Python | Superfactorial of a number.

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Given a number, the task is to find the Superfactorial of a number. The result of multiplying the product of first n factorials is called Superfactorial of a number.

Superfactorial(n)= 1 ^ n * 2 ^ (n-1) * 3 ^ (n-2) * . . . . . * n ^ 1

Examples:

Input : 3
Output : 12 H(3) = 1! * 2! * 3! = 12

Input : 4
Output : 288 H(4) = 1^4 * 2^3 * 3^2 * 4^1 = 288

An efficient approach is to compute all factorial iteratively till n, then compute the product of all factorial till n.

Python3

# Python3 program to find the
# Superfactorial of a number
 
# function to calculate the
# value of Superfactorial 
def superfactorial(n):
 
    # initialise the
    # val to 1
    val = 1
    ans = []
    for i in range(1, n + 1):
        val = val * i
        ans.append(val)
    # ans is the list with
    # all factorial till n.
    arr = [1]
    for i in range(1, len(ans)):
        arr.append((arr[-1]*ans[i]))
 
    return arr
 
# Driver Code
n = 5
arr = superfactorial(n)
print(arr[-1])

Output

Using built-in functions:

Time complexity: O(n)
Auxiliary Space: O(n)

Python3

import math
 
# Function to calculate the Superfactorial of a number
def superfactorial(n):
    # Initialize an empty list to store the factorials
    factorials = []
    # Use a loop to calculate the factorial of each number till n
    for i in range(1, n+1):
        factorials.append(math.factorial(i))
    # Initialize the result to 1
    result = 1
    # Use a loop to compute the product of all factorials
    for factorial in factorials:
        result *= factorial
    return result
 
# Test the function
n = 5
print(superfactorial(n))
#This code is contributed by Edula Vinay Kumar Reddy

Output

Explanation: Using built-in math.factorial() function, we can calculate the factorial of each number till n and then compute the product of all factorials using a loop. This method is useful when we have to do many mathematical operations on large dataset.

Time-complexity:O(N) Since super-factorials of number can be huge, hence the numbers will overflow. We can use boost libraries in C++ or BigInteger in Java to store the super-factorial of a number N.

Auxiliary Space: O(N)

Using Recursion

Python3

import math
 
def superfactorial(n, acc=1):
    if n == 1:
        return acc
    else:
        return superfactorial(n-1, acc*math.factorial(n))
 
# Driver code
n = 5
result = superfactorial(n)
print(result)
#this code is contributed by Jyothi Pinjala.