OEIS A000108: Catalan numbers

We explore the Catalan numbers C_n = binom(2n, n)/(n+1) (equivalently (2n)!/(n!(n+1)!)) and the remarkable variety of objects they count: balanced parentheses, Dyck paths, non-crossing partitions, and triangulations of polygons. We also touch on their recurrences and asymptotics (C_n ~ 4^n/(n^{3/2} sqrt(pi))), primality patterns (only C_2=2 and C_3=5 are prime), and deeper algebraic connections such as the Catalan monoid.

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