Program to find the nth row of Pascal's Triangle in Python

Pascal's Triangle is a triangular array of numbers where each row represents the binomial coefficients. We need to find the nth row (0-indexed) of this mathematical structure.

Understanding Pascal's Triangle

Pascal's Triangle follows these rules:

• The top row contains only [1]
• Each subsequent row starts and ends with 1
• Each interior number is the sum of the two numbers above it

Algorithm Approach

We'll build each row iteratively by adding adjacent elements from the previous row ?

def pascal_triangle_row(n):
    if n == 0:
        return [1]
    if n == 1:
        return [1, 1]
    
    previous_row = [1, 1]
    
    for i in range(2, n + 1):
        current_row = [1]  # Start with 1
        
        # Add adjacent elements from previous row
        for j in range(len(previous_row) - 1):
            current_row.append(previous_row[j] + previous_row[j + 1])
        
        current_row.append(1)  # End with 1
        previous_row = current_row
    
    return previous_row

# Test the function
print(pascal_triangle_row(4))
print(pascal_triangle_row(0))
print(pascal_triangle_row(3))

[1, 4, 6, 4, 1]
[1]
[1, 3, 3, 1]

Using Mathematical Formula

We can also calculate each element directly using the combination formula: C(n,k) = n! / (k! * (n-k)!) ?

def pascal_triangle_formula(n):
    row = []
    
    for k in range(n + 1):
        # Calculate C(n, k) = n! / (k! * (n-k)!)
        value = 1
        for i in range(k):
            value = value * (n - i) // (i + 1)
        row.append(value)
    
    return row

# Test the formula approach
print(pascal_triangle_formula(4))
print(pascal_triangle_formula(5))

[1, 4, 6, 4, 1]
[1, 5, 10, 10, 5, 1]

Comparison of Methods

Conclusion

Pascal's Triangle can be generated iteratively by summing adjacent elements from the previous row. The mathematical approach using combinations is more efficient for calculating a specific row directly.