Question: 2017 Fall Final - 2

Let $n \geq 2$ be an even integer. A permutation $a_1,a_2,\dots,a_n$ of the set $\{1,2,\dots,n\}$ is called awesome if $a_2 = 2 \cdot a_1$.
For example, if $n = 6$, then the permutation $3,6,4,1,5,2$ is awesome, whereas the permutation $3,5,4,1,6,2$ is not awesome.
How many awesome permutations of the set $\{1,2,\dots,n\}$ are there?