Number of Catalan

We know that any positive n-side shape can be divided into (n-2) triangles with (n-3) diagonal lines, so how many options are there? (Don't say I'm bored)
In fact, it's simple:
We use HN to represent the number of partition schemes of n-side shape, and define h2=1,h3=1;
Then for an n-side shape (at least one diagonal line at any point of it), we can use a diagonal line of a point to divide it into a
(i) A side shape and a (n-i+2) edge (2<i<n), then the number of schemes at this point (that is, the diagonal) is (Hi) * (hn-i+2);
Plus h2=1, that is, the total scheme number is hn=h2*hn-1+h3*hn-2+ +HN-1*H2;
We write it in the form of hn+1: hn+1=h2*hn+h3*hn-1+ +hn*h3;
Redefine cn=hn+1 (c1=1,c2=1), you can get: cn=c1*cn-1+c2*cn-2+ +CN-1*C1;
This is the famous Catalan number, which is named by Belgian mathematician Catalan (Eugen Charlie Cartland, 1814-1894), and it is widely used in combinatorial mathematics for various counting problems.
Finish......
Short, but can be rotated ...

Number of Catalan