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