# Definition:Catalan Number

## Definition

The **Catalan Numbers** $C_n$ are a sequence of natural numbers defined by:

- $C_n = \dfrac 1 {n + 1} \dbinom {2 n} n $

## Sequence of Catalan Numbers

The sequence of Catalan numbers begins:

- $(1,) \, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, \ldots$

where the inclusion or not of the initial $1$ depends on whether $n$ starts from $0$ or $1$.

This sequence is A000108 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

The **Catalan numbers** are also known as **Segner numbers** for Johann Andreas Segner.

## Source of Name

This entry was named for Eugène Charles Catalan.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $42$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $42$