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A179470 G.f. satisfies A(x) = exp( Sum_{n>=1} A(2*x^n)*x^n/n ).

%I A179470 #14 Jun 01 2023 14:54:39
%S A179470 1,1,3,15,138,2370,78532,5110472,659436845,169486506217,
%T A179470 86947958127377,89122003350193045,182611160539104099261,
%U A179470 748158103862060509908713,6129659711065116858192667033,100434475863953990317790200253757
%N A179470 G.f. satisfies A(x) = exp( Sum_{n>=1} A(2*x^n)*x^n/n ).
%C A179470 Compare to the g.f. of A000081: G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ).
%H A179470 Seiichi Manyama, <a href="/A179470/b179470.txt">Table of n, a(n) for n = 0..81</a>
%F A179470 From _Seiichi Manyama_, Jun 01 2023: (Start)
%F A179470 A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2^k * a(k)).
%F A179470 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(d-1) * a(d-1) ) * a(n-k). (End)
%e A179470 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 138*x^4 + 2370*x^5 +...
%e A179470 log(A(x)) = A(2x)*x + A(2x^2)*x^2/2 + A(2x^3)*x^3/3 + A(2x^4)*x^4/4 + A(2x^5)*x^5/5 +...
%e A179470 More generally, if F(x,q) = exp( Sum_{n>=1} F(q*x^n,q)*x^n/n )
%e A179470 then coefficients in F(x,q) = Sum_{n>=0} c(n,q)*x^n begin:
%e A179470 c(0,q) = 1; c(1,q) = 1; c(2,q) = q + 1;
%e A179470 c(3,q) = q^3 + q^2 + q + 1;
%e A179470 c(4,q) = q^6 + q^5 + q^4 + 2*q^3 + 3/2*q^2 + 3/2*q + 1;
%e A179470 c(5,q) = q^10 + q^9 + q^8 + 2*q^7 + 5/2*q^6 + 5/2*q^5 + 3*q^4 + 3*q^3 + 3/2*q^2 + 3/2*q + 1; ...
%e A179470 where C(n,q) are integers for integer values of q.
%o A179470 (PARI) {a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A,x,2*x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)}
%Y A179470 Cf. A000081, A083413.
%K A179470 nonn
%O A179470 0,3
%A A179470 _Paul D. Hanna_, Jul 15 2010