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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:

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