MEMO 2014 I4

For integers $n \ge k \ge 0$ we define the bibinomial coefficient $\left( \binom{n}{k} \right)$ by $$\left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .$$ Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer.

Remark: 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 $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$ .