Factorial, Double Factorial, and Multifactorial
Factorial
For a natural number $n$, $n!$ is read as $n$factorial and is defined as follows.
$$ n!=n\cdot(n-1)\cdot(n-2)\cdots 2\cdot 1 =\prod\limits_{k=1}^n k $$
Description
It is used in many places to neatly express equations. The factorial of $0$ is defined as $0!:=1$. By generalizing the domain of definition of factorial, one can also define something called a Gamma function.
Double Factorial
For a natural number $n$, $n!!$ is read as $n$double factorial and is defined as follows.
$$ n!!=n\cdot (n-2)\cdot (n-4) \cdot (n-6) \cdots $$
Description
If a factorial is the product of values subtracted by $1$ from $n$, then the double factorial is the product of values subtracted by $2$ from $n$. Therefore, if $n$ is even, the multiplication ends at $2$, and if it is odd, the multiplication ends at $1$. If $n$ is even,
$$ n!!=\prod \limits_{k=1}^{\frac{n}{2}}(2k)=n\cdot(n-2)\cdots 4\cdot 2 $$
If $n$ is odd,
$$ n!!=\prod \limits_{k=1}^{\frac{n+1}{2}}(2k-1)=n\cdot(n-2)\cdots 3\cdot 1 $$
For example, if $7!!=7\cdot 5\cdot 3\cdot 1=105$ and $10!!=10\cdot 8\cdot 6\cdot 4\cdot 2=3840$
$n!!$ is not often used because it can be confused with the notation of $(n!)!$. Of course, there aren’t many practical uses for it either. However, it is used for convenience when dealing with complex equations in quantum mechanics, among other areas. Like factorial, it is defined as $0!!=1$.
Multifactorial
For a natural number $n>k$, $n!^{(k)}=n!_{k}$ is defined as follows and called multifactorial.
$$ n!^{(k)}=n\cdot(n-k)\cdot (n-2k) \cdot (n-3k)\cdots $$
Description
If the Gamma function extends the domain of definition of factorial, then the multifactorial can be seen as an extension of the properties of factorial itself. Since double factorial is seldom encountered, naturally, multifactorials are even rarer. Start from $n$ and multiply by subtracting the number of exclamation marks until a negative value is obtained. For example,
$$ 9!!!!=9\cdot 5\cdot 1,\quad 8!!!=8\cdot 5\cdot 2 $$
In the case of $0<n \le k$, it is defined as $n!^{(k)}=n$, and in the case of $-k < n\le 0$, it is defined as $n!^{(k)}=1$.
Joke
