I shall denote, as usual, the factorial product of the numbers up to n {\displaystyle n} by n ! {\displaystyle n!} , but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently define n ! ! = n . ( n − 2 ) ! ! . {\displaystyle n!!=n.(n-2)!!.} Starting with 1 ! ! = 1 , 2 ! ! = 2 , {\displaystyle 1!!=1,\quad 2!!=2,} it follows that, for positive values of n {\displaystyle n} , n ! ! = n ⋅ n − 2 ⋅ n − 4 ⋯ , {\displaystyle n!!=n\cdot n-2\cdot n-4\cdots ,} where the last factor is either 1 {\displaystyle 1} or 2 {\displaystyle 2} , according as n {\displaystyle n} is odd or even.
1903, Arthur Schuster, “On some Definite Integrals, and a New Method of Reducing a Function of Spherical Co-ordintes to a Series of Spherical Harmonics”, in Philosophical Transactions of the Royal Society A, volume 200, page 182