%I #11 Nov 05 2012 12:01:38

%S 1,1,5,51,817,18562,576687,24203258,1375038677,106708683355,

%T 11435867474152,1708844338589752,358640659116617571,

%U 106261016900832212139,44607231638918264608274,26598477338494285370797703,22569718290467849884279856477

%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

%C Compare to the LambertW identity:

%C Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

%e O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...

%e where

%e A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...

%e simplifies to a power series in x with integer coefficients.

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A218672, A218673, A218674, A218676.

%Y Cf. A217900, A218670, A218667, A218668, A218669, A134055.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2012