  We initiate a study of the growth and matrix-valued Hilbert series of
non-negatively graded twisted Calabi-Yau algebras that are homomorphic images
of path algebras of weighted quivers, generalizing techniques previously used
to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras.
Several results are proved without imposing any assumptions on the degrees of
generators or relations of the algebras. We give particular attention to
twisted Calabi-Yau algebras of dimension d at most 3, giving precise
descriptions of their matrix-valued Hilbert series and partial results
describing which underlying quivers yield algebras of finite GK-dimension. For
d = 2, we show that these are algebras with mesh relations. For d = 3, we show
that the resulting algebras are a kind of derivation-quotient algebra arising
from an element that is similar to a twisted superpotential.
