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The double factorial is an extension onto the normal factorial function. It is denoted with two exclamation points: .

Definition: Double factorial
The double factorial of an integer is defined recursively as:

The double factorial is not defined when n is a negative even integer.

Do not confuse the double factorial for a factorial computed twice.

The double in double factorial represents the increment between the values of the terms when the factorial is expanded into a product. In the case of a regular factorial, each factor is decremented by one, from the number 'a' to 1. In the case of a double factorial, each factor is decremented by two.

The double factorial terminates with the sequence of evens, for example: or the sequence of odds: eg


The following properties hold:

for any integer

There also exists the triple factorial, which is not as commonly known or used as the double, and with it a set of of analogous properties.

See also

  • A006882 - Double factorials in OEIS