I am stuck trying to find $[z^n]$ of the following EGF:
$$Â(z) = e^{z + z^2}$$
Well, I know this is a convolution between $e^z$ and $e^{z^2}$. So, we can rewrite it as
$$Â(z) = e^z.e^{z^2}$$
Finding the $nth$ term is
$$n = n!(\sum_{k=0}^n[z^k]e^{z^2} . [z^{n-k}]e^z)$$
Writing down $e$ as a series
$$n = n!\sum_{k=0}^n([z^k](\frac{z^0}{0!} + \frac{z^2}{1!} + \frac{z^4}{2!} + \frac{z^6}{3!} ...) . [z^{n-k}](\frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + ...))$$
Which I think is
$$n = n!\sum_{k=0}^n(\frac{(1^k + (-1)^k)}{2(\frac{k}{2}!)} . \frac{1}{(n-k)!})$$
Is this right? I feel like I am doing something wrong here..