w3resource

Python Math: nth Catalan Number

Python Math: Exercise-25 with Solution

Write a Python program for nth Catalan Number.

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

Catalan number

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)

Catalan number

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:

Flowchart: nth Catalan Number

Visualize Python code execution:

The following tool visualize what the computer is doing step-by-step as it executes the said program:

Python Code Editor:

Have another way to solve this solution? Contribute your code (and comments) through Disqus.

Previous: Write a Python program to convert a float to ratio.
Next: Write a Python program to print number with commas as thousands separators (from right side)?

What is the difficulty level of this exercise?

Test your Programming skills with w3resource's quiz.



Share this Tutorial / Exercise on : Facebook and Twitter

Python: Tips of the Day

Lambdas for Sorting:

>>> students = [{'name': 'John', 'score': 98}, {'name': 'Mike', 'score': 94}, {'name': 'Jennifer', 'score': 99}]
>>> sorted(students, key=lambda x: x['score'])
[{'name': 'Mike', 'score': 94}, {'name': 'John', 'score': 98}, {'name': 'Jennifer', 'score': 99}]