Sum of all elements up to Nth row in a Pascal triangle

Given a row number n, the task is to calculate the sum of all elements of each row up to n^th row.

Examples:

Input  : 2
Output : 7 
Explanation:  row 0 have element 1
              row 1 have elements 1, 1
              row 2 have elements 1, 2, 1
              so, sum will be ((1) + (1 + 1) + (1 + 2 + 1)) = 7

Input  : 4
Output : 31
Explanation:  row 0 have element 1
              row 1 have elements 1, 1
              row 2 have elements 1, 2, 1
              row 3 have elements 1, 3, 3, 1
              row 4 have elements 1, 4, 6, 4, 1
              so, sum will be ((1) + (1 + 1) + (1 + 2 + 1)
              + (1 + 3 + 3 + 1) + (1 + 4 + 6 + 4 + 1)) = 31

Below is the example of Pascal triangle having 11 rows: 

                             Pascal's triangle
                                                                  
0th row                             1                                
1st row                           1   1                              
2nd row                         1   2   1                            
3rd row                       1   3   3   1                          
4th row                     1   4   6   4   1                        
5th row                   1   5   10  10  5   1                     
6th row                 1   6   15  20  15  6   1                   
7th row               1   7   21  35  35  21  7   1                  
8th row             1  8   28   56  70   56  28  8  1                
9th row           1   9  36  84  126  126  84  36  9  1              
10th row        1  10  45  120 210  256  210 120 45 10  1            

Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. So a simple solution is to generating all row elements up to nth row and adding them. But this approach will have O(n³) time complexity. However, it can be optimized up to O(n²) time complexity. Refer the following article to generate elements of Pascal’s triangle: 

• Pascal’s triangle

Better Solution: Let’s have a look on pascal’s triangle pattern 

                                                               sum of elements in ith row
0th row                             1                                1    -> 20
1st row                           1   1                              2    -> 21
2nd row                         1   2   1                            4    -> 22
3rd row                       1   3   3   1                          8    -> 23
4th row                     1   4   6   4   1                        16   -> 24
5th row                   1   5   10  10  5   1                      32   -> 25
6th row                 1   6   15  20  15  6   1                    64   -> 26
7th row               1   7   21  35  35  21  7   1                  128  -> 27
8th row             1  8   28   56  70   56  28  8  1                256  -> 28
9th row           1   9  36  84  126  126  84  36  9  1              512  -> 29
10th row        1  10  45  120 210  256  210 120 45 10  1            1024 -> 210

As shown above, the sum of elements in the i^th row is equal to 2ⁱ. Now it can be easily calculated the sum of all elements up to n^th row by adding powers of 2.

Below is the implementation of above approach: 

C++

// C++ program to find sum of all elements
// upto nth row in Pascal triangle.
#include <bits/stdc++.h>
using namespace std;

// Function to find sum of all elements
// upto nth row.
long long int calculateSum(int n)
{

    // Initialize sum with 0
    long long int sum = 0;

    // Loop to calculate power of 2
    // upto n and add them
    for (int row = 0; row < n; row++) {
        sum = sum + (1 << row);
    }

    return sum;
}

// Driver function
int main()
{
    int n = 10;
    cout << " Sum of all elements:" << calculateSum(n);
    return 0;
}

Java

// Java program to find sum of all elements
// upto nth row in Pascal triangle.
import java.io.*;

class GFG {

    // Function to find sum of all elements
    // upto nth row.
    static long calculateSum(int n)
    {

        // Initialize sum with 0
        long sum = 0;

        // Loop to calculate power of 2
        // upto n and add them
        for (int row = 0; row < n; row++) {
            sum = sum + (1 << row);
        }

        return sum;
    }

    // Driver code
    public static void main(String[] args)
    {
        int n = 10;
        System.out.println("Sum of all elements:"
                           + calculateSum(n));
    }
}

Python3

# Python program to find sum of all elements
# upto nth row in Pascal triangle.

# Function to find sum of all elements
# upto nth row.
def calculateSum(n) :

    # Initialize sum with 0
    sum = 0

    # Loop to calculate power of 2
    # upto n and add them
    for row in range(n):
        sum = sum + (1 << row)

    return sum

# Driver code
n = 10
print("Sum of all elements:", calculateSum(n))

C#

// C# program to find sum of all elements
// upto nth row in Pascal triangle.
using System;

public class GFG {

    // Function to find sum of all elements
    // upto nth row.
    static long calculateSum(int n)
    {

        // Initialize sum with 0
        long sum = 0;

        // Loop to calculate power of 2
        // upto n and add them
        for (int row = 0; row < n; row++) {
            sum = sum + (1 << row);
        }

        return sum;
    }

    static public void Main()
    {
        int n = 10;
        Console.WriteLine("Sum of all elements:"
                          + calculateSum(n));
    }
}

PHP

<?php
// PHP program to find sum of
// all elements upto nth row
// in Pascal triangle.

// Function to find sum of
// all elements upto nth row.
function calculateSum($n)
{

    // Initialize sum with 0
    $sum = 0;

    // Loop to calculate power
    // of 2 upto n and add them
    for ($row = 0; $row < $n; $row++)
    {
        $sum = $sum + (1 << $row);
    }

    return $sum;
}

// Driver Code
$n = 10;
echo " Sum of all elements : " .
               calculateSum($n);

// This code is contributed by Mahadev.
?>

Javascript

<script>
// Javascript program to find sum of
// all elements upto nth row
// in Pascal triangle.

// Function to find sum of
// all elements upto nth row.
function calculateSum(n)
{

    // Initialize sum with 0
    let sum = 0;

    // Loop to calculate power
    // of 2 upto n and add them
    for (let row = 0; row < n; row++)
    {
        sum = sum + (1 << row);
    }

    return sum;
}

// Driver Code
let n = 10;
document.write(" Sum of all elements : " + calculateSum(n));

// This code is contributed by _saurabh_jaiswal.
</script>

Output:

Sum of all elements:1023

Time Complexity: O(n) 
Auxiliary Space: O(1) as it is using constant space for variables

Efficient solution: 

2ⁿ can be expressed as 
2ⁿ = ( 2⁰ + 2¹ + 2² + 2³ +. . . + 2^(n-1) ) + 1 
For Example:
2⁶ = ( 20 + 2¹ + 2² + 2³ + 2⁴ + 2⁵ ) + 1 
64 = ( 1 + 2 + 4 + 8 +16 + 32 ) + 1 
64 = 63 + 1 

So, calculate 2ⁿ instead of calculating every power of 2 up to (n – 1) and from above example the sum of the power of 2 up to (n – 1) will be (2ⁿ – 1).

C++

// C++ program to find sum of all elements
// upto nth row in Pascal triangle.
#include <bits/stdc++.h>
using namespace std;

// Function to find sum of all elements
// upto nth row.
long long int calculateSum(int n)
{

    // Initialize sum with 0
    long long int sum = 0;

    // Calculate 2^n
    sum = 1 << n;

    return (sum - 1);
}

// Driver function
int main()
{

    int n = 10;
    cout << " Sum of all elements:" << calculateSum(n);
    return 0;
}

Java

// Java program to find sum of all elements
// upto nth row in Pascal triangle.
import java.io.*;

class GFG {

    // Function to find sum of all elements
    // upto nth row.
    static long calculateSum(int n)
    {

        // Initialize sum with 0
        long sum = 0;

        // Calculate 2^n
        sum = 1 << n;

        return (sum - 1);
    }

    // Driver code
    public static void main(String[] args)
    {
        int n = 10;
        System.out.println("Sum of all elements:"
                           + calculateSum(n));
    }
}

Python3

# Python3 program to find sum of all elements
# upto nth row in Pascal triangle.

# Function to find sum of all elements
# upto nth row.
def calculateSum(n) :

    # Initialize sum with 0
    sum = 0

    # Calculate 2 ^ n
    sum = 1 << n;

    return (sum - 1)

# Driver unicode
n = 10
print("Sum of all elements:", calculateSum(n))

C#

// C# program to find sum of all elements
// upto nth row in Pascal triangle.
using System;

public class GFG {

    // Function to find sum of all elements
    // upto nth row.
    static long calculateSum(int n)
    {

        // Initialize sum with 0
        long sum = 0;

        // Calculate 2^n
        sum = 1 << n;

        return (sum - 1);
    }

    // Driver code
    static public void Main()
    {
        int n = 10;
        Console.WriteLine("Sum of all elements:"
                          + calculateSum(n));
    }
}

PHP

<?php
// PHP program to find sum
// of all elements upto nth
// row in Pascal triangle.

// Function to find
// sum of all elements
// upto nth row.

function calculateSum($n)
{
    // Initialize sum with 0
    $sum = 0;

    // Calculate 2^n
    $sum = 1 << $n;

    return ($sum - 1);
}

// Driver Code
$n = 10;
echo " Sum of all elements:" ,
             calculateSum($n);

// This code is contributed
// by akt_mit
?>

Javascript

// Javascript program to find sum
// of all elements upto nth
// row in Pascal triangle.

// Function to find
// sum of all elements
// upto nth row.

function calculateSum(n)
{
    // Initialize sum with 0
    sum = 0;

    // Calculate 2^n
    sum = 1 << n;

    return (sum - 1);
}

// Driver Code
let n = 10;
document.write(" Sum of all elements:" + calculateSum(n));

// This code is contributed _saurabh_jaiswal

Output:

Sum of all elements:1023

Time Complexity: O(1)
Auxiliary Space: O(1) because constant space has been used