%I #5 Mar 30 2012 18:37:26

%S 1,1,2,10,39,147,639,2857,12725,58081,270250,1268444,6009439,28736727,

%T 138401100,670641714,3268021317,16004012529,78716657052,388701645264,

%U 1926266491659,9576792342099,47753368809171,238759903786041

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

%F G.f. satisfies:

%F (1) A(x) = Sum_{n>=0} x^(2n)*A(x)^n*[Sum_{k>=0} C(n+k,k)^3*x^k].

%F (2) A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(3n)*A(x)^n/(1-x-x^2*A(x))^(3n+1).

%e G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 39*x^4 + 147*x^5 + 639*x^6 +...

%e where g.f. A(x) satisfies:

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

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

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^3*x^k))); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3,(3*m)!/m!^3*x^(3*m)*A^m/(1-x-x^2*A+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

%Y Cf. A181665, A183876, A181543.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2011