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## Section: New Results

### Formal description of catalan numbers

Participant : José Grimm.

Catalan number can be defined by a recurrence, or by explicit formulas using binomial numbers. An important property is ${C}_{n+1}={\sum }_{k\le n}{C}_{k}{C}_{n-k}$. The easiest way to prove this formula is to use Dyck paths.

A Dyck path of size $2n$ is a sequence $l$ of integers $+1$ and $-1$ so that the sum ${s}_{k}$ of the $k$ first terms is $\ge 0$ for $k\le 2n$ and ${s}_{2n}=0$. The relation between Dyck paths and Catalan numbers is easy to prove and then properties of Dyck paths are quite simple to state and verify.

The proofs have been done with the Math-Components library.