Find the functional equation for the generating function whose coefficients satisfy $$
a\_n = \sum\_{i=1}^{n-1}2^ia\_{n-i}, \text{ for } n\ge 2, a\_0 = a\_1 = 1
$$

This is what I've tried so far:

$$
\begin{align}
g(x) -1 -x &= \sum\_{n\ge2} \sum\_{i=1}^{n-1} 2^i a\_{n-i}\\
&=\sum\_{n\ge2} \sum\_{i=1}^n 2^i a\_{n-i}x^n\\
\end{align}
$$
at which point I'm stuck. How do I proceed?