Solving recurrence by using an exponential generating function

Question

I need to solve the reccurence $g_0 = 0, g_1 = 1, g_n = -2n g_{n-1} + \sum \binom{n}{k} g_k g_{k-1}, n > 1$, by using an exponential generating function. I've written it as $g_n = -2ng_{n-1} + \sum \binom{n}{k} g_k g_{k-1} + [n=1]$, for $n\in \mathbb{N}$, using the Iverson notation. I got $\hat{G}(z) = -2z\hat{G}(z) + \hat{G}(z)^2 + z$, but I don't know how to choose between the two roots.

Answer 1

This implies $G^2-(2z+1)G+z=0$, so $G=z+\frac12\pm \sqrt{(z+\frac12)^2-z} =z+\frac12\pm \frac12\sqrt{4z^2+1}$

One of the two signs leads to $G\to 1$ as $z\to 0$, so it cannot give the answer searched for because that contradicts $g_0=0$.

BTW: the range of summation of the $k$ is missing in the recurrence. Is $k=n$ included?