We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the n-th power of a tridiagonal matrix and the
enumeration of weighted paths of n steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte