This problem is about counting pairs of lattice paths from $(0,0)$ to $(n,n)$.
We say two paths have $k$ shared steps if they have $k$ edges in common. For example, in the following figure, the two paths (red and blue) have $3$ shared steps.
$a(n)$ is the number of pairs of lattice paths from $(0,0)$ to $(n,n)$ that do not share steps.
Find $a(n)$.