n, a(n) for n = 0..380</a>
%F E.g.f. satisfies: A(x) = Sum_{n>=0} (A(x)^n + A(-x)^n)/2 * x^n/n!.
%F E.g.f.: A(x) = log(F(x))/x = (F(x) + 1/F(-x))/2 where F(x) is the e.g.f. of A199202.
%F E.g.f. satisfies: log(-exp(x*y) + 2*y) + (x*y)/(exp(x*y)*(exp(x*y) - 2*y)) = 0, where y = A(x). - _Vaclav Kotesovec_, Feb 26 2014
%F a(n) ~ c * n! * d^n / n^(3/2), where d = 1.9126860724609002014... (see Mathematica program), and c = 0.8469455456348576... if n is even, and c = 0.9471934761885996... if n is odd. - _Vaclav Kotesovec_, Feb 26 2014
%e E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 216*x^5/5! + 1561*x^6/6! +...
%e Related expansions begin:
%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! + 3607*x^6/6! +...+ A199202(n)*x^n/n! +...
%e exp(x*A(-x)) = 1 + x - x^2/2! - 2*x^3/3! - 3*x^4/4! + 56*x^5/5! - 485*x^6/6! +...
%t (* Output from following program is constant d *) 1/r/.FindRoot[{E^(3*s)*r^2 + 2*r*s^2 - 2*E^(2*s)*r*(1+s) + E^s*(r^2*(1-2*s) + 4*s) == 0,(r*s)/(E^s*(E^s*r-2*s)) + Log[-E^s+(2*s)/r] == 0}, {{r, 1/2}, {s, 1}}, WorkingPrecision -> 50] (* _Vaclav Kotesovec_, Feb 26 2014 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(exp(x*A)+exp(x*subst(A,x,-x+x*O(x^n))))/2);n!*polcoeff(A,n)}
%Y Cf. A195510, A199202.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Oct 22 2011