The smith normal form of the walk matrix of the Dynkin graph $A_n$

In this paper, we give the rank of the walk matrix of the Dynkin graph $A_n$, and prove that its Smith normal form is $$\text{diag}(\underset{\lceil \frac{n}{2}\rceil }{\underbrace{1,\dots ,1}},0,\dots ,0)\mathrm{.}$$