A permutation $\pi$ is alternating if $\pi_1 > \pi_2 < \pi_3 > \pi_4 < \dots$. Let $a(n)$ be the number of alternating permutations of size $n$.
(a) Find a recurrence relation for $a(n)$. (b) Evaluate the exponential generating function for $a$.
If I'm understanding this problem correctly, then
$n=1$: $1$ permutation
$n=2$: $1$ permutation
$n=3$: $1$ permutation
$n=4$: $2$ permutations
$n=5$: $5$ permutations
I can't figure out a recurrence relation from this. Any help with both parts of the question is appreciated.