The Catalan numbers, or Catalan sequence, have many interesting applications in combinatorics.

The $n$ Catalan number is given by: $$C_n = \frac{\binom{2n}{n}}{n+1},$$

where $\binom{n}{r}$ represents the binomial coefficient. The first several Catalan numbers are $1$ $1$ $2$ $5$ $14$ $42$ $132$ $429$ $1430$ $4862$ ,...(see OEIS sequence A000108 for more terms). The Catalan numbers are also generated by the recurrence relation

\begin{equation*} C_0=1,\qquad C_n=\sum_{i=0}^{n-1} C_i C_{n-1-i}. \end{equation*}For example, $C_3=1\cdot 2+ 1\cdot 1+2\cdot 1=5$ $C_4 = 1\cdot 5 + 1\cdot 2 + 2\cdot 1 + 5\cdot 1 = 14$ etc.

The ordinary generating function for the Catalan numbers is \begin{equation*} \sum_{n=0}^\infty C_n z^n=\frac{1-\sqrt{1-4z}}{2z}. \end{equation*} Interpretations of the $n$ Catalan number include:

1. The number of ways to arrange $n$ pairs of matching parentheses, e.g.: $$()$$ $$(())\text{ } ()()$$ $$((()))\text{ } (()())\text{ } ()(())\text{ } (())()\text{ } ()()()$$
2. The number of ways a convex polygon of $n+2$ sides can be split into $n$ triangles.
3. The number of rooted binary trees with exactly $n+1$ leaves.

The Catalan sequence is named for Eugène Charles Catalan, but it was discovered in 1751 by Euler when he was trying to solve the problem of subdividing polygons into triangles.

Bibliography

1

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.
Concrete Mathematics.
Addison-Wesley, 1998.
Zbl 0836.00001.