Bibinomial coefficient integer For integers $n \ge k \ge 0$ we define the bibinomial coefficient.   $\left( \binom{n}{k} \right)$ by
$$ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .$$
What are all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer?
(Note: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.)
The question is from a European math competition, which is already over.

Hint: 
Use that
$$
n!! = 2^kk!
$$
when $n=2k$ and that
$$
n!! = \frac{n!}{2^k k!}
$$
when $n=2k+1$.
