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A202519 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) + (-1)^n)^n * x^n/n ).

%I A202519 #27 Mar 30 2012 18:37:33
%S A202519 1,1,7,27,165,877,5451,32887,210505,1347865,8859695,58647219,
%T A202519 393704205,2662542565,18166847507,124738843247,861922384657,
%U A202519 5986483380145,41780493605719,292817777533259,2060138522838645,14544377538584925,103007560370361691,731635362026777831
%N A202519 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) + (-1)^n)^n * x^n/n ).
%F A202519 G.f. satisfies: A(x) = 1/(1-2*x*A(x)) * exp( Sum_{n>=1} (-1)^n/(1 - (-1)^n*2*x*A(x))^n * x^n/n ).
%F A202519 G.f. satisfies: A(x) = sqrt( (1 - (2*A(x)-1)^2*x^2)/(1 - (2*A(x)+1)^2*x^2) ) / (1 - (2*A(x)-1)*x).
%F A202519 G.f. satisfies: 0 = -(1-x) - 2*x*A(x) + (1-x)*(1+x)^2*A(x)^2 - 2*x*(1+x)^2*A(x)^3 - 2^2*x^2*(1-x)*A(x)^4 + 2^3*x^3*A(x)^5.
%e A202519 G.f.: A(x) = 1 + x + 7*x^2 + 27*x^3 + 165*x^4 + 877*x^5 + 5451*x^6 +...
%e A202519 where
%e A202519 log(A(x)) = (2*A(x) - 1)*x + (2*A(x) + 1)^2*x^2/2 + (2*A(x) - 1)^3*x^3/3 + (2*A(x) + 1)^4*x^4/4 +...
%e A202519 log(A(x)*(1-2*x*A(x))) = -1/(1 + 2*x*A(x))*x + 1/(1 - 2*x*A(x))^2*x^2/2 - 1/(1 + 2*x*A(x))^3*x^3/3 + 1/(1 - 2*x*A(x))^4*x^4/4 +...
%o A202519 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2*A+(-1)^m+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}
%Y A202519 Cf. A185385, A163138, A202669, A155200.
%K A202519 nonn
%O A202519 0,3
%A A202519 _Paul D. Hanna_, Dec 22 2011