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

Enumeration of simple random walks and tridiagonal matrices