-th Catalan number
can be defined as
Catalan numbers have many combinatorial interpretations.
For example, is the number of ways a regular -gon can be
divided into triangles taking into account different orientations as distinct. See also the picture aside.
The only odd Catalan numbers are those where .
Two interesting sums involving Catalan numbers:
The first Catalan numbers are
1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670 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 Catalan numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p