%I #9 Sep 03 2024 15:02:40

%S 1,1,3,11,42,174,763,3457,16075,76351,368767,1805682,8943948,44736096,

%T 225646033,1146461185,5862224756,30144922281,155791900727,

%U 808773877919,4215675455503,22054576750972,115765182718467,609508331610920,3218059655553030,17034314889643633

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 174*x^5 + 763*x^6 +...

%e where the logarithm of the g.f. A = A(x) equals the series:

%e log(A(x)) = (A + x)*x + (A^2 + 2^3*x*A + x^2)*x^2/2 +

%e (A^3 + 3^3*x*A^2 + 3^3*x^2*A + x^3)*x^3/3 +

%e (A^4 + 4^3*x*A^3 + 6^3*x^2*A^2 + 4^3*x^3*A + x^4)*x^4/4 +

%e (A^5 + 5^3*x*A^4 + 10^3*x^2*A^3 + 10^3*x^3*A^2 + 5^3*x^4*A + x^5)*x^5/5 +

%e (A^6 + 6^3*x*A^5 + 15^3*x^2*A^4 + 20^3*x^3*A^3 + 15^3*x^4*A^2 + 6^3*x^5*A + x^6)*x^6/6 +...

%e more explicitly,

%e log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 117*x^4/4 + 581*x^5/5 + 2987*x^6/6 + 15499*x^7/7 + 81213*x^8/8 +...

%o (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 Cf. A192131, A166896, A198944, A199875, A166990, A198950, A181143, A181543, A200074.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 14 2011