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A196531 E.g.f. satisfies: A(x) = Sum_{n>=0} exp(A(x)^n - 1)*A(x)^n*x^n/n!.

%I A196531 #10 Mar 30 2012 18:37:31
%S A196531 1,1,5,52,853,19291,557719,19657667,817847321,39236975668,
%T A196531 2132767035851,129548751144077,8696909403827077,639445475893738749,
%U A196531 51105213341360790655,4411321463887034379616,409004744372281965629617,40539225718259037965353203
%N A196531 E.g.f. satisfies: A(x) = Sum_{n>=0} exp(A(x)^n - 1)*A(x)^n*x^n/n!.
%F A196531 E.g.f. satisfies: A(x) = exp(-1) * Sum_{n>=0} exp(x*A(x)^(n+1))/n!.
%F A196531 E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A195895; thus, A(x) = (1/x)*Series_Reversion(x/G(x)).
%e A196531 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 52*x^3/3! + 853*x^4/4! + 19291*x^5/5! +...
%e A196531 where
%e A196531 A(x) = 1 + exp(A(x)-1)*A(x)*x + exp(A(x)^2-1)*A(x)^2*x^2/2! + exp(A(x)^3-1)*A(x)^3*x^3/3! +...
%e A196531 Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = e.g.f. of A195895(n):
%e A196531 G(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 2996*x^5/5! + 57613*x^6/6! +...+ A195895(n)*x^n/n! +...
%o A196531 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, exp(A^m-1+x*O(x^n))*A^m*x^m/m!)); n!*polcoeff(A, n)}
%o A196531 (PARI) /* Alternate e.g.f. (requires high precision): */
%o A196531 {a(n)=local(A=1+x); for(i=1, n, A=exp(-1)*sum(m=0, 2*n+10, exp(x*A^(m+1)+x*O(x^n))/m!)); round(n!*polcoeff(A, n))}
%Y A196531 Cf. A195895.
%K A196531 nonn
%O A196531 0,3
%A A196531 _Paul D. Hanna_, Oct 03 2011