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%I #4 Mar 30 2012 18:37:26
%S 1,1,2,11,50,216,1062,5457,28088,147686,792716,4303036,23587816,
%T 130552984,728316462,4090107939,23106799662,131234374020,748839982332,
%U 4290960328482,24681334371228,142454368143300,824786385860748
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^3 *x^k* A(x)^(2k)].
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k>=0} C(n+k,k)^3*x^k].
%F (2) A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(3n)*A(x)^(2n)/(1-x-x^2*A(x)^2)^(3n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 50*x^4 + 216*x^5 + 1062*x^6 +...
%e where g.f. A(x) satisfies:
%e * A(x) = 1 + x*(1 + x*A(x)^2) + x^2*(1 + 8*x*A(x)^2 + x^2*A(x)^4) + x^3*(1 + 27*x*A(x)^2 + 27*x^2*A(x)^4 + x^3*A(x)^6) + x^4*(1 + 64*x*A(x)^2 + 216*x^2*A(x)^4 + 64*x^3*A(x)^6 + x^4*A(x)^8) +...;
%e * A(x) = 1/(1-x-x^2*A(x)^2) + 6*x^3*A(x)^2/(1-x-x^2*A(x)^2)^4 + 90*x^6*A(x)^4/(1-x-x^2*A(x)^2)^7 + 1680*x^9*A(x)^6/(1-x-x^2*A(x)^2)^10 + 34650*x^12*A(x)^8/(1-x-x^2*A(x)^2)^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))^(2*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))^(2*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^(2*m)/(1-x-x^2*A^2+x*O(x^n))^(3*m+1))); polcoeff(A, n)}
%Y Cf. A186097, A187001.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 01 2011
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