The m! and factorial(m) commands return the factorial of m.

If m is a positive integer, Maple returns the product of the numbers from 1 to m. If m is $0$ (zero), Maple returns $1$ (one).

If m is a (real or complex) floating-point number,  Maple returns the generalized factorial function result calculated using GAMMA(m+1).

If m is a negative integer, Maple returns an error.

The doublefactorial(n) command returns the double factorial of n, defined in terms of the generalized factorial as

FunctionAdvisor( definition, doublefactorial );

$\left[\mathrm{doublefactorial}\left(n\right)=2^{\frac{n}{2}}{\left(\frac{2}{\mathrm{\pi }}\right)}^{\frac{1}{4}-\frac{\cos \left(\mathrm{\pi }n\right)}{4}}\left(\frac{n}{2}\right)!\,\mathrm{with\; no\; restrictions\; on}\left(n\right)\right]$

When n is a positive integer, this definition is equivalent to the product:

 $n\left(n-2\right)\mathrm{...}\left(6\right)\left(4\right)\left(2\right)$ if n is an even, positive integer

 $n\left(n-2\right)\mathrm{...}\left(5\right)\left(3\right)\left(1\right)$ if n is an odd, positive integer

Note: In Maple, !! is used for repeated factorials and so it does not indicate the double factorial.

The type function perceives the factorial function as of type function and as of type "!", while it perceives doublefactorial as of type function only.

The internal representation of an unevaluated factorial uses the standard representation of functions, with the function name factorial. Thus to the op function, the 0th operand of $m!$ is factorial.

$5!=\mathrm{\Gamma }\left(6\right)$

$120=120$

$3.5!$

$11.63172840$

$m!$

$m!$

$\left(-2\right)!$

Error, numeric exception: division by zero

$\left(-2.1\right)!$

$9.714806383$

$\left(-3.\mathrm{I}\right)!$

$0.01929275896-0.03389601054\mathrm{I}$

$\mathrm{doublefactorial}\left(5\right)$

$15$

$\left(5!\right)!$

$6689502913449127057588118054090372586752746333138029810295671352301633557244962989366874165271984981308157637893214090552534408589408121859898481114389650005964960521256960000000000000000000000000000$

$\mathrm{doublefactorial}\left(10\right)$

$3840$

$\mathrm{doublefactorial}\left(-7\right)$

$-\frac{1}{15}$

$\mathrm{doublefactorial}\left(1.+2.\mathrm{I}\right)$

$3.636406167\times {10}^{\mathrm{-14}}+4.100313151\times {10}^{\mathrm{-14}}\mathrm{I}$

$\mathrm{type}\left(m!\,\mathrm{function}\right)$

$\mathrm{true}$

$\mathrm{type}\left(m!\,\mathrm{`!`}\right)$

$\mathrm{true}$

$\mathrm{op}\left(0\,m!\right)$

$\mathrm{factorial}$