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%I #6 Dec 24 2012 17:26:33
%S 1,1,2,65,3524,1364432,1445333132,7913299718555,162327934705456532,
%T 14083866155101076361024,5251111824344114834186373747,
%U 7956883819596423111541696080219295,51760975171209084256721290749117849746987,1424616119143714906580708999710589586791029920856
%N O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^4*x)^n/n! * exp(-n*x*A(n^4*x)).
%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 + 2*x^2 + 65*x^3 + 3524*x^4 + 1364432*x^5 +...
%e where
%e A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^4*x)^2/2!*exp(-2*x*A(2^4*x)) + 3^3*x^3*A(3^4*x)^3/3!*exp(-3*x*A(3^4*x)) + 4^4*x^4*A(4^4*x)^4/4!*exp(-4*x*A(4^4*x)) + 5^5*x^5*A(5^4*x)^5/5!*exp(-5*x*A(5^4*x)) +...
%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, x, k^4*x)^k/k!*exp(-k*x*subst(A, x, k^4*x)+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0,16,print1(a(n),", "))
%Y Cf. A218672, A218681, A219342.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 24 2012
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