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Double Factorial

Tool for calculating the double factorial. The double factorial n!! is the product of non-zero positive integers less than or equal to n that have the same parity as n (even or odd).

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Double Factorial -

Tag(s) : Arithmetics

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# Double Factorial

## Double Factorial Calculator N!!

### What is Double Factorial? (Definition)

The double factorial (also called semifactorial) of a number $n$, often denoted $n!!$, is a mathematical operation applied to a positive integer $n$ which consists of the multiplication (the product) of all non-zero integers less than or equal to $n$ which have the same parity as $n$.

### How to calculate a double factorial?

The formula for the double factorial is: $$n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots$$

If $n$ is an even number (multiple of 2) then $n!!$ is the multiplication of all multiples of $2$ less or equal than $n$ (and greater than $0$)

If $n$ is an odd number (not a multiple of 2) then $n!!$ is the multiplication of all non-multiple numbers of $2$ less or equal than $n$ (and greater than $0$).

Example: $$8!! = 2 \times 4 \times 6 \times 8 = 384$$

Example: $$5!! = 1 \times 3 \times 5 = 15$$

Be careful not to confuse the double factorial $n!!$ with the factorial of factorial $(n!)!$

### What is the value of double factorial of 0?

By convention, the double factorial of zero is equal to 1: $0!! = 1$

### What are the first values of the double factorial function?

The values of the first double factorials: $$0!! = 1 \\ 1!! = 1 \\ 2!! = 2 \\ 3!! = 3 \\ 4!! = 8 \\ 5!! = 15 \\ 6!! = 48 \\ 7!! = 105 \\ 8!! = 384 \\ 9!! = 945 \\ 10!! = 3840$$

### What are the properties of double factorial?

The remarkable relations of the double factorial with the factorial are:

$$n! = n!! (n-1)!!$$

$$n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!}$$

### What is !!n? (left side exclamation marks)

When the two exclamation points are to the left of the number, it may be the subfactorial or the double subfactorial.

$$!n = n!\sum_{k=0}^n \frac{(-1)^k}{k!}$$

$$!!n= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor }\,n!! \sum_{i=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \frac{(-1)^i}{(n-2 i)!!}$$

### What is the algorithm for programming the double factorial?

A non-recursive function to calculate the double factorial of a number N is: // Pseudo-codefunction doubleFactorial(n) { if (n == 0 OR n == 1) return 1 result = 1 for i from n down to 2 by 2 { result = result * i } return result}

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