  For an element $w$ in the Weyl algebra generated by $D$ and $U$ with relation
$DU=UD+1$, the normally ordered form is $w=\sum c_{i,j}U^iD^j$. We demonstrate
that the normal order coefficients $c_{i,j}$ of a word $w$ are rook numbers on
a Ferrers board. We use this interpretation to give a new proof of the rook
factorization theorem, which we use to provide an explicit formula for the
coefficients $c_{i,j}$. We calculate the Weyl binomial coefficients: normal
order coefficients of the element $(D+U)^n$ in the Weyl algebra. We extend all
these results to the $q$-analogue of the Weyl algebra. We discuss further
generalizations using $i$-rook numbers.
