  The normal ordering coefficients of strings consisting of $V,U$ which satisfy
$UV=qVU+hV^s$ ($s\in\mathbb N$) are considered. These coefficients are studied
in two contexts: first, as a multiple of a sequence satisfying a generalized
recurrence, and second, as $q$-analogues of rook numbers under the row creation
rule introduced by Goldman and Haglund. A number of properties are derived,
including recurrences, expressions involving other $q$-analogues and explicit
formulas. We also give a Dobinsky-type formula for the associated Bell numbers
and the corresponding extension of Spivey's Bell number formula. The
coefficients, viewed as rook numbers, are extended to the case $s\in\mathbb R$
via a modified rook model.
