have many combinatorial interpretations.
is the total number of ways
in which it is possible to draw non-intersecting chords between
points on a circle.
For example, for , it is possible to draw such chords in 127 ways.
In the picture aside I display only the 16 ones which are distinct by rotation or reflection.
Motzkin numbers can be computed with the recurrence
Several sums are know for Motzkin numbers, for example,
The first Motzkin numbers are
1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many Motzkin numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p