Semifactorial

With the symbol n#33;#33; we indicate the semifactorial of n, that is the product 1.3.5#92;dotsn if n is odd, and the product 2.4.6#92;dotsn if n is even.

Source: wiktionary

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 n#33;#33;#61;#92;begin#123;cases#125;#92;displaystyle 1#92;times 3#92;times 5#92;times#92;dots#92;timesnamp;#92;text#123;if#125;n#92;text#123;isodd#125;,#92;#92;#91;5pt#93;#92;displaystyle 2#92;times 4#92;times 6#92;times#92;dots#92;timesnamp;#92;text#123;if#125;n#92;text#123;iseven#125;.#92;end#123;cases#125;By convention, 0#33;#33;#61;(-1)#33;#33;#61;1.

Source: wiktionary

The double factorial or semifactorial function is defined byn#33;#33;#61;#92;begin#123;cases#125;#92;displaystyle 1amp;n#61;-1,0#92;#92;#91;5pt#93;#92;displaystylen#92;times(n-2)#92;times(n-4)#92;times#92;dots#92;times 5#92;times 3#92;times 1amp;n#61;1,3,5,#92;dots#92;#92;#91;5pt#93;#92;displaystylen#92;times(n-2)#92;times(n-4)#92;times#92;dots#92;times 6#92;times 4#92;times 2amp;n#61;2,4,6,#92;dots#92;end#123;cases#125;

Source: wiktionary