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A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^k)^n) ).

%I A218551 #13 Nov 16 2012 00:35:53
%S A218551 1,1,2,5,13,37,106,322,987,3119,9985,32499,106910,355524,1191960,
%T A218551 4026739,13689783,46807685,160842381,555175377,1923970425,6691769948,
%U A218551 23351250882,81729943060,286842588316,1009256119760,3559337691300,12579738946685,44549347255523,158058591860684
%N A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^k)^n) ).
%C A218551 Compare to the dual g.f. G(x) of A219231:
%C A218551 G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*G(x^n)^k) ).
%e A218551 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 106*x^6 + 322*x^7 +...
%e A218551 where
%e A218551 log(A(x)) = x/(1*(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
%e A218551 x^2/(2*(1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
%e A218551 x^3/(3*(1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
%e A218551 x^4/(4*(1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
%e A218551 Explicitly,
%e A218551 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 342*x^6/6 + 1198*x^7/7 + 4071*x^8/8 + 14356*x^9/9 + 50408*x^10/10 +...
%o A218551 (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/(1-x^(m*k)*subst(A,x,x^k +x*O(x^n))^m)))));polcoeff(A,n)}
%o A218551 for(n=0,30,print1(a(n),", "))
%Y A218551 Cf. A219231, A001383, A218552, A218575.
%K A218551 nonn
%O A218551 0,3
%A A218551 _Paul D. Hanna_, Nov 01 2012