1. What is the Nth Catalan Number?

The Nth Catalan Number is the nth number in the sequence of Catalan numbers, which are a sequence of natural numbers that occur in various counting problems in combinatorics. They are named after the Belgian mathematician Eugène Charles Catalan.

2. How Does the Calculator Work?

The calculator uses the Catalan number formula:

Where:

• \( C_n \) — The nth Catalan number
• \( n \) — The position in the sequence (natural number)
• \( \binom{2n}{n} \) — Binomial coefficient "2n choose n"

Explanation: The formula calculates the number of valid parenthesis expressions, binary trees, and other combinatorial structures that can be formed with n elements.

3. Importance of Catalan Numbers

Details: Catalan numbers have numerous applications in combinatorics, computer science, and mathematics. They count the number of valid parenthesis expressions, full binary trees, triangulations of polygons, and many other combinatorial structures.

4. Using the Calculator

Tips: Enter a natural number n (non-negative integer) to calculate the nth Catalan number. The calculator will compute the result using the binomial coefficient formula.

5. Frequently Asked Questions (FAQ)

Q1: What are the first few Catalan numbers?
A: C₀ = 1, C₁ = 1, C₂ = 2, C₃ = 5, C₄ = 14, C₅ = 42, C₆ = 132, C₇ = 429, C₈ = 1430

Q2: What are some applications of Catalan numbers?
A: Counting valid parenthesis expressions, binary trees, triangulations of polygons, paths that don't cross the diagonal in a grid, and many other combinatorial structures.

Q3: Are Catalan numbers always integers?
A: Yes, despite the division by (n+1) in the formula, Catalan numbers are always integers because the binomial coefficient is divisible by (n+1).

Q4: How do Catalan numbers grow with n?
A: Catalan numbers grow exponentially, approximately as \( C_n \sim \frac{4^n}{n^{3/2}\sqrt{\pi}} \).

Q5: Can I calculate Catalan numbers for large n?
A: For very large n, the numbers become extremely large and may exceed typical integer limits in programming languages. This calculator handles moderate values of n.