  In this paper, we first introduce the new class of vertically-recurrent
matrices, using a generalization of "the Hockey stick and Puck theorem" in
Pascal's triangle. Then, we give an interesting formula for the lower
triangular decomposition of these matrices. We also deal with the $m$-th power
of these matrices in some special cases. Furthermore, we present two important
applications of these matrices for decomposing \emph{admissible matrices} and
matrices which arise in the theory of \emph{ladder networks}. Finall,y we pose
some open problems and conjectures about these new kind of matrices.
