  Asinowski, Bacher, Banderier and Gittenberger (A. Asinowski, A. Bacher, C.
Banderier and B. Gittenberger. Analytic combinatorics of lattice paths with
forbidden patterns, the vectorial kernel method, and generating functions for
pushdown automata. Algorithmica, pp. 1-43, 2019.) recently developed the
vectorial kernel method - a powerful extension of the classical kernel method
that can be used for paths that obey constraints that can be described by
finite automata, e.g. avoid a fixed pattern, avoid several patterns at once,
stay in a horizontal strip and many others more. However, they only considered
walks with steps of length one. In this paper we will generalize their results
to walks with longer steps. We will also give some applications of this
extension and prove a conjecture about the asymptotic behavior of the expected
number of ascents in Schroeder paths.
