Factorial2

The double factorial $n!!$, the product of every second integer from n down to 1 or 2.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDouble factorial↗MathWorldDoubleFactorial↗WikidataQ13520655↗nLabdouble factorial↗FungrimDoubleFactorial↗Fungrim entrya68f0e↗b6b014↗OEISA006882↗Wolfram LanguageFactorial2✓↗

Factorial2(n)the double factorial $n!!$.

SageMathInteger(n).multifactorial(2)SymPyfactorial2(n)mpmathfac2(n)JuliaCombinatorics.doublefactorial(big(n))

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always
(any) -> any	number-theoryreplaces analytic	always

Domain: Combinatorics

Details

• Skips every other factor: $n!! = n(n-2)(n-4)\cdots$, stopping at 1 (n odd) or 2 (n even).
• Recurrence: $n!! = n\,(n-2)!!$, with $0!! = 1$.
• Splits an ordinary factorial into its even and odd parts: $n! = n!!\,(n-1)!!$.
• $(2n-1)!!$ counts the perfect matchings of $2n$ objects into pairs.
• The recurrence run downwards extends it to negative odd integers, $(-2k-1)!! = \dfrac{(-1)^k}{(2k-1)!!}$; at negative even integers it hits a pole (a division by the $0!!$ term) and returns NaN.

Examples

Basic #

7 × 5 × 3 × 1

live ↗

In$7!!$Out$105$

8 × 6 × 4 × 2

live ↗

In$8!!$Out$384$

live ↗

In$10!!$Out$3\,840$

live ↗

In$0!!$Out$1$

live ↗

In$20!!$Out$3\,715\,891\,200$

The recurrence $n!! = n \cdot (n-2)!!$ run downwards extends it to negative odd integers: $(-1)!! = 1$, $(-3)!! = -1$, $(-5)!! = \frac{1}{3}$

live ↗

In$(-1)!!$Out$1$

Threads element-wise over a list

live ↗

In$\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack!!$Out$\bigl\lbrack1, 2, 3, 8, 15\bigr\rbrack$

Negative odd integers follow the recurrence backwards: $(-3)!! = -1$

live ↗

In$(-3)!!$Out$-1$

$(-5)!! = \frac{1}{3}$ by the backward recurrence

live ↗

In$(-5)!!$Out$\frac{1}{3}$

Real arguments use the analytic continuation $2^{(1+2x-\cos\pi x)/4}\,\pi^{(\cos\pi x-1)/4}\,\Gamma(1+\frac{x}{2})$

live ↗

In$2.5!!$Out$2.407\,069\,456\,116\,044$

The number of ways to pair up 10 people into 5 couples: $(2n-1)!!$ counts perfect matchings of $2n$ objects

live ↗

In$9!!$Out$945$

Splits an ordinary factorial into even and odd parts: $n! = n!!\,(n-1)!!$

live ↗

In$8!=8!!×7!!$Out$\mathrm{True}$

Even double factorials: $(2n)!! = 2^n\,n!$

live ↗

In$10!!=2^5×5!$Out$\mathrm{True}$

Odd double factorials: $(2n-1)!! = \frac{(2n)!}{2^n\,n!}$

live ↗

In$9!!=\frac{10!}{2^5×5!}$Out$\mathrm{True}$

Negative even arguments hit a pole in the recurrence, so compute-engine returns ComplexInfinity

live ↗

In$(-2)!!$Out$\tilde∞$

$15!! = 2{,}027{,}025$, already past two million despite skipping every other factor

live ↗

In$15!!$Out$2\,027\,025$

Possible issues #

$(-\infty)!! $ — the trig-based analytic continuation isn't defined at $\pm\infty$; native declines cleanly to Indeterminate

live ↗

In$-∞!!$Out$\operatorname{Indeterminate}$

About

Examples	17
Checked against	Wolfram Language 17 agreeSageMath 14 agree3 errorSymPy 14 agree3 errormpmath 16 agree1 errormathlib4 10 agree6 errorJulia 11 agree6 error
Crosswalk	WikipediaMathWorldWikidatanLabFungrimFungrim entry ×2OEISWolfram LanguageSageMathSymPympmathJulia

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Nat.doubleFactorial n, n!! with 0!! = 1!! = 1; natural n only, so the negative odd extension (-1)!! = 1, (-3)!! = -1 is out of its domain.

See also: Factorial