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A219262 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n / Product_{k>=1} (1 - x^(n*k)*A(x^k)^n) ).

%I A219262 #6 Nov 16 2012 18:41:05
%S A219262 1,1,3,11,43,179,772,3446,15737,73315,346869,1662593,8055433,39390631,
%T A219262 194145569,963490818,4810435187,24145458246,121771909878,616744946740,
%U A219262 3135657413451,15997777629900,81877262767137,420263466778106,2162874177974610,11158378099471260,57696623896231494
%N A219262 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n / Product_{k>=1} (1 - x^(n*k)*A(x^k)^n) ).
%C A219262 Compare to the dual g.f. G(x) of A219263:
%C A219262 G(x) = exp( Sum_{n>=1} x^n*G(x^n)/n / Product_{k>=1} (1 - x^(n*k)*G(x^n)^k) ).
%e A219262 G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 772*x^6 + 3446*x^7 +...
%e A219262 where
%e A219262 log(A(x)) = x*A(x)/1/(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
%e A219262 x^2*A(x)^2/2/((1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
%e A219262 x^3*A(x)^3/3/((1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
%e A219262 x^4*A(x)^4/4/((1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
%e A219262 Explicitly,
%e A219262 log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 121*x^4/4 + 601*x^5/5 + 2999*x^6/6 + 15247*x^7/7 + 78057*x^8/8 + 403279*x^9/9 + 2095425*x^10/10 +...
%o A219262 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
%o A219262 for(n=0, 30, print1(a(n), ", "))
%Y A219262 Cf. A218551, A219260, A219263.
%K A219262 nonn
%O A219262 0,3
%A A219262 _Paul D. Hanna_, Nov 16 2012