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| [[File:Chord diagrams K6 matchings.svg|thumb|360px|The fifteen different chord diagrams on six points or equivalently the fifteen different [[perfect matching]]s on a six-vertex [[complete graph]]]]
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| [[File:Unordered binary trees with 4 leaves.svg|thumb|300px|The fifteen different [[rooted binary tree]]s (with unordered children) on a set of four labeled leaves]]
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| In [[mathematics]], the product of all the odd integers up to some odd positive integer ''n'' is called the '''double factorial''' or '''odd factorial''' of ''n'', and denoted by ''n''<nowiki>!!</nowiki>.<ref name="callan">{{citation|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|arxiv=0906.1317|year=2009}}.</ref> That is,
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| :<math>(2k-1)!! = \prod_{i=1}^k (2i-1).</math>
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| For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.
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| The sequence of double factorials for ''n'' = 1, 3, 5, 7, ... starts as
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| : 1, 3, 15, 105, 945, 10395, 135135, .... {{OEIS|id=A001147}}
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| {{harvtxt|Merserve|1948}}<ref name="meserve"/> (possibly the earliest publication to use double factorial notation)<ref name="gq12"/> states that the double factorial was originally introduced in order to simplify the expression of certain [[List of integrals of trigonometric functions|trigonometric integrals]] arising in the derivation of the [[Wallis product]]. Double factorials also arise in the expression of the volume of a [[hypersphere]], and they have many applications in [[enumerative combinatorics]].<ref name="callan"/><ref name="dm93"/>
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| ==Relation to the factorial==
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| 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 ''n''!, and much smaller than the iterated factorial (''n''!)!.
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| For an odd positive integer ''n'' = 2''k'' − 1, ''k'' ≥ 1, the double factorial may be expressed in terms of either factorials or [[Permutation#Counting sequences without repetition|''k''-permutations of ''2k'']] as<ref name="callan"/><ref name="gq12">{{citation
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| | last1 = Gould | first1 = Henry
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| | last2 = Quaintance | first2 = Jocelyn
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| | doi = 10.4169/math.mag.85.3.177
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| | issue = 3
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| | journal = [[Mathematics Magazine]]
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| | mr = 2924154
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| | pages = 177–192
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| | title = Double fun with double factorials
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| | volume = 85
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| | year = 2012}}.</ref>
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| :<math>(2k-1)!! = \frac{(2k)!}{2^k k!} = \frac {_{2k}P_k} {2^k} = \frac {{(2k)}^{\underline k}} {2^k}.</math>
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| In this expression, the first denominator equals <math>\displaystyle\prod_{i=1}^k 2i</math> and cancels the unwanted even factors from the numerator.
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| ==Extensions==
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| ===Negative arguments===
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| The ordinary factorial, when extended to the [[Gamma function]], has a [[Pole (complex analysis)|pole]] at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial may be extended to any negative odd integer argument by inverting its [[recurrence relation]] | |
| :<math>n!! = n \times (n-2)!!</math> | |
| to give
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| :<math>n!! = \frac{(n+2)!!}{n+2}.</math>
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| Using this inverted recurrence, −1!! = 1, −3!! = −1, and −5!! = 1/3; negative odd numbers with greater magnitude have fractional double factorials.<ref name="callan"/> In particular, this gives, when ''n'' is an odd number,
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| :<math>(-n)!! \times n!! = (-1)^{(n-1)/2} \times n.</math>
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| ===Even arguments===
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| Sometimes ''n''<nowiki>!!</nowiki> is defined for non-negative even integers as well. One choice is a definition similar to the one for odd values:<ref name="callan"/><ref name="meserve">{{citation
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| | last = Meserve | first = B. E.
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| | doi = 10.2307/2306136
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| | issue = 7
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| | journal = [[The American Mathematical Monthly]]
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| | mr = 1527019
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| | pages = 425–426
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| | title = Classroom Notes: Double Factorials
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| | volume = 55
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| | year = 1948}}</ref>
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| :<math>(2k)!!= \prod_{i=1}^k (2i) = 2^k k!.</math>
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| One consequence of this definition is that (as an [[empty product]])
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| :<math>0!! = 1.</math>
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| The sequence of double factorials for ''n'' = 0, 2, 4, 6, 8, ... starts as
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| : 1, 2, 8, 48, 384, 3840, 46080, 645120, .... {{OEIS|id=A000165}}
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| These double factorials for even and odd integers share the following property for all positive integers n:
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| : <math>n! = n!! \times (n-1)!!.</math>
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| However, this definition does not match the expression for odd arguments of the double factorial in terms of the ordinary factorial, and is also inconsistent with the extension of the definition of <math>n!!</math> to complex numbers <math>n</math> that is achieved via the [[Gamma function]] below.
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| ===Complex arguments===
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| Disregarding the above definition of ''n''<nowiki>!!</nowiki> for even values of ''n'', the double factorial for odd integers can be extended to most real and complex numbers ''z'' by noting that when ''z'' is a positive odd integer then
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| <ref>{{citation|title=Mathematical Methods: For Students of Physics and Related Fields|series=Undergraduate Texts in Mathematics|first=Sadri|last=Hassani|publisher=Springer|year=2000|isbn=9780387989587|page=266|url=http://books.google.com/books?id=dxSOzeLMij4C&pg=PA266}}.</ref>
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| <ref>{{citation|title=Double factorial: Specific values (formula 06.02.03.0005) |publisher=Wolfram Research|date=2001-10-29 |url=http://functions.wolfram.com/06.02.03.0005 |accessdate=2013-03-23}}.</ref>
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| :<math>z!! = z(z-2)\cdots (3)
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| = 2^{(z-1)/2}\left(\frac{z}{2}\right)\left(\frac{z-2}{2}\right)\cdots \left(\frac{3}{2}\right)</math>
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| :<math> = 2^{(z-1)/2} \frac{\Gamma\left(\frac{z}{2}+1\right)}{\Gamma\left(\frac{1}{2}+1\right)}
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| = \sqrt{\frac{2^{z+1}}{\pi}} \Gamma\left(\frac{z}{2}+1\right) = \left(\frac{z}{2}\right)!\sqrt{\frac{2^{z+1}}{\pi}} \,.</math>
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| From this one can derive an alternative definition of ''z''<nowiki>!!</nowiki> for non-negative even integer values of ''z'':
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| :<math>(2k)!!= \sqrt{ \frac{2}{\pi} } \prod_{i=1}^k (2i) = 2^k k! \sqrt{ \frac{2}{\pi} } \,,</math>
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| with the value for 0!! in this case being
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| :<math>0!! = \sqrt{ \frac{2}{\pi} } \approx 0.79788456... \,.</math>
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| The expression found for ''z''<nowiki>!!</nowiki> is defined for all complex numbers except the negative even integers. Using it as the definition, the [[Volume of an n-ball|volume]] of an ''n''-[[dimension]]al [[hypersphere]] of radius ''R'' can be expressed as<ref>{{citation|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8}}.</ref>
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| :<math>V_n=\frac{2 (2\pi)^{(n-1)/2}}{n!!} R^n.</math>
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| ==Applications in enumerative combinatorics==
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| Double factorials are motivated by the fact that they occur frequently in [[enumerative combinatorics]] and other settings. For instance, ''n''!! counts
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| *[[Perfect matching]]s of the [[complete graph]] ''K''<sub>''n'' + 1</sub>. In such a graph, any single vertex ''v'' has ''n'' possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. For instance, a complete graph with four vertices ''a'', ''b'', ''c'', and ''d'' has three perfect matchings: ''ab'' and ''cd'', ''ac'' and ''bd'', and ''ad'' and ''bc''.<ref name="callan"/> Perfect matchings may be described in several other equivalent ways, including [[Involution (mathematics)|involutions]] without fixed points on a set of ''n'' + 1 items ([[permutations]] in which each cycle is a pair)<ref name="callan"/> or chord diagrams (sets of chords of a set of ''n'' + 1 points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called [[Richard Brauer|Brauer]] diagrams).<ref name="dm93"/><ref>{{citation|title=Patterns in Permutations and Words|series=EATCS Monographs in Theoretical Computer Science|first=Sergey|last=Kitaev|publisher=Springer|year=2011|url=9783642173332|page=96|url=http://books.google.com/books?id=JgQHtgR5N60C&pg=PA96}}.</ref><ref>{{citation
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| | last1 = Dale | first1 = M. R. T.
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| | last2 = Narayana | first2 = T. V.
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| | doi = 10.1016/0378-3758(86)90161-8
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| | issue = 2
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| | journal = Journal of Statistical Planning and Inference
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| | mr = 852528
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| | pages = 245–249
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| | title = A partition of Catalan permuted sequences with applications
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| | volume = 14
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| | year = 1986}}.</ref> The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the [[Telephone number (mathematics)|telephone number]]s, which may be expressed as a summation involving double factorials.<ref>{{citation
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| | last1 = Tichy | first1 = Robert F.
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| | last2 = Wagner | first2 = Stephan
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| | doi = 10.1089/cmb.2005.12.1004
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| | issue = 7
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| | journal = [[Journal of Computational Biology]]
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| | pages = 1004–1013
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| | title = Extremal problems for topological indices in combinatorial chemistry
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| | url = http://www.math.tugraz.at/fosp/pdfs/tugraz_main_0052.pdf
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| | volume = 12
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| | year = 2005}}.</ref>
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| *[[Stirling permutation]]s, permutations of the [[multiset]] of numbers 1, 1, 2, 2, ..., ''k'', ''k'' in which each pair of equal numbers is separated only by larger numbers, where ''k'' = (''n'' + 1)/2. The two copies of ''k'' must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is ''k'' − 1, with ''n'' positions into which the adjacent pair of ''k'' values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction.<ref name="callan"/> Alternatively, instead of the restriction that values between a pair may be parger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted order; such a permutation defines a matching on the 2''k'' positions of the permutation, so again the number of permutations may be counted by the double permutations.<ref name="dm93">{{citation
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| | last1 = Dale | first1 = M. R. T.
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| | last2 = Moon | first2 = J. W.
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| | doi = 10.1016/0378-3758(93)90035-5
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| | issue = 1
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| | journal = Journal of Statistical Planning and Inference
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| | mr = 1209991
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| | pages = 75–87
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| | title = The permuted analogues of three Catalan sets
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| | volume = 34
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| | year = 1993}}.</ref>
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| *[[Heap (data structure)|heap-ordered tree]]s, trees with ''k'' + 1 nodes labeled 0, 1, 2, ... ''k'', such that the root of the tree has label 0, each other node has a larger label than its parent, and such that the children of each node have a fixed ordering. An [[Euler tour technique|Euler tour]] of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.<ref name="callan"/><ref>{{citation
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| | last = Janson | first = Svante | authorlink = Svante Janson
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| | arxiv = 0803.1129
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| | contribution = Plane recursive trees, Stirling permutations and an urn model
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| | mr = 2508813
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| | pages = 541–547
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| | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
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| | series = Discrete Math. Theor. Comput. Sci. Proc., AI
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| | title = Fifth Colloquium on Mathematics and Computer Science
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| | year = 2008}}.</ref>
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| *[[Unrooted binary tree]]s with (''n'' + 5)/2 labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the ''n'' tree edges and making the new vertex be the parent of a new leaf.
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| *[[Rooted binary tree]]s with (''n'' + 3)/2 labeled leaves. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf.<ref name="callan"/><ref name="dm93"/>
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| {{harvtxt|Callan|2009}} and {{harvtxt|Dale|Moon|1993}} list several additional objects with the same [[combinatorial class|counting sequence]], including "trapezoidal words" ([[numeral system|numerals]] in a [[mixed radix]] system with increasing odd radixes), height-labeled [[Dyck path]]s, height-labeled ordered trees, "overhang paths", and certain vectors describing the lowest-numbered leaf descendant of each node in a rooted binary tree. For [[bijective proof]]s that some of these objects are equinumerous, see {{harvtxt|Rubey|2008}} and {{harvtxt|Marsh|Martin|2011}}.<ref>{{citation
| |
| | last = Rubey | first = Martin
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| | contribution = Nestings of matchings and permutations and north steps in PDSAWs
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| | mr = 2721495
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| | pages = 691–704
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| | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
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| | series = Discrete Math. Theor. Comput. Sci. Proc., AJ
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| | title = 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
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| | year = 2008}}.</ref><ref>{{citation
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| | last1 = Marsh | first1 = Robert J.
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| | last2 = Martin | first2 = Paul
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| | arxiv = 0906.0912
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| | doi = 10.1007/s10801-010-0252-6
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| | issue = 3
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| | journal = [[Journal of Algebraic Combinatorics]]
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| | mr = 2772541
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| | pages = 427–453
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| | title = Tiling bijections between paths and Brauer diagrams
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| | volume = 33
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| | year = 2011}}.</ref>
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| The even double factorials also have applications in enumerative combinatorics; in particular they give the numbers of elements of the [[hyperoctahedral group]]s (signed permutations or symmetries of a [[hypercube]])
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| ==Additional identities==
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| Using the extension of the double factorial to even arguments, for even values of ''n'',
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| :<math>\int_{0}^{\pi/2}\sin^n x\,dx=\int_{0}^{\pi/2}\cos^n x\,dx=\frac{(n-1)!!}{n!!}\cdot\frac{\pi}{2},</math> | |
| and for odd values of ''n'', the same formula applies without the final <math> \ \frac{\pi}{2} \ </math> factor. Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.<ref name="meserve"/><ref>{{citation
| |
| | last1 = Dassios | first1 = George
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| | last2 = Kiriaki | first2 = Kiriakie
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| | issue = part A
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| | journal = Bulletin de la Société Mathématique de Grèce
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| | mr = 935868
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| | pages = 40–43
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| | title = A useful application of Gauss theorem
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| | volume = 28
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| | year = 1987}}.</ref>
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| Some additional identities involving double factorials are:
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| :<math>(2n-1)!! = \sum_{k=1}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!!.</math><ref name="callan"/> | |
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| :<math>(2n-1)!! = \sum_{k=0}^{n} \binom{2n-k-1}{k-1} \frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!!.</math><ref name="callan"/> | |
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| :<math>(2n-1)!! = \sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!} k(2k-3)!!.</math><ref name="callan"/>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{mathworld|id=DoubleFactorial|title=Double Factorial}}
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| [[Category:Integer sequences]]
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| [[Category:Enumerative combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[fr:Analogues de la factorielle#Multifactorielles]]
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