Školjka

%V0 For integers $n \geq k \geq 0$ we define the [i]bibinomial coefficient[/i] $\displaystyle \left(\!\!\binom{n}{k}\!\!\right)$ by $$ \left(\!\!\binom{n}{k}\!\!\right) = \frac{n!!}{k!!(n - k)!!} \text{.} $$ Determine all pairs $(n, k)$ of integers with $n \geq k \geq 0$ such that the corresponding bibinomial coefficient is an integer. [i]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 $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]