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Special Functions

["Factorial", n]

n!
$$n!$$
["Factorial", 5]
// -> 120

["Factorial2", n]

The double factorial of n: 

 n!! = n \cdot (n-2) \cdot (n-4) \times
\cdots
, that is the product of all the positive integers up to n that have the same parity (odd or even) as n.

n!!
$$n!!$$
["Factorial2", 5]
// -> 15

It can also be written in terms of the \Gamma function:

n!! = 2^{\frac{n}{2}+\frac{1}{4}(1-\cos(\pi n))}\pi^{\frac{1}{4}(\cos(\pi
n)-1)}\Gamma\left(\frac{n}{2}+1\right)

This is not the same as the factorial of the factorial of n (i.e. ((n!)!)).

Reference

["Gamma", z]

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

The Gamma Function is an extension of the factorial function, with its argument shifted by 1, to real and complex numbers.

\operatorname{\Gamma}\left(z\right) = \int\limits_{0}^{\infty} t^{z-1}
\mathrm{e}^{-t} \, \mathrm{d}t
["Gamma", 5]
// 24

["GammaLn", z]

\\ln(\\gamma(z))
$$\\ln(\\gamma(z))$$

This function is called gammaln in MatLab and SciPy and LogGamma in Mathematica.

See also Statistics for the Error Functions