A sequence is defined as a zig-zag sequence if every even-numbered "spot" is greater than the one before it and every odd numbered spot is less than the one spot before it. Also, a sequence must use every number from 1 to its size once in a sequence (no repeats).
For example: ${1,3,2,5,4}$ is a zig-zag sequence of length $5$. ${1,4,2,3}$ is a zig-zag of length $4$.
My question is, for any $n$, what is the number of possible zig-zag sequences of length $n$.