SILK-ROAD 2012 Problem 3

Let $n > 1$ be an integer. Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix} 2n 2i+1 \end{matrix} \right):0 \le i \le n-1 \right\}$ i.e. the largest positive integer, dividing $\left( \begin{matrix} 2n 2i+1 \end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n-1$ . (Here $\left( \begin{matrix} m l \end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)