Editing Double factorial (section)

==Relation to the factorial==
Because the double factorial only involves about half the factors of the ordinary [[factorial]], its value is not substantially larger than the square root of the factorial {{math|''n''!}}, and it is much smaller than the iterated factorial {{math|(''n''!)!}}.

The factorial of a positive {{mvar|n}} may be written as the product of two double factorials:<ref name=":0" />
<math display="block">n! = n!! \cdot (n-1)!!\,,</math>
and therefore
<math display="block">n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!}\,,</math>
where the denominator cancels the unwanted factors in the numerator.  (The last form also applies when {{math|1=''n'' = 0}}.)

For an even non-negative integer {{math|''n'' {{=}} 2''k''}} with {{math|''k'' ≥ 0}}, the double factorial may be expressed as
<math display="block">(2k)!! = 2^k k!\,.</math>

For odd {{math|''n'' {{=}} 2''k'' − 1}} with {{math|''k'' ≥ 1}}, combining the two previous formulas yields
<math display="block">(2k-1)!! = \frac{(2k)!}{2^k k!} = \frac{(2k-1)!}{2^{k-1} (k-1)!}\,.</math>

For an odd positive integer {{math|''n'' {{=}} 2''k'' − 1}} with {{math|''k'' ≥ 1}}, the double factorial may be expressed in terms of [[Permutation#Permutations without repetitions|{{mvar|k}}-permutations of {{math|2''k''}}]]<ref name="callan"/><ref name="gq12">{{cite journal
 | last1 = Gould | first1 = Henry
 | last2 = Quaintance | first2 = Jocelyn
 | doi = 10.4169/math.mag.85.3.177
 | issue = 3
 | journal = [[Mathematics Magazine]]
 | mr = 2924154
 | pages = 177–192
 | title = Double fun with double factorials
 | volume = 85
 | year = 2012| s2cid = 117120280
 }}</ref> or a [[Falling and rising factorials|falling factorial]] as
<math display="block">(2k-1)!! = \frac {_{2k}P_k} {2^k} = \frac {(2k)^{\underline k}} {2^k}\,.</math>