Definition:Catalan Number

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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