n!! denotes the double factorial of n and is defined recursively by

For example, 8!! = 2 · 4 · 6 · 8 = 384 and 9!! = 1 · 3 · 5 · 7 · 9 = 945. The sequence of double factorials for n = 0, 1, 2, ... starts as

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, ...

The above definition can be rewritten to define double factorials of negative odd numbers:

The sequence of double factorials for n = −1, −3, −5, −7, ... starts as

while the double factorial of negative even integers is undefined.

Some identities involving double factorials are:

where Γ is the Gamma function. The last equation above can be used to define the double factorial as a function of any complex number n ≠ 0, just as the Gamma function generalizes the factorial function. One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number (for n > 2).

MULTIFACTORIALS

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more. The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. In general, the k^th factorial, denoted by n!^(k), is defined recursively as

Some mathematicians have suggested an alternative notation of n!₂ for the double factorial and similarly n!_k for other multifactorials, but this has not come into general use.

In the same way that ( − n)! is not defined for integers, and ( − n)!! is not defined for even integers, ( − n)!^(k) is not defined for .

Also, (kn)!^(k) = kⁿn!

RELATED FUNCTIONS

Dwyer Function

Definition

for x, y integer and positive.

History

This function was first introduced in 2005 as an upper bound on the number of Euler circuits in a complete graph by researchers at Richmond, The American International University in London. It has since been studied in comparison with other-rapidly increasing functions in the context of algorithm analysis applications. More recently, it has been shown to comprise part of an asymptotic limit function for the number of Euler circuits in complete graphs.

Applications

1. In graph theory, the number of Hamilton circuits, H(n), in the complete graph K{n} is given by D(0, n − 1).

2. In graph theory, for n odd, the number of Euler circuits, E(n), in the complete graph K{n} is estimated by

[((n² − 3n+1)n − 1)D((n − 3)/2, n − 2)]/ (n − 2)³

г(x) & D(x,1) Values

MULTIFACTORIALS REFERENCES

http://mathworld.wolfram.com/DoubleFactorial.html

http://www.algana.co.uk/Research/Counting.pdf

http://www.answers.com/topic/factorial

RELATED FUNCTIONS

• Ackermann function

• Dwyer function

• Exponential function

• Gamma function

• Reciprocal Beta function

• Sudan function

• Power Tower function

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