The Motzkin numbers — 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188 — count a surprising variety of things, much like the [[catalan]] numbers they're related to. \(M_n\) is the number of ways to draw non-crossing chords between \(n\) points on a circle (not all points need be connected), the number of lattice paths from \((0,0)\) to \((n,0)\) using up, down, and level steps that never dip below the axis, and the number of ways to build certain rooted trees.
Named after Theodore Motzkin, they satisfy a clean recurrence and grow like \(3^n / n^{3/2}\). They interleave with the Catalan numbers in many combinatorial identities.