Python Math: nth Catalan Number

Last update on April 24 2025 12:44:37 (UTC/GMT +8 hours)

25. Nth Catalan Number Calculator

Write a Python program for the nth Catalan numbers.

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894).

Using zero-based numbering, the nth Catalan number is given directly in terms of binomial coefficients by

The first Catalan numbers for n = 0, 1, 2, 3, … are
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, ....

Pictorial Presentation

The C5 = 42 noncrossing partitions of a 5-element set (below, the other 10 of the 52 partitions)

Image Credits: Watchduck

Sample Solution:

Python Code:

def catalan_number(num):
    if num <=1:
         return 1
   
    res_num = 0
    for i in range(num):
        res_num += catalan_number(i) * catalan_number(num-i-1)
    return res_num
 
for n in range(10):
    print(catalan_number(n))
	

Sample Output:

1                                                                                                             
1                                                                                                             
2                                                                                                             
5                                                                                                             
14                                                                                                            
42                                                                                                            
132                                                                                                           
429                                                                                                           
1430                                                                                                          
4862 

Flowchart:

For more Practice: Solve these Related Problems:

• Write a Python program to compute the nth Catalan number using a recursive formula, and print the result.
• Write a Python function that calculates Catalan numbers iteratively up to n and then returns the nth Catalan number.
• Write a Python script to generate a list of the first n Catalan numbers and print the sequence.
• Write a Python program to compare the recursive and iterative approaches for computing the nth Catalan number and print both results.

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