The (b, c)-Motzkin paths are paths that start from the origin and end on the x-axis, not going below the x-axis, using up steps \(U=(1,1)\) , level steps \(L=(1,0)\) and down steps \(D=(1,-1)\) where each level step L can be one of b possible colors and each down step D be one of c possible colors. The free (b, c)-Motzkin paths are defined similarly but with no restriction of staying above the x-axis. The generalized central trinomial coefficient \( T_n(b,c)\) counts the number of free (b, c)-Motzkin paths from (0, 0) to (n, 0), the generalized sub-central trinomial coefficient \( T_{n,1}(b,c)\) counts the number of partial free (b, c)-Motzkin paths from (0, 0) to (n, 1), and the generalized Motzkin number \( M_n(b,c)\) counts the number of (b, c)-Motzkin paths from (0, 0) to (n, 0). The numbers \( M_n(b,c)\) , \( T_n(b,c)\) , \( T_n(b,c)+ T_{n,1}(b,c)\) , and \( T_n(b,c)- T_{n,1}(b,c)\) are collectively referred to as the generalized Motzkin family. In particular, when \(b=c=1\) we have the classical Motzkin family defined by Barcucci, Pinzani and Sprugnoli [4]. Using Riordan arrays, we investigate their mutual relationships and connections to the generalized trinomial coefficients.
