Meaning of double factorial | Babel Free

Definitions

For an odd integer, the result of multiplying all the odd integers from 1 to the given number; or for an even integer, the result of multiplying all the even integers from 2 to the given number; symbolised by a double exclamation mark (!!). For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.

Equivalents

Examples

“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, n#92;cdotn-2#92;cdotn-4#92;cdots 1, if n be odd, or n#92;cdotn-2#92;cdots 2 if n be odd^([sic – meaning even]). I propose to write n#33;#33; for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial." Full advantage of the new symbol is only gained by extending its meaning to the negative values of n. Its complete definition may then be included in the equationsn#33;#33;#61;n(n-2)#33;#33;,#92;quad 1#33;#33;#61;1,#92;quad 2#33;#33;#61;2.”

“The double factorial notation #92;begin#123;align#125;(2n)#33;#33;amp;#61;2#92;cdot 4#92;cdot 6#92;cdots(2n-2)(2n)#61;2#123;n#125;n#33;#92;#92;(2n#43;1)#33;#33;amp;#61;1#92;cdot 3#92;cdot 5#92;cdots(2n-1)(2n#43;1)#61;#92;frac#123;(2n#43;1)#33;#125;#123;2#123;n#125;n#33;#125;#92;end#123;align#125;may be considered as a generalization of n#33;#61;1#92;cdot 2#92;cdot 3#92;cdotsn.”

“We prefer now to write the expansion in a slightly different way in order to exhibit more clearly the symmetry properties of the expansion coefficients:f#95;k(m,m')#61;(#92;tfrac#123;1#125;#123;2#125;#92;pi)#92;tfrac#123;1#125;#123;2#125;2#123;-2#92;bar#123;m#125;-3#125;#92;sum#95;#123;#92;sigma#125;(2#92;bar#123;m#125;-2#92;sigma#43;1)#33;#33;#92;,T#95;k#123;#92;,2#92;sigma#125;(m,m')J#95;#123;2#92;sigma#125;,where, as above, #92;bar#123;m#125; is the average of m and m', #92;bar#123;m#125;#61;#92;tfrac#123;1#125;#123;2#125;(m#43;m'), and the double factorial notation is used, (2n#43;1)#33;#33;#61;1#92;cdot 3#92;cdot 5#92;cdots(2n#43;1).”

“The symbol #33;#33; in Eq. (9) denotes the double factorial given by n#33;#33;#61;n(n-2)(n-4)#92;dots#92;kappa#95;n, where #92;kappa#95;n is 1 for odd n and 2 for even n.”

“Double factorials can also be defined recursively. Just as we can define the ordinary factorial by n#33;#61;n#92;cdot(n-1)#33; for n#92;geq 1 with 0#33;#61;1, we can define the double factorial byn#33;#33;#61;n#92;cdot(n-2)#33;#33;for n#92;geq 2 with initial values 0#33;#33;#61;1#33;#33;#61;1. With our convention that (-1)#33;#33;#61;1, the recursion is valid for all positive integers n.”

CEFR level