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%I #9 Mar 06 2014 12:10:59
%S 1,1,13,7330,185307558,201002187396362,9357300769149011773697,
%T 18775362849239140086719414696830,
%U 1631039199744298058694966065590003308698494,6159916689356522044764167426829149420348399496664634288
%N G.f.: exp( Sum_{n>=1} binomial(3*n,n)^n/3^n * x^n/n ).
%C Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x) = exp( Sum_{n>=1} binomial(3*n,n)/3 * x^n/n ).
%F a(n) = (1/n) * Sum_{k=1..n} binomial(3*k,k)^k/3^k * a(n-k) for n>0 with a(0)=1.
%e G.f.: A(x) = 1 + x + 13*x^2 + 7330*x^3 + 185307558*x^4 + 201002187396362*x^5 +...
%e where
%e log(A(x)) = x + 5^2*x^2/2 + 28^3*x^3/3 + 165^4*x^4/4 + 1001^5*x^5/5 + 6188^6*x^6/6 + 38760^7*x^7/7 +...+ A025174(n)^n*x^n/n +...
%t nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k,k]^k/3^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* _Vaclav Kotesovec_, Mar 06 2014 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m, m)^m/3^m*x^m/m)+x*O(x^n)), n)}
%o for(n=0,15,print1(a(n),", "))
%Y Cf A213409, A001764, A025174, A200002.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 10 2012
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