Multifactorials Calculator

What is a Multifactorial?

A multifactorial is a generalization of the factorial operation. It includes:

• n! - factorial
• n!! - double factorial
• n!!! - triple factorial
• n!!!! - quadruple factorial
• n!!!!! - quintuple factorial

Each type of multifactorial involves multiplying a decreasing sequence of numbers, with the number of factors skipped between each term increasing as the number of exclamation points increases.

How to Calculate Multifactorials

The calculation method depends on the type of multifactorial:

• For n!, multiply n by all positive integers less than n down to 1.
• For n!!, multiply n by all positive integers of the same parity (odd or even) less than n down to 1 or 2.
• For n!!!, multiply n by every third number less than n down to 1, 2, or 3.
• This pattern continues for higher-order multifactorials.

Formulas

The general formula for a k-th order multifactorial of n is:

\[ n^{(k)} = n \times (n-k) \times (n-2k) \times ... \times (n \bmod k + k) \]

Where k is the number of exclamation points and n mod k + k is the last term greater than 0.

Calculation Steps

1. Determine the order of the multifactorial (k)
2. Start with the given number n
3. Multiply by (n-k)
4. Continue multiplying by numbers decreasing by k each time
5. Stop when you reach the last number greater than 0 in the sequence

Examples

Let's calculate some multifactorials:

1. 5! = 5 × 4 × 3 × 2 × 1 = 120
2. 5!! = 5 × 3 × 1 = 15
3. 6!! = 6 × 4 × 2 = 48
4. 7!!! = 7 × 4 × 1 = 28
5. 10!!!! = 10 × 6 × 2 = 120

These examples demonstrate how the calculation changes based on the number of exclamation points in the multifactorial.