  We introduce a natural weighted enumeration of lattice points in a polytope,
and give a Brion-type formula for the corresponding generating function. The
weighting has combinatorial significance, and its generating function may be
viewed as a generalization of the Rogers-Szeg\H{o} polynomials. It also arises
from the geometry of the toric arc scheme associated to the normal fan of the
polytope. We show that the asymptotic behavior of the coefficients at $q = 1$
is Gaussian.
