"Beautiful" permutations

A permutation of the integers from $1$ to $n$ is called good if and only if the absolute values of the differences between each pair of consecutive numbers form a permutation of the integers from $1$ to n - 1.

For example, the permutation $123$ is not good, because the absolute differences are $1$ $1$, which do not form a permutation of $1$ to $2$. However, the permutation $312$ is good, because the differences $2$ $1$ form a valid permutation.

In this problem, you are given an integer $n$, and you are required to find the number of good permutations of $n$ numbers.