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A179469 G.f. satisfies A(x) = exp( Sum_{n>=1} 2^n*A(x^n)*x^n/n ).

%I A179469 #10 Jun 02 2023 10:19:21
%S A179469 1,2,8,32,140,624,2928,14048,69200,347040,1768120,9122144,47572128,
%T A179469 250341312,1327718272,7089595552,38082093120,205638343552,
%U A179469 1115635692576,6078058719232,33239328613648,182402290944576,1004073853702320
%N A179469 G.f. satisfies A(x) = exp( Sum_{n>=1} 2^n*A(x^n)*x^n/n ).
%H A179469 Seiichi Manyama, <a href="/A179469/b179469.txt">Table of n, a(n) for n = 0..1000</a>
%F A179469 From _Seiichi Manyama_, Jun 02 2023: (Start)
%F A179469 A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-2*x^(k+1))^a(k).
%F A179469 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(k/d) * a(d-1) ) * a(n-k). (End)
%e A179469 G.f.: A(x) = 1 + 2*x + 8*x^2 + 32*x^3 + 140*x^4 + 624*x^5 + +...
%e A179469 log(A(x)) = 2*A(x) + 4*A(x^2)*x^2/2 + 8*A(x^3)*x^3/3 + 16*A(x^4)*x^4/4 +...
%o A179469 (PARI) {a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A,x,x^m+x*O(x^n))*2^m*x^m/m)));polcoeff(A,n)}
%Y A179469 Cf. A054598, A179470.
%K A179469 nonn
%O A179469 0,2
%A A179469 _Paul D. Hanna_, Jul 15 2010