Notes on Motzkin and Schr Oder Numbers

The Motzkin numberMn is the cardinality of the set of sequencesMn = f(xi)ni=1 : xi 2 f 1; 0; 1g; all partial sums P k i=1 xi are nonnegative and P n i=1 xi = 0g. These sequences have a pictorial representation as Motzkin paths: lattice paths of upsteps (corresponding to +1), downsteps ( 1) and atsteps (0) that begin and end at, but never dip below, \ground level". For example, the M3 = 4 Motzkin 3-paths are pictured in Figure 1.