How to calculate catalan numbers with the method of Binominal Coefficients using Python?

Catalan numbers are defined as a sequence of natural numbers that can be used to find the number of possibilities of various combinations. The n-th Catalan number is calculated using the binomial coefficient formula:

This can also be expressed as: C(n) = (1/(n+1)) × C(2n, n), where C(2n, n) is the binomial coefficient.

For example, if n = 3:

C(3) = (1/4) × C(6, 3) = (1/4) × 20 = 5

Calculate Catalan Numbers Using math.comb()

The math.comb() function calculates binomial coefficients efficiently. We can use it directly to compute Catalan numbers using the formula C(n) = C(2n, n) / (n+1).

Example

Here's how to calculate Catalan numbers for the first 10 natural numbers −

import math

def catalan_number(n):
    # math.comb(2 * n, n) calculates binomial coefficient 
    # // (n + 1) performs integer division to get Catalan number
    return math.comb(2 * n, n) // (n + 1)

# Calculate first 10 Catalan numbers
for i in range(10):
    print(f"Catalan({i}) = {catalan_number(i)}")

Catalan(0) = 1
Catalan(1) = 1
Catalan(2) = 2
Catalan(3) = 5
Catalan(4) = 14
Catalan(5) = 42
Catalan(6) = 132
Catalan(7) = 429
Catalan(8) = 1430
Catalan(9) = 4862

Calculate Catalan Numbers By Defining a Custom Binomial Function

You can also implement the binomial coefficient calculation manually to understand the underlying mathematics better.

Example

This approach calculates the binomial coefficient iteratively without computing large factorials −

def binomial_coeff(n, k):
    """Calculate binomial coefficient C(n, k) efficiently"""
    if k > n - k:  # Take advantage of symmetry
        k = n - k
    
    result = 1
    for i in range(k):
        result *= (n - i)
        result //= (i + 1)
    return result

def catalan_number(n):
    """Calculate n-th Catalan number using custom binomial function"""
    return binomial_coeff(2 * n, n) // (n + 1)

# Test with multiple values
for i in range(6):
    print(f"Catalan({i}) = {catalan_number(i)}")

Catalan(0) = 1
Catalan(1) = 1
Catalan(2) = 2
Catalan(3) = 5
Catalan(4) = 14
Catalan(5) = 42

Comparison of Methods

Conclusion

Catalan numbers can be efficiently calculated using binomial coefficients with the formula C(n) = C(2n, n) / (n+1). The math.comb() method is recommended for practical applications due to its optimization and simplicity.