double factorials

The double factorial of an integer

0$">, denoted by $n!!$

, is the product of all the integers from 1 to $n$

which have the same parity as $n$

. By definition, $0!! =1$

. The double factorial is not to be confused with the double application of factorial, i.e. $n!! \neq (n!)!$ .

For example, $8!! = 2\cdot4\cdot6\cdot8$ and $7!!=1\cdot3\cdot5\cdot7$ .

The first double factorials are 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many double factorials are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Double Factorial
Wikipedia, Double factorial
OEIS, Sequence A006882

Double factorials can also be... (you may click on names or numbers)

ABA 384 aban 15 48 105 384 945 abundant 48 384 945 3840 10395 46080 135135 645120 2027025 10321920 34459425 admirable 945 alternating 105 945 amenable 48 105 384 945 3840 46080 645120 2027025 10321920 34459425 185794560 apocalyptic 384 3840 10395 arithmetic 15 105 945 10395 135135 645120 astonishing 15 Bell 15 betrothed 48 binomial 15 105 brilliant 15 cake 15 congruent 15 384 3840 10395 46080 135135 645120 constructible 15 48 384 3840 Cunningham 15 48 Curzon 105 34459425 cyclic 15 D-number 15 deficient 15 105 dig.balanced 15 384 10395 economical 15 105 384 3840 10321920 emirpimes 15 equidigital 15 105 384 3840 evil 15 48 105 384 945 3840 46080 135135 645120 2027025 10321920 34459425 185794560 frugal 10321920 185794560 gapful 105 3840 10395 46080 135135 645120 2027025 10321920 34459425 185794560 654729075 3715891200 13749310575 81749606400 happy 135135 645120 2027025 Harshad 48 3840 46080 645120 10321920 185794560 654729075 3715891200 hexagonal 15 highly composite 48 idoneal 15 48 105 inconsummate 945 interprime 15 105 3840 10395 Jordan-Polya 48 384 3840 46080 645120 10321920 185794560 3715891200 81749606400 1961990553600 51011754393600 junction 105 645120 2027025 10321920 Lehmer 15 Leyland 945 lucky 15 105 135135 Lynch-Bell 15 48 384 magic 15 metadrome 15 48 nude 15 48 384 135135 oban 15 odious 10395 654729075 panconsummate 15 pandigital 15 partition 15 pernicious 48 384 10395 plaindrome 15 48 practical 48 384 3840 46080 645120 prim.abundant 945 pseudoperfect 48 384 945 3840 10395 46080 135135 645120 repunit 15 Ruth-Aaron 15 105 Saint-Exupery 3840 3715891200 self 34459425 654729075 semiprime 15 sphenic 105 super Niven 48 super-d 105 135135 superabundant 48 tau 384 645120 10321920 tetranacci 15 triangular 15 105 uban 15 48 Ulam 48 46080 unprimeable 46080 645120 wasteful 48 945 10395 46080 135135 645120 2027025 Zeisel 105 Zuckerman 15 384 Zumkeller 48 384 945 3840 10395 46080