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We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.

Abstract. We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of a production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method of Riordan matrices. Finally we treat the case of rational production matrices, i.e. those leading to rational generating functions.

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