In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n.[1] That is,
n ! ! = ∏ ∏ --> k = 0 ⌈ n 2 ⌉ − − --> 1 ( n − − --> 2 k ) = n ( n − − --> 2 ) ( n − − --> 4 ) ⋯ ⋯ --> . {\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}(n-2k)=n(n-2)(n-4)\cdots .}
Restated, this says that for even n, the double factorial[2] is n ! ! = ∏ ∏ --> k = 1 n 2 ( 2 k ) = n ( n − − --> 2 ) ( n − − --> 4 ) ⋯ ⋯ --> 4 ⋅ ⋅ --> 2 , {\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,} while for odd n it is n ! ! = ∏ ∏ --> k = 1 n + 1 2 ( 2 k − − --> 1 ) = n ( n − − --> 2 ) ( n − − --> 4 ) ⋯ ⋯ --> 3 ⋅ ⋅ --> 1 . {\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.} For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.[3][4]
The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as
The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as
The term odd factorial is sometimes used for the double factorial of an odd number.[5][6]
The term semifactorial is also used by Knuth as a synonym of double factorial.[7]
In a 1902 paper, the physicist Arthur Schuster wrote:[8]
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, n ⋅ ⋅ --> n − − --> 2 ⋅ ⋅ --> n − − --> 4 ⋯ ⋯ --> 1 {\displaystyle n\cdot n-2\cdot n-4\cdots 1} , if n {\displaystyle n} be odd, or n ⋅ ⋅ --> n − − --> 2 ⋯ ⋯ --> 2 {\displaystyle n\cdot n-2\cdots 2} if n {\displaystyle n} be odd [sic]. I propose to write n ! ! {\displaystyle n!!} for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial".
Meserve (1948)[9] states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics.[1][10] They occur in Student's t-distribution (1908), though Gosset did not use the double exclamation point notation.
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:[3] n ! = n ! ! ⋅ ⋅ --> ( n − − --> 1 ) ! ! , {\displaystyle n!=n!!\cdot (n-1)!!\,,} and therefore n ! ! = n ! ( n − − --> 1 ) ! ! = ( n + 1 ) ! ( n + 1 ) ! ! , {\displaystyle n!!={\frac {n!}{(n-1)!!}}={\frac {(n+1)!}{(n+1)!!}}\,,} 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 ( 2 k ) ! ! = 2 k k ! . {\displaystyle (2k)!!=2^{k}k!\,.}
For odd n = 2k − 1 with k ≥ 1, combining the two previous formulas yields ( 2 k − − --> 1 ) ! ! = ( 2 k ) ! 2 k k ! = ( 2 k − − --> 1 ) ! 2 k − − --> 1 ( k − − --> 1 ) ! . {\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}k!}}={\frac {(2k-1)!}{2^{k-1}(k-1)!}}\,.}
For an odd positive integer n = 2k − 1 with k ≥ 1, the double factorial may be expressed in terms of k-permutations of 2k[1][11] or a falling factorial as ( 2 k − − --> 1 ) ! ! = 2 k P k 2 k = ( 2 k ) k _ _ --> 2 k . {\displaystyle (2k-1)!!={\frac {_{2k}P_{k}}{2^{k}}}={\frac {(2k)^{\underline {k}}}{2^{k}}}\,.}
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
Callan (2009) and Dale & Moon (1993) list several additional objects with the same counting sequence, including "trapezoidal words" (numerals in a mixed radix system with increasing odd radixes), height-labeled 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).[16][17]
The even double factorials give the numbers of elements of the hyperoctahedral groups (signed permutations or symmetries of a hypercube)
Stirling's approximation for the factorial can be used to derive an asymptotic equivalent for the double factorial. In particular, since n ! ∼ ∼ --> 2 π π --> n ( n e ) n , {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n},} one has as n {\displaystyle n} tends to infinity that
n ! ! ∼ ∼ --> { π π --> n ( n e ) n / 2 if n is even , 2 n ( n e ) n / 2 if n is odd . {\displaystyle n!!\sim {\begin{cases}\displaystyle {\sqrt {\pi n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is even}},\\[5pt]\displaystyle {\sqrt {2n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is odd}}.\end{cases}}}
The ordinary factorial, when extended to the gamma function, has a 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 n ! ! = n × × --> ( n − − --> 2 ) ! ! {\displaystyle n!!=n\times (n-2)!!} to give n ! ! = ( n + 2 ) ! ! n + 2 . {\displaystyle n!!={\frac {(n+2)!!}{n+2}}\,.} 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 ( − − --> n ) ! ! × × --> n ! ! = ( − − --> 1 ) n − − --> 1 2 × × --> n . {\displaystyle (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,.}
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[18][19]
z ! ! = z ( z − − --> 2 ) ⋯ ⋯ --> 5 ⋅ ⋅ --> 3 = 2 z − − --> 1 2 ( z 2 ) ( z − − --> 2 2 ) ⋯ ⋯ --> ( 5 2 ) ( 3 2 ) = 2 z − − --> 1 2 Γ Γ --> ( z 2 + 1 ) Γ Γ --> ( 1 2 + 1 ) = 2 π π --> 2 z 2 Γ Γ --> ( z 2 + 1 ) , {\displaystyle {\begin{aligned}z!!&=z(z-2)\cdots 5\cdot 3\\[3mu]&=2^{\frac {z-1}{2}}\left({\frac {z}{2}}\right)\left({\frac {z-2}{2}}\right)\cdots \left({\frac {5}{2}}\right)\left({\frac {3}{2}}\right)\\[5mu]&=2^{\frac {z-1}{2}}{\frac {\Gamma \left({\tfrac {z}{2}}+1\right)}{\Gamma \left({\tfrac {1}{2}}+1\right)}}\\[5mu]&={\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)\,,\end{aligned}}} where Γ Γ --> ( z ) {\displaystyle \Gamma (z)} is the gamma function.
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 logarithmically convex in the sense of the Bohr–Mollerup theorem. Asymptotically, n ! ! ∼ ∼ --> 2 n n + 1 e − − --> n . {\textstyle n!!\sim {\sqrt {2n^{n+1}e^{-n}}}\,.}
The generalized formula 2 π π --> 2 z 2 Γ Γ --> ( z 2 + 1 ) {\displaystyle {\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)} 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: ( 2 k ) ! ! = 2 π π --> 2 k Γ Γ --> ( k + 1 ) = 2 π π --> ∏ ∏ --> i = 1 k ( 2 i ) , {\displaystyle (2k)!!={\sqrt {\frac {2}{\pi }}}2^{k}\Gamma \left(k+1\right)={\sqrt {\frac {2}{\pi }}}\prod _{i=1}^{k}(2i)\,,} with the value for 0!! in this case being
Using this generalized formula as the definition, the volume of an n-dimensional hypersphere of radius R can be expressed as[20]
V n = 2 ( 2 π π --> ) n − − --> 1 2 n ! ! R n . {\displaystyle V_{n}={\frac {2\left(2\pi \right)^{\frac {n-1}{2}}}{n!!}}R^{n}\,.}
For integer values of n, ∫ ∫ --> 0 π π --> 2 sin n --> x d x = ∫ ∫ --> 0 π π --> 2 cos n --> x d x = ( n − − --> 1 ) ! ! n ! ! × × --> { 1 if n is odd π π --> 2 if n is even. {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}} Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is ∫ ∫ --> 0 π π --> 2 sin n --> x d x = ∫ ∫ --> 0 π π --> 2 cos n --> x d x = ( n − − --> 1 ) ! ! n ! ! π π --> 2 . {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}{\sqrt {\frac {\pi }{2}}}\,.}
Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.[9][21]
Double factorials of odd numbers are related to the gamma function by the identity:
( 2 n − − --> 1 ) ! ! = 2 n ⋅ ⋅ --> Γ Γ --> ( 1 2 + n ) π π --> = ( − − --> 2 ) n ⋅ ⋅ --> π π --> Γ Γ --> ( 1 2 − − --> n ) . {\displaystyle (2n-1)!!=2^{n}\cdot {\frac {\Gamma \left({\frac {1}{2}}+n\right)}{\sqrt {\pi }}}=(-2)^{n}\cdot {\frac {\sqrt {\pi }}{\Gamma \left({\frac {1}{2}}-n\right)}}\,.}
Some additional identities involving double factorials of odd numbers are:[1]
( 2 n − − --> 1 ) ! ! = ∑ ∑ --> k = 0 n − − --> 1 ( n k + 1 ) ( 2 k − − --> 1 ) ! ! ( 2 n − − --> 2 k − − --> 3 ) ! ! = ∑ ∑ --> k = 1 n ( n k ) ( 2 k − − --> 3 ) ! ! ( 2 ( n − − --> k ) − − --> 1 ) ! ! = ∑ ∑ --> k = 0 n ( 2 n − − --> k − − --> 1 k − − --> 1 ) ( 2 k − − --> 1 ) ( 2 n − − --> k + 1 ) k + 1 ( 2 n − − --> 2 k − − --> 3 ) ! ! = ∑ ∑ --> k = 1 n ( n − − --> 1 ) ! ( k − − --> 1 ) ! k ( 2 k − − --> 3 ) ! ! . {\displaystyle {\begin{aligned}(2n-1)!!&=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\binom {n}{k}}(2k-3)!!(2(n-k)-1)!!\\&=\sum _{k=0}^{n}{\binom {2n-k-1}{k-1}}{\frac {(2k-1)(2n-k+1)}{k+1}}(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\frac {(n-1)!}{(k-1)!}}k(2k-3)!!\,.\end{aligned}}}
An approximation for the ratio of the double factorial of two consecutive integers is ( 2 n ) ! ! ( 2 n − − --> 1 ) ! ! ≈ ≈ --> π π --> n . {\displaystyle {\frac {(2n)!!}{(2n-1)!!}}\approx {\sqrt {\pi n}}.} This approximation gets more accurate as n increases, which can be seen as a result of the Wallis Integral.
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 α α --> {\displaystyle \alpha } :
n ! ( α α --> ) = { n ⋅ ⋅ --> ( n − − --> α α --> ) ! ( α α --> ) if n > α α --> ; n if 1 ≤ ≤ --> n ≤ ≤ --> α α --> ; and ( n + α α --> ) ! ( α α --> ) / ( n + α α --> ) if n ≤ ≤ --> 0 and is not a negative multiple of α α --> ; {\displaystyle n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}}
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
z ! ( α α --> ) = z ( z − − --> α α --> ) ⋯ ⋯ --> ( α α --> + 1 ) = α α --> z − − --> 1 α α --> ( z α α --> ) ( z − − --> α α --> α α --> ) ⋯ ⋯ --> ( α α --> + 1 α α --> ) = α α --> z − − --> 1 α α --> Γ Γ --> ( z α α --> + 1 ) Γ Γ --> ( 1 α α --> + 1 ) . {\displaystyle {\begin{aligned}z!_{(\alpha )}&=z(z-\alpha )\cdots (\alpha +1)\\&=\alpha ^{\frac {z-1}{\alpha }}\left({\frac {z}{\alpha }}\right)\left({\frac {z-\alpha }{\alpha }}\right)\cdots \left({\frac {\alpha +1}{\alpha }}\right)\\&=\alpha ^{\frac {z-1}{\alpha }}{\frac {\Gamma \left({\frac {z}{\alpha }}+1\right)}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}\,.\end{aligned}}}
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.
A class of generalized Stirling numbers of the first kind is defined for α > 0 by the following triangular recurrence relation:
[ n k ] α α --> = ( α α --> n + 1 − − --> 2 α α --> ) [ n − − --> 1 k ] α α --> + [ n − − --> 1 k − − --> 1 ] α α --> + δ δ --> n , 0 δ δ --> k , 0 . {\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }=(\alpha n+1-2\alpha )\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{\alpha }+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.}
These generalized α-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or α-factorial functions, (x − 1)!(α), as
( x − − --> 1 | α α --> ) n _ _ --> := ∏ ∏ --> i = 0 n − − --> 1 ( x − − --> 1 − − --> i α α --> ) = ( x − − --> 1 ) ( x − − --> 1 − − --> α α --> ) ⋯ ⋯ --> ( x − − --> 1 − − --> ( n − − --> 1 ) α α --> ) = ∑ ∑ --> k = 0 n [ n k ] ( − − --> α α --> ) n − − --> k ( x − − --> 1 ) k = ∑ ∑ --> k = 1 n [ n k ] α α --> ( − − --> 1 ) n − − --> k x k − − --> 1 . {\displaystyle {\begin{aligned}(x-1|\alpha )^{\underline {n}}&:=\prod _{i=0}^{n-1}\left(x-1-i\alpha \right)\\&=(x-1)(x-1-\alpha )\cdots {\bigl (}x-1-(n-1)\alpha {\bigr )}\\&=\sum _{k=0}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right](-\alpha )^{n-k}(x-1)^{k}\\&=\sum _{k=1}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }(-1)^{n-k}x^{k-1}\,.\end{aligned}}}
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
σ σ --> n ( α α --> ) ( x ) := [ x x − − --> n ] ( α α --> ) ( x − − --> n − − --> 1 ) ! x ! {\displaystyle \sigma _{n}^{(\alpha )}(x):=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]_{(\alpha )}{\frac {(x-n-1)!}{x!}}}
for 0 ≤ n ≤ x. These polynomials have a particularly nice closed-form ordinary generating function given by
∑ ∑ --> n ≥ ≥ --> 0 x ⋅ ⋅ --> σ σ --> n ( α α --> ) ( x ) z n = e ( 1 − − --> α α --> ) z ( α α --> z e α α --> z e α α --> z − − --> 1 ) x . {\displaystyle \sum _{n\geq 0}x\cdot \sigma _{n}^{(\alpha )}(x)z^{n}=e^{(1-\alpha )z}\left({\frac {\alpha ze^{\alpha z}}{e^{\alpha z}-1}}\right)^{x}\,.}
Other combinatorial properties and expansions of these generalized α-factorial triangles and polynomial sequences are considered in Schmidt (2010).[22]
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 Pochhammer symbol and the generalized, rational-valued binomial coefficients as
( α α --> n − − --> 1 ) ! ( α α --> ) = ∑ ∑ --> k = 0 n − − --> 1 ( n − − --> 1 k + 1 ) ( − − --> 1 ) k × × --> ( 1 α α --> ) − − --> ( k + 1 ) ( 1 α α --> − − --> n ) k + 1 × × --> ( α α --> ( k + 1 ) − − --> 1 ) ! ( α α --> ) ( α α --> ( n − − --> k − − --> 1 ) − − --> 1 ) ! ( α α --> ) = ∑ ∑ --> k = 0 n − − --> 1 ( n − − --> 1 k + 1 ) ( − − --> 1 ) k × × --> ( 1 α α --> + k − − --> n k + 1 ) ( 1 α α --> − − --> 1 k + 1 ) × × --> ( α α --> ( k + 1 ) − − --> 1 ) ! ( α α --> ) ( α α --> ( n − − --> k − − --> 1 ) − − --> 1 ) ! ( α α --> ) , {\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times \left({\frac {1}{\alpha }}\right)_{-(k+1)}\left({\frac {1}{\alpha }}-n\right)_{k+1}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\\&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times {\binom {{\frac {1}{\alpha }}+k-n}{k+1}}{\binom {{\frac {1}{\alpha }}-1}{k+1}}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\,,\end{aligned}}}
and moreover, we similarly have double sum expansions of these functions given by
( α α --> n − − --> 1 ) ! ( α α --> ) = ∑ ∑ --> k = 0 n − − --> 1 ∑ ∑ --> i = 0 k + 1 ( n − − --> 1 k + 1 ) ( k + 1 i ) ( − − --> 1 ) k α α --> k + 1 − − --> i ( α α --> i − − --> 1 ) ! ( α α --> ) ( α α --> ( n − − --> 1 − − --> k ) − − --> 1 ) ! ( α α --> ) × × --> ( n − − --> 1 − − --> k ) k + 1 − − --> i = ∑ ∑ --> k = 0 n − − --> 1 ∑ ∑ --> i = 0 k + 1 ( n − − --> 1 k + 1 ) ( k + 1 i ) ( n − − --> 1 − − --> i k + 1 − − --> i ) ( − − --> 1 ) k α α --> k + 1 − − --> i ( α α --> i − − --> 1 ) ! ( α α --> ) ( α α --> ( n − − --> 1 − − --> k ) − − --> 1 ) ! ( α α --> ) × × --> ( k + 1 − − --> i ) ! . {\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (n-1-k)_{k+1-i}\\&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}{\binom {n-1-i}{k+1-i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (k+1-i)!.\end{aligned}}}
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).
( 2 n − − --> 1 ) ! ! = ∑ ∑ --> k = 0 n − − --> 1 ( n k + 1 ) ( 2 k − − --> 1 ) ! ! ( 2 n − − --> 2 k − − --> 3 ) ! ! . {\displaystyle (2n-1)!!=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!.}
Similar identities can be obtained via context-free grammars.[23] 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).[24]
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