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%I #6 Mar 30 2012 18:37:31
%S 1,1,2,7,23,78,291,1126,4436,17910,73773,308188,1303402,5573133,
%T 24050795,104620985,458324429,2020417339,8956142180,39899217350,
%U 178549985024,802275736073,3618237414959,16373514195570,74325340129430,338356926399193,1544406450870590
%N G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k] * x^n/n ).
%C Compare to a g.f. G(x) of A036765 (rooted trees with a degree constraint):
%C G(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*G(x)^k] * x^n/n ).
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 23*x^4 + 78*x^5 + 291*x^6 +...
%e where
%e log(A(x)) = (1 + x*A(x))*x + (1 + 2^3*x*A(x) + x^2*A(x)^2)*x^2/2 +
%e (1 + 3^3*x*A(x) + 3^3*x^2*A(x)^2 + x^3*A(x)^3)*x^3/3 +
%e (1 + 4^3*x*A(x) + 6^3*x^2*A(x)^2 + 4^3*x^3*A(x)^3 + x^4*A(x)^4)*x^4/4 +
%e (1 + 5^3*x*A(x) + 10^3*x^2*A(x)^2 + 10^3*x^3*A(x)^3 + 5^3*x^4*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
%e more explicitly,
%e log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 63*x^4/4 + 251*x^5/5 + 1110*x^6/6 +...
%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*A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}
%Y Cf. A036765, A192131.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 31 2011
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