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A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

%I A218676 #9 Nov 05 2012 12:02:09
%S A218676 1,1,6,71,1311,34146,1207717,57298282,3653975784,316252925221,
%T A218676 37596625187796,6206102367103899,1434418185304457039,
%U A218676 466995106832397752352,215051811411620578152401,140491107719613466192347681,130481943378389095603359529403
%N A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).
%C A218676 Compare to the LambertW identity:
%C A218676 Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
%e A218676 O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
%e A218676 where
%e A218676 A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
%e A218676 simplifies to a power series in x with integer coefficients.
%o A218676 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,x,k*x)+x*O(x^n))));polcoeff(A,n)}
%o A218676 for(n=0,25,print1(a(n),", "))
%Y A218676 Cf. A218672, A218673, A218674, A218675.
%Y A218676 Cf. A217900, A218670, A218667, A218668, A218669, A134055.
%K A218676 nonn
%O A218676 0,3
%A A218676 _Paul D. Hanna_, Nov 04 2012