What are Catalan numbers?

Question

I have read the Wikipedia article for Catalan number and a number of other websites, but still couldn't understand it. Please explain it in simple terms, or using some examples. Thanks in advance!

Answer 1

I'd read the Wikipedia article in a different section order than it is currently written:

Applications in combinatorics

The $n$th Catalan number counts the number of different ways $n$ pairs of brackets can be correctly matched.

E.g. for $n = 3$ there are these distinct correctly matched pairs of brackets: $$((()))\space\space()(())\space\space()()()\space\space(())()\space\space(()())$$ (Etc.)

Properties

$C_0 = 1 \text{ and } C_{n+1} = \sum_{i=0}^nC_iC_{n−i} \text{ for } n\ge0$

Etc.