Category Archives: Catalan numbers

Integrality of the Catalan Numbers via Kummer’s Theorem

Why is the nth Catalan number, , an integer?  If you know one of its combinatorial interpretations, then the answer is clear, but how do you get integrality strictly from this formula?  In this post I’m going to discuss how one can … Continue reading

Posted in binomial coefficients, Catalan numbers, number theory | Leave a comment

The Catalan Numbers from Their Generating Function

Deriving the expression for the Catalan numbers from the Catalan recurrence is a classic exercise in generating function manipulation, although the details are sometimes omitted in textbooks (and on the Wikipedia page).  This post fills in the details. The Catalan numbers … Continue reading

Posted in Catalan numbers, combinatorics | 9 Comments