It is well-known that the number of linear extensions of the following Hasse diagram is counted by Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$.
Now we consider the number of linear extensions of following diagrams $D_n$. $D_1$, $D_2$ and $D_3$ are drawn as follows. One can easily figure out what $D_n$ looks like.
$a(n)$ is the number of linear extensions of $D_n$.
Find $a(n)$.