I'm trying to find information about a specific lattice, which is proving difficult since I am not sure what its standard name is.

Consider the regular $n$-Simplex embedded in $\mathbb{R}^n$ with one of the $(n+1)$ vertices centered at the origin. Does the lattice generated by the position vectors of the remaining vertices have a name? Is this the root lattice $A_n$?

Any help would be greatly appreciated.