%I #15 Nov 05 2012 12:00:46

%S 1,1,3,20,209,3173,67292,1970761,79764057,4490097388,354111363537,

%T 39360693851404,6193012446752244,1383433132321835172,

%U 439684769985895688173,199116777197880585373014,128631139424158036273736167,118640007280899188486618513612

%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(2*n)/n! * exp(-n*x*A(n*x)^2).

%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 + 3*x^2 + 20*x^3 + 209*x^4 + 3173*x^5 + 67292*x^6 +...

%e where

%e A(x) = 1 + x*A(x)^2*exp(-x*A(x)^2) + 2^2*x^2*A(2*x)^4/2!*exp(-2*x*A(2*x)^2) + 3^3*x^3*A(3*x)^6/3!*exp(-3*x*A(3*x)^2) + 4^4*x^4*A(4*x)^8/4!*exp(-4*x*A(4*x)^2) + 5^5*x^5*A(5*x)^10/5!*exp(-5*x*A(5*x)^2) +...

%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^2,x,k*x)^k/k!*exp(-k*x*subst(A^2,x,k*x)+x*O(x^n))));polcoeff(A,n)}

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

%Y Cf. A218672, A218674, A218675, A218676.

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2012