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G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3*A(x)^k) * x^n/n ).
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%I #8 Mar 30 2012 18:37:27

%S 1,2,9,53,357,2611,20180,162276,1344455,11400944,98498545,864068233,

%T 7677040177,68947431898,624960856374,5710352911097,52542826413590,

%U 486458467209032,4528570067254485,42365044032385154,398081015128641213

%N G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3*A(x)^k) * x^n/n ).

%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 53*x^3 + 357*x^4 + 2611*x^5 + 20180*x^6 +...

%e where the g.f. satisfies:

%e log(A(x)) = (1 + A(x))*x + (1 + 8*A(x) + A(x)^2)*x^2/2 + (1 + 27*A(x) + 27*A(x)^2 + A(x)^3)*x^3/3 + (1 + 64*A(x) + 216*A(x)^2 + 64*A(x)^3 + A(x)^4)*x^4/4 +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

%Y Cf. A007863 (variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 24 2011