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)`.