Summary: The generalized Motzkin numbers are common generalizations of the Motzkin numbers and the Catalan numbers. We investigate their combinatorial properties, including the combinatorial interpretation, the recurrence relation, the binomial transform, the Hankel transform, the log-convexity, the continued fraction of the generating function, and the total positivity of the corresponding Hankel matrix.

generalized Motzkin number, Motzkin number, Catalan number, Hankel matrix, total positivity, log-convexity