Graded Betti numbers of powers of path ideals of paths

Let $I_{n,m}=(x_1\cdots x_m,x_2\cdots x_{m+1},\dots ,x_{n+1}{x}_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n+m-1$ over a polynomial ring $S=\mathrm{k}[x_1,\dots ,x_{n+m}]$. We compute all the graded Betti numbers of all powers of $I_{n,m}$.