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Double Factorial — Definition, Formula & Examples

Double factorial of a positive integer nn, written n!!n!!, is the product of all positive integers from 1 (or 2) up to nn that have the same parity (odd or even) as nn.

For a positive integer nn, the double factorial is defined as n!!=n⋅(n−2)⋅(n−4)⋯n!! = n \cdot (n-2) \cdot (n-4) \cdots, where the product continues down in steps of 2 until reaching 1 (if nn is odd) or 2 (if nn is even). By convention, 0!!=10!! = 1 and (−1)!!=1(-1)!! = 1.

Key Formula

n!!=∏k=0⌈n/2⌉−1(n−2k)n!! = \prod_{k=0}^{\lceil n/2 \rceil - 1} (n - 2k)
Where:
  • nn = A non-negative integer
  • kk = Index running from 0 up to $\lceil n/2 \rceil - 1$

How It Works

To compute n!!n!!, start at nn and multiply by every second integer going downward. If nn is odd, you multiply all odd numbers from nn down to 1. If nn is even, you multiply all even numbers from nn down to 2. The notation n!!n!! does not mean taking the factorial twice — it is a separate operation with its own definition.

Worked Example

Problem: Compute 7!! (the double factorial of 7).
Identify parity: Since 7 is odd, multiply all odd integers from 7 down to 1.
7!!=7⋅5⋅3⋅17!! = 7 \cdot 5 \cdot 3 \cdot 1
Multiply step by step: Compute the product from left to right.
7⋅5=35,35⋅3=105,105⋅1=1057 \cdot 5 = 35, \quad 35 \cdot 3 = 105, \quad 105 \cdot 1 = 105
Answer: 7!!=1057!! = 105

Why It Matters

Double factorials appear in combinatorics when counting perfect matchings and in probability when computing moments of the normal distribution. For instance, the number of ways to pair up 2n2n objects into nn pairs is (2n−1)!!(2n-1)!!. They also show up in physics and certain integral identities involving powers of trigonometric functions.

Common Mistakes

Mistake: Interpreting n!!n!! as (n!)!(n!)!, i.e., taking the factorial of nn and then taking the factorial of that result.
Correction: The double factorial n!!n!! is a single operation that multiplies every other integer down from nn. It is not the factorial applied twice. For example, 6!!=6⋅4⋅2=486!! = 6 \cdot 4 \cdot 2 = 48, whereas (6!)!=720!(6!)! = 720!, an astronomically larger number.