Ordinary generating function of powers of 2

Question

Is there a good closed form expression for the generating function of the formal power series $$ A(z) := \sum_{n=0}^\infty z^{2^n} = z + z^2 + z^4 + z^8 + z^{16} + \cdots. $$ Is there a tractable way to retrieve the coefficient of $z^m$ in powers of $A(z)$, say in $A(z)^k$ for $k \geq 1$? Thanks.

Answer 1

How about $A(z)=\frac{z^2}{1-z^2}$? This works if $|z|<1$.

I got this idea from expanding $\frac{1}{1-z}$.

Answer 2

The value $A(1/2)=\kappa$ is known as the Kempner number, and was proven transcendental in 1916. The paper "The Many Faces of the Kempner Number", by Adamczewski, may provide some insight for you.