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A196959 E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} (-1)^(n-k) * C(n,k) * (1 + x*A(x)^(2*k))^k.

%I A196959 #5 Mar 30 2012 18:37:31
%S A196959 1,1,9,193,6721,326001,20316937,1548374129,139576777921,
%T A196959 14530808439073,1715928199384521,226652340142349793,
%U A196959 33113449456084235905,5302086923264289694225,923349950199153833740105,173761214485224395469845521,35139709415689684107278235265
%N A196959  E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} (-1)^(n-k) * C(n,k) * (1 + x*A(x)^(2*k))^k.
%F A196959  E.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^(2*n^2) * exp(A(x)^(2*n) - 1)*x^n/n!.
%e A196959  E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6721*x^4/4! + 326001*x^5/5! +...
%e A196959 where:
%e A196959 A(x) = 1 + A(x)^2*exp(A(x)^2 - 1)*x + A(x)^8*exp(A(x)^4 - 1)*x^2/2! + A(x)^18*exp(A(x)^6 - 1)*x^3/3! + A(x)^32*exp(A(x)^8 - 1)*x^4/4! +...
%e A196959 Also, e.g.f. A = A(x) satisfies:
%e A196959 A(x) = 1 - (1 - (1+x*A^2)) + 1/2!*(1 - 2*(1+x*A^2) + (1+x*A^4)^2) -
%e A196959 1/3!*(1 - 3*(1+x*A^2) + 3*(1+x*A^4)^2 - (1+x*A^6)^3) +
%e A196959 1/4!*(1 - 4*(1+x*A^2) + 6*(1+x*A^4)^2 - 4*(1+x*A^6)^3 + (1+x*A^8)^4) -
%e A196959 1/5!*(1 - 5*(1+x*A^2) + 10*(1+x*A^4)^2 - 10*(1+x*A^6)^3 + 5*(1+x*A^8)^4 - (1+x*A^10)^5) +-...
%o A196959  (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^(2*m)-1)*A^(2*m^2)*X^m/m!)); n!*polcoeff(A, n)}
%o A196959 (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^(2*k))^k))); n!*polcoeff(A, n)}
%Y A196959  Cf. A195947, A196958.
%K A196959 nonn
%O A196959 0,3
%A A196959 _Paul D. Hanna_, Oct 08 2011