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A200215 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k) * x^n*A(x)^n/n ).

%I A200215 #10 Jul 06 2025 21:38:40
%S A200215 1,1,3,13,61,306,1623,8937,50565,292283,1718827,10250916,61854848,
%T A200215 376949934,2316738789,14343701657,89379109846,560108223900,
%U A200215 3527723269978,22318890516413,141778326349191,903936594232782,5782447430948438,37102633354583532,238729798670985104
%N A200215 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k) * x^n*A(x)^n/n ).
%e A200215 G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 61*x^4 + 306*x^5 + 1623*x^6 +...
%e A200215 where the logarithm of the g.f. A = A(x) equals the series:
%e A200215 log(A(x)) = (1 + x*A)*x*A + (1 + 2^3*x*A + x^2*A^2)*x^2*A^2/2 +
%e A200215 (1 + 3^3*x*A + 3^3*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
%e A200215 (1 + 4^3*x*A + 6^3*x^2*A^2 + 4^3*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
%e A200215 (1 + 5^3*x*A + 10^3*x^2*A^2 + 10^3*x^3*A^3 + 5^3*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
%e A200215 (1 + 6^3*x*A + 15^3*x^2*A^2 + 20^3*x^3*A^3 + 15^3*x^4*A^4 + 6^3*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
%e A200215 more explicitly,
%e A200215 log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 185*x^4/4 + 1126*x^5/5 + 7043*x^6/6 + 44689*x^7/7 + 286241*x^8/8 +...
%o A200215 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
%Y A200215 Cf. A198944, A200212, A192131, A166896, A199875, A166990, A198950, A181143, A181543.
%K A200215 nonn
%O A200215 0,3
%A A200215 _Paul D. Hanna_, Nov 14 2011