Proof of the Catalan recurrence from the definition

Proof

Let $n$ be a natural number.
The node Catalan numbers defines the sequence $(C_m)$ by $C_0 = 1$ and, for every natural number $m$ , $C_{m+1} = \sum_{i=0}^{m} C_i\, C_{m-i}$ .
Taking $m = n$ in this defining rule gives $C_{n+1} = \sum_{i=0}^{n} C_i\, C_{n-i}$ , which is the claim.

Idea

Here the Catalan numbers are defined by this recursion, so the recurrence is part of the definition. Its real content lies in the families that satisfy it, such as Dyck words and binary trees.

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Catalan numbers definition
Proof of the Catalan recurrence from the definition proof

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Catalan recurrence

Tag	`4FYR5VFZGH`
URI	`m2://4FYR5VFZGH`
Kind	proof
Writing	W0 stub
MSC	05A15
Proves	Catalan recurrence
Uses	Catalan numbers
Changed	4 November 2026

Proof of the Catalan recurrence from the definition

Proof

Idea

Reading path

Where this leads

Further reading

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