In how many ways can the elements of [n] be permuted so that the sum of every two consecutive elements in the permutation is odd?

[n] is the set of integers from 1 to n.

My attempt:

If n is even:

$\displaystyle n\frac{n}{2}\left(\frac{n}{2} - 1\right)\left(\frac{n}{2}-1\right)\left(\frac{n}{2}-2\right)\left(\frac{n}{2}-2\right)...=2\left(\frac{n}{2}!\right)^2$

If n is odd, wemuststart with an odd element:

$\displaystyle \left(\frac{n+1}{2}\right)\left(\frac{n-1}{2}\right)\left(\frac{n-1}{2}\right)\left(\frac{n-3}{2}\right)\left(\frac{n-3}{2}\right)...$$\displaystyle =\left(\frac{n+1}{2}\right)\left(\frac{n-1}{2}!\right)^2$

Is this right?