In this paper we consider combinatorial numbers C m,k for m ≥ 1 and k ≥ 0 which unifies the entries of the Catalan triangles B n,k and A n,k for appropriate values of parameters m and k, i.e., B n,k = C 2n,n−k and A n,k = C 2n+1,n+1−k . In fact, some of these numbers are the well-known Catalan numbers Cn that is C2n,n−1 = C2n+1,n = Cn.We present new identities for recurrence relations, linear sums and alternating sum of C m,k . After that, we check sums (and alternating sums) of squares and cubes of C m,k and, consequently, for B n,k and A n,k . In particular, one of these equalities solves an open problem posed in [8]. We also present some linear identities involving harmonic numbers Hn and Catalan triangles numbers C m,k . Finally, in the last section new open problems and identities involving Cn are conjectured.