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%I #4 Nov 18 2012 12:53:59
%S 1,2,14,238,6636,273354,15920706,1292724636,146453417488,
%T 23281175674462,5218509363479914,1654434832566803018,
%U 743482275590960686464,474454676244907785390480,430533281246889283353506596,556106522019612061492965277720,1023417606325146596408758881753232
%N O.g.f. satisfies: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * (n*x)^n * A(n*x)^n/n! * exp(-(n+2)*n*x*A(n*x)).
%C Compare the g.f. to the LambertW identity:
%C 1 = Sum_{n>=0} 2*(n+2)^(n-1) * exp(-(n+2)*x) * x^n/n!.
%e O.g.f.: A(x) = 1 + 2*x + 14*x^2 + 238*x^3 + 6636*x^4 + 273354*x^5 +...
%e where
%e A(x) = 1 + 2*3^0*1^1*x*A(x)*exp(-3*1*x*A(x)) + 2*4^1*2^2*x^2*A(2*x)^2*exp(-4*2*x*A(2*x))/2! + 2*5^2*3^3*x^3*A(3*x)^3*exp(-5*3*x*A(3*x))/3! + 2*6^3*4^4*x^4*A(4*x)^4*exp(-6*4*x*A(4*x))/4! + 2*7^4*5^5*x^5*A(5*x)^5*exp(-7*5*x*A(5*x))/5! +...
%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,2*(k+2)^(k-1)*k^k*x^k*subst(A,x,k*x)^k/k!*exp(-(k+2)*k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A219345, A218102, A219220, A217900.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 18 2012
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