Editing Catalan number (section)

{{Short description|Recursive integer sequence}}
{{Distinguish|Catalan's constant}}
{{Use American English|date = March 2019}}
[[File:Noncrossing partitions 5.svg|thumb|The {{math|1=C<sub>5</sub> = 42}} [[noncrossing partition]]s of a 5-element set (below, the other 10 of the [[Bell number|52]] [[Partition of a set|partitions]])]]
The '''Catalan numbers''' are a [[sequence]] of [[natural number]]s that occur in various [[Enumeration|counting problems]], often involving [[recursion|recursively]] defined objects. They are named after [[Eugène Charles Catalan|Eugène Catalan]], though they were previously discovered in the 1730s by [[Minggatu]].

The {{mvar|n}}-th Catalan number can be expressed directly in terms of the [[central binomial coefficient]]s by

:<math>C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} \qquad\text{for }n\ge 0.</math>

The first Catalan numbers for {{math|1=''n'' = 0, 1, 2, 3, ...}} are

:{{math|1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...}} {{OEIS|id=A000108}}.