A note on the geometry of figurate numbers

Abstract

We give a short proof of the formula $n^p={\sum }_{\ell =0}^{p-1}(-1{)}^{\ell }{c}_{p,\ell }{F}_n^{p-\ell }$, where $F_n^{p-\ell }$ is the figurate number and $c_{p,\ell }$ is the number of $(p-\ell )$-dimensional facets of $p$-dimensional simplices obtained by cutting the $p$-dimensional cube.