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A195947 E.g.f. satisfies: A(x) = Sum_{n>=0} (-1)^n/n! * Sum_{k=0..n} (-1)^k*C(n,k)*(1 + x*A(x)^k)^k.

%I A195947 #13 Mar 30 2012 18:37:29
%S A195947 1,1,5,58,1093,28731,971719,40236449,1972617385,111779567596,
%T A195947 7189852342091,517600784497237,41237095369088029,3602389000897583001,
%U A195947 342422738142493542031,35186740743134660359186,3887047020291801938191057,459397561144034558519708403
%N A195947 E.g.f. satisfies: A(x) = Sum_{n>=0} (-1)^n/n! * Sum_{k=0..n} (-1)^k*C(n,k)*(1 + x*A(x)^k)^k.
%F A195947 E.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^(n^2)*exp(A(x)^n - 1)*x^n/n!.
%e A195947 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1093*x^4/4! + 28731*x^5/5! +...
%e A195947 where:
%e A195947 A(x) = 1 + A(x)*exp(A(x) - 1)*x + A(x)^4*exp(A(x)^2 - 1)*x^2/2! + A(x)^9*exp(A(x)^3 - 1)*x^3/3! + A(x)^16*exp(A(x)^4 - 1)*x^4/4! +...
%e A195947 Also, e.g.f. A = A(x) satisfies:
%e A195947 A(x) = 1 - (1 - (1+x*A)) + 1/2!*(1 - 2*(1+x*A) + (1+x*A^2)^2) -
%e A195947 1/3!*(1 - 3*(1+x*A) + 3*(1+x*A^2)^2 - (1+x*A^3)^3) +
%e A195947 1/4!*(1 - 4*(1+x*A) + 6*(1+x*A^2)^2 - 4*(1+x*A^3)^3 + (1+x*A^4)^4) -
%e A195947 1/5!*(1 - 5*(1+x*A) + 10*(1+x*A^2)^2 - 10*(1+x*A^3)^3 + 5*(1+x*A^4)^4 - (1+x*A^5)^5) +-...
%o A195947 (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1,n,exp(A^m-1)*A^(m^2)*X^m/m!)); n!*polcoeff(A, n)}
%o A195947 (PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, 1/m!*sum(k=0, m, binomial(m, k)*(-1)^(m-k)*(1+X*A^k)^k))); n!*polcoeff(A, n)}
%Y A195947 Cf. A196022, A195895.
%K A195947 nonn
%O A195947 0,3
%A A195947 _Paul D. Hanna_, Sep 27 2011