<p>In this article, we introduce a new scale of weighted Orlicz-Sobolev sequence spaces and prove the existence of various continuous and compact embeddings for them. Whereas continuous embeddings are relatively easy to come by as a result of pointwise comparisons between the generating Orlicz functions, the way to compactness is more tortuous as it follows from the combination of the existence of a Schauder basis in the spaces under consideration with a condition on the Orlicz functions regarding their local behavior in a small neighborhood of the origin, coupled with constraints on the so-called generalized order of differentiability. We illustrate our results by means of several concrete examples and also mention some open questions along the way.</p>