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A213409 G.f.: exp( Sum_{n>=1} binomial(3*n^2,n^2) * x^n/n ).

%I A213409 #16 Feb 04 2017 01:46:22
%S A213409 1,3,252,1563022,563716946982,10517711119760250261,
%T A213409 9692061982207456039533424586,430311348543725825536505706371595438684,
%U A213409 906895928239445077568583988067142630846220290783969,89857639488565787203362892584824012528872539028234934088960440084
%N A213409 G.f.: exp( Sum_{n>=1} binomial(3*n^2,n^2) * x^n/n ).
%C A213409 Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x)^3 = exp( Sum_{n>=1} binomial(3*n,n) * x^n/n ).
%H A213409 G. C. Greubel, <a href="/A213409/b213409.txt">Table of n, a(n) for n = 0..34</a>
%F A213409 a(n) = (1/n) * Sum_{k=1..n} binomial(3*k^2,k^2) * a(n-k) for n>0 with a(0)=1.
%F A213409 a(n) ~ sqrt(3) * (27/4)^(n^2) / (2*sqrt(Pi)*n^2). - _Vaclav Kotesovec_, Mar 06 2014
%e A213409 G.f.: A(x) = 1 + 3*x + 252*x^2 + 1563022*x^3 + 563716946982*x^4 +...
%e A213409 where
%e A213409 log(A(x)) = 3*x + 495*x^2/2 + 4686825*x^3/3 + 2254848913647*x^4/4 + 52588547141148893628*x^5/5 +...+ C(3*n^2,n^2)*x^n/n +...
%t A213409 nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* _Vaclav Kotesovec_, Mar 06 2014 *)
%o A213409 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m^2, m^2)*x^m/m)+x*O(x^n)), n)}
%o A213409 (PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(3*k^2, k^2)*a(n-k)))}
%Y A213409 Cf. A201556, A213410, A001764, A245245 (log).
%K A213409 nonn
%O A213409 0,2
%A A213409 _Paul D. Hanna_, Jun 10 2012