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A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n) ).

%I A218552 #13 Nov 16 2012 00:24:40
%S A218552 1,1,2,4,9,20,46,107,253,604,1463,3573,8812,21901,54837,138145,350068,
%T A218552 891529,2281092,5860471,15113614,39109461,101521521,264286160,
%U A218552 689820642,1804890193,4733051924,12437565725,32746931264,86375236835,228212881032,603915863737,1600500761487
%N A218552 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n) ).
%C A218552 Compare to the dual g.f. G(x) of A219232:
%C A218552 G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^n)^k) ).
%e A218552 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 107*x^7 +...
%e A218552 where
%e A218552 log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
%e A218552 x^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
%e A218552 x^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
%e A218552 x^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
%e A218552 Explicitly,
%e A218552 log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 117*x^6/6 + 295*x^7/7 + 755*x^8/8 + 1933*x^9/9 + 5048*x^10/10 +...
%o A218552 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*prod(k=1,n\m+1,1+x^(m*k)*subst(A,x,x^k +x*O(x^n))^m))));polcoeff(A,n)}
%o A218552 for(n=0,30,print1(a(n),", "))
%Y A218552 Cf. A219232, A218551, A218153, A218576.
%K A218552 nonn
%O A218552 0,3
%A A218552 _Paul D. Hanna_, Nov 01 2012