The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, , if be odd, or if be odd [sic]. I propose to write for such products, and if a name be required for the product to call it the "alternate factorial" or the "double 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 n!, and it is much smaller than the iterated factorial (n!)!.
The factorial of a positive n may be written as the product of two double factorials:[2]
and therefore
where the denominator cancels the unwanted factors in the numerator. (The last form also applies when n = 0.)
For an even non-negative integer n = 2k with k ≥ 0, the double factorial may be expressed as
For odd n = 2k − 1 with k ≥ 1, combining the two previous formulas yields
Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, n‼ for odd values of n counts
permutations in which each cycle is a pair)[1] 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 Brauer diagrams).[9][11][12] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the telephone numbers, which may be expressed as a summation involving double factorials.[13]
Stirling permutations, 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.[1] Alternatively, instead of the restriction that values between a pair may be larger 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 2k positions of the permutation, so again the number of permutations may be counted by the double permutations.[9]
Heap-ordered trees, 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 of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.[1][14]
Unrooted binary trees 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.
Rooted binary trees 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.[1][9]
Dyck paths, 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 proofs that some of these objects are equinumerous, see Rubey (2008) and Marsh & Martin (2011).[15][16]
The even double factorials give the numbers of elements of the hyperoctahedral groups (signed permutations or symmetries of a hypercube)
pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation
to give
Using this inverted recurrence, (−1)!! = 1, (−3)!! = −1, and (−5)!! = 1/3; negative odd numbers with greater magnitude have fractional double factorials.[1] In particular, when n is an odd number, this gives
Complex arguments
Disregarding the above definition of n!! 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[17][18]
The final expression is defined for all complex numbers except the negative even integers and satisfies (z + 2)!! = (z + 2) · z!! everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is
The generalized formula does not agree with the previous product formula for z!! for non-negative even integer values of z. Instead, this generalized formula implies the following alternative:
Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is
Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.[8][20]
Double factorials of odd numbers are related to the gamma function by the identity:
Some additional identities involving double factorials of odd numbers are:[1]
An approximation for the ratio of the double factorial of two consecutive integers is
This approximation gets more accurate as n increases, which can be seen as a result of the Wallis Integral.
Generalizations
Definitions
In the same way that the double factorial generalizes the notion of the single factorial, the following definition of the integer-valued multiple factorial functions (multifactorials), or α-factorial functions, extends the notion of the double factorial function for positive integers :
Alternative extension of the multifactorial
Alternatively, the multifactorial z!(α) can be extended to most real and complex numbers z by noting that when z is one more than a positive multiple of the positive integer α then
This last expression is defined much more broadly than the original. In the same way that z! is not defined for negative integers, and z‼ is not defined for negative even integers, z!(α) is not defined for negative multiples of α. However, it is defined for and satisfies (z+α)!(α) = (z+α)·z!(α) for all other complex numbers z. This definition is consistent with the earlier definition only for those integers z satisfying z ≡ 1 mod α.
In addition to extending z!(α) to most complex numbers z, this definition has the feature of working for all positive real values of α. Furthermore, when α = 1, this definition is mathematically equivalent to the Π(z) function, described above. Also, when α = 2, this definition is mathematically equivalent to the alternative extension of the double factorial.
Generalized Stirling numbers expanding the multifactorial functions
These generalized α-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or α-factorial functions, (x − 1)!(α), as
The distinct polynomial expansions in the previous equations actually define the α-factorial products for multiple distinct cases of the least residues x ≡ n0 mod α for n0 ∈ {0, 1, 2, ..., α − 1}.
The generalized α-factorial polynomials, σ(α) n(x) where σ(1) n(x) ≡ σn(x), which generalize the
Stirling convolution polynomials
from the single factorial case to the multifactorial cases, are defined by
for 0 ≤ n ≤ x. These polynomials have a particularly nice closed-form
ordinary generating function
given by
Other combinatorial properties and expansions of these generalized α-factorial triangles and polynomial sequences are considered in Schmidt (2010).[21]
Suppose that n ≥ 1 and α ≥ 2 are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or α-factorial functions, (αn − 1)!(α), in terms of the
binomial coefficients
as
and moreover, we similarly have double sum expansions of these functions given by
The first two sums above are similar in form to a known non-round combinatorial identity for the double factorial function when α := 2 given by Callan (2009).
Similar identities can be obtained via context-free grammars.[22] Additional finite sum expansions of congruences for the α-factorial functions, (αn − d)!(α), modulo any prescribed integer h ≥ 2 for any 0 ≤ d < α are given by Schmidt (2018).[23]