Cyclic

Consider regular cyclic polygon with $2n$ number of sides. ( $n$ is an integer.) We want to connect with a line segment each vertex with an other vertex, such that each vertex is connected exactly one times, and the segments don't intersect.

For example, if $n=2$ , then this can be done in $2$ ways:

How many ways can it be done, if $n=8$ ?

Bonus : Generalize for $n=k$ .