Add Catalan Number DP implementation in Java

[bhumikasahai]

Catalan numbers are a sequence of natural numbers that frequently appear in various counting problems in combinatorics. They are named after the Belgian mathematician Eugène Charles Catalan.

The sequence starts: 1, 1, 2, 5, 14, 42, 132, ...

In essence, the n
th
Catalan number provides the solution to many problems that involve a recursive structure. Some classic examples include:

Balanced Parentheses: The number of ways to form a correctly matched sequence with n pairs of parentheses. For n=3, there are 5 ways: ((())), ()()(), (())(), ()(()), (()()).

Binary Search Trees: The number of unique binary search trees that can be formed with n keys.

Polygon Triangulation: The number of ways to divide a convex polygon with n+2 sides into triangles by connecting its vertices.

The calculation for each number in the sequence depends on all the numbers that came before it, which is why it is often solved using dynamic programming.

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[Pradeepsingh61]

Hi @bhumikasahai fix this errors Testing: Java/dynamic_programming/Knapsack-bhumika.java

📦 Compiling Java file...
Java/dynamic_programming/Knapsack-bhumika.java:1: error: class Knapsack is public, should be declared in a file named Knapsack.java
public class Knapsack {
^
1 error
❌ Java/dynamic_programming/Knapsack-bhumika.java failed to compile