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Concept
Double factorial
Summary
In mathematics, the double factorial of a number n, denoted by n!!, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is,
Restated, this says that for even n, the double factorial is
while for odd n it is
For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0!! = 1 as an empty product.
The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as
The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as
The term odd factorial is sometimes used for the double factorial of an odd number.
In a 1902 paper, the physicist Arthur Schuster wrote:
The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, , if be odd, or if be odd [sic]. I propose to write for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial."
states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics. They occur in Student's t-distribution (1908), though Gosset did not use the double exclamation point notation.
Because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial n!, and it is much smaller than the iterated factorial (n!)!.
The factorial of a positive n may be written as the product of two double factorials:
and therefore
where the denominator cancels the unwanted factors in the numerator. (The last form also applies when n = 0.
Official source
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