G.f. puzzle

[Ralf Stephan]

Hello,

I'm at a complete loss and any hint is very much welcome.


Let F(z) be the o.g.f. of A080100 = 2^e0(n), where e0(n) counts zeros
in the binary representation of n. F(z) satisfies

  (1) the functional equation F(z^2) = (1 + F(z))/(2 + z);

  (2) the ...what? continued product(?)

      F(z) =   -1 + (z+2) (-1 + (z^2+2) (-1 + (z^4+2) (-1 + ... )));

  (3) F(z) = -1 + lim(n->oo,
             prod(k=0,n, 2+z^2^k) - sum(k=0,n-1, prod(l=0,k, 2+z^2^k))).

Also, the coefficients

  (4) [z^n] F(z) = [z^n] prod(k=0,[log2(n)], 2+z^2^k).
(This is not the same as the infinite product since that doesn't converge)


Questions: can F(z) be stated in a more closed form? Where can I read
  about handling of such formal series? What does your CAS software say?


Thanks for your time,
R. Stephan