Factorials

Factorial of a natural number $n$, written $n!$ and read "$n$ factorial" (or "factorial of $n$"), is the product of the first $n$ natural numbers:

The notation $n!$ was introduced by the French mathematician Christian Kramp in 1808. For a positive integer $n$, factorial unwinds recursively:

Small values: $1!=1,\ 2!=2,\ 3!=6,\ 4!=24,\ 5!=120$, and $22!=1124000727777607680000$ — the number 22 (Ramanujan's birth date) is the least integer $N>1$ whose factorial has exactly $N$ digits (finding the next such $N$ is left as an exercise for students and teachers alike!).

Why $0!=1$. Substituting $n=0$ into the recursion $(n+1)!=(n+1)\times n!$ gives $1!=(0+1)\times0! \Rightarrow 1=1\times0! \Rightarrow 0!=1$. This convention extends the idea of factorial to non-negative integers (factorial can in fact be extended further, to certain negative and complex numbers, but that is beyond this course).

A useful identity (Example 4.24). $\dfrac{(2n)!}{n!}=2^n\big(1\cdot3\cdot5\cdots(2n-1)\big)$ — proved by splitting $(2n)!$ into its odd factors $1\cdot3\cdot5\cdots(2n-1)$ and its even factors $2\cdot4\cdot6\cdots(2n)=2^n\cdot n!$, so $(2n)!=\big(1\cdot3\cdots(2n-1)\big)\times2^n\, n!$, and dividing by $n!$ gives the stated identity. (This identity resurfaces later to prove $^{2n}C_n=\dfrac{2^n\times1\times3\times5\cdots(2n-1)}{n!}$.)

Double Factorial. Factorial can be viewed as a function $f:\mathbb N\cup\{0\}\to\mathbb N$, $f(0)=1$, $f(n)=n(n-1)(n-2)\cdots3\cdot2\cdot1$ for $n\ne0$. The double factorial $n!!$ is defined by …