%I #4 Mar 30 2012 18:37:26

%S 1,1,2,12,62,300,1635,9413,54505,321655,1938621,11834305,72975115,

%T 454385175,2852742151,18034439209,114709370133,733605250447,

%U 4714351916849,30426720332601,197141722168571,1281835943761551

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

%F G.f. satisfies:

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

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 62*x^4 + 300*x^5 + 1635*x^6 +...

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

%e * A(x) = 1 + x*(1 + x*A(x)^3) + x^2*(1 + 8*x*A(x)^3 + x^2*A(x)^6) + x^3*(1 + 27*x*A(x)^3 + 27*x^2*A(x)^6 + x^3*A(x)^9) + x^4*(1 + 64*x*A(x)^3 + 216*x^2*A(x)^6 + 64*x^3*A(x)^9 + x^4*A(x)^12) +...;

%e * A(x) = 1/(1-x-x^2*A(x)^3) + 6*x^3*A(x)^3/(1-x-x^2*A(x)^3)^4 + 90*x^6*A(x)^6/(1-x-x^2*A(x)^3)^7 + 1680*x^9*A(x)^9/(1-x-x^2*A(x)^3)^10 + 34650*x^12*A(x)^12/(1-x-x^2*A(x)^3)^13 +...

%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))^(3*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))^(3*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^(3*m)/(1-x-x^2*A^3+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

%Y Cf. A186097, A187000.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 01 2011