The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation:

$$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$

where, $[x]$ is the nearest integer to $x$ not exceeding it.

How do we investigate the asymptotics of this type of recursion relation?