  Let $\sigma$ be a permutation on $n$ letters. We say that a permutation
$\tau$ is an even (resp. odd) $k$th root of $\sigma$ if $\tau^k=\sigma$ and
$\tau$ is an even (resp. odd) permutation. In this article, we obtain
generating functions for the number of even and odd $k$th roots of
permutations. Our result implies know generating functions of Moser and Wyman
and also some generating functions for sequences in The On-line Encyclopedia of
Integer Sequences (OEIS).
