Families of Generalized Catalan Numbers

Binomial coefficients, when defined as quotients of partial products of the natural numbers, can be generalized to any nonzero integer sequence, but the resulting ratios cannot be expected to be integers. As will be demonstrated in another talk, if a sequence is both multiplicative and divisible, the associated generalized binomial coefficients will be integers. Once one has integral generalized binomial coefficients, it is natural to ask about analogs of Catalan numbers. The classical Catalan numbers can be defined in terms of central binomial coefficients. In this talk, we use that definition to define generalized Catalan numbers and describe a proof showing that these will also be integers when the original sequence is both multiplicative and divisible. Along the way we will mention a handful of relevant examples of sequences having integer generalized Catalan numbers.