semifactorial

semi- +‎ factorial

semifactorial (plural semifactorials)

1. (mathematics, combinatorics, rare) Synonym of double factorial
   • 1942 February, Giulio Racah, “Theory of Complex Spectra. I”, in Physical Review, volume 61, numbers 3–4, →DOI, page 188:

     With the symbol ${\displaystyle n!!}$ we indicate the semifactorial of ${\displaystyle n}$, that is the product ${\displaystyle 1.3.5\dots n}$ if ${\displaystyle n}$ is odd, and the product ${\displaystyle 2.4.6\dots n}$ if ${\displaystyle n}$ is even.

   • 1994 January, Alan Edelman, Eric Kostlan, Michael Shub, “How many eigenvalues of a random matrix are real?”, in Journal of the American Mathematical Society, volume 7, number 1, →DOI, pages 250–251:

     In the formulas above we use the Euler beta function, a Jacobi polynomial evaluated at three, and also the familiar double factorial (also known as the semifactorial) notation defined by

     $${\displaystyle n!!={\begin{cases}\displaystyle 1\times 3\times 5\times \dots \times n&{\text{if }}n{\text{ is odd}},\\[5pt]\displaystyle 2\times 4\times 6\times \dots \times n&{\text{if }}n{\text{ is even}}.\end{cases}}}$$

     

     By convention, ${\displaystyle 0!!=(-1)!!=1}$.

   • 2009, Keith B. Oldham, Jan C. Myland, Jerome Spanier, An Atlas of Functions, 2nd edition, Springer, →DOI, →ISBN, page 25:

     The double factorial or semifactorial function is defined by

     $${\displaystyle n!!={\begin{cases}\displaystyle 1&n=-1,0\\[5pt]\displaystyle n\times (n-2)\times (n-4)\times \dots \times 5\times 3\times 1&n=1,3,5,\dots \\[5pt]\displaystyle n\times (n-2)\times (n-4)\times \dots \times 6\times 4\times 2&n=2,4,6,\dots \end{cases}}}$$