# What does the !! mean in trigonometric identity?

What does the $$!!$$ mean in:

$$\int_0^x \sin^n(t) \mathrm dt = \begin{cases} \frac{(n-1)\color{red}{!!}}{n\color{red}{!!}}\Big[1-\cos(x)\sum_{j=0}^{(n-1)/2}\frac{(2j-1)\color{red}{!!}}{(2j)\color{red}{!!}}\sin^{2j}(x)\Big]&\text{for n odd}\\ \frac{(n-1)\color{red}{!!}}{n\color{red}{!!}}\Big[x-\cos(x)\sum_{j=0}^{(n-2)/2}\frac{(2j)\color{red}{!!}}{(2j+1)\color{red}{!!}}\sin^{2j+1}(x)\Big]&\text{for n even}\\ \end{cases}.$$

Is it factorial applied twice?

This is from page 317 of An Atlas of Functions, Second edition: with Equator, the Atlas Function Calculator by Keith B. Oldham, Jan Myland, Jerome Spanier

• That looks like being the double-factorial. It's smaller than the factorial: every other number multiplied together, not every number multiplied together. Jan 12 '20 at 17:25
• @Patrick Stevens thanks I'll look it up and try to apply. Jan 12 '20 at 17:25
• This is not quite a duplicate of The double factorial notation Jan 12 '20 at 17:56
• actually there is a symbolic index in that book on page 733 where it is described as double factorial function [2:13] which is on page 25 where the function is fully explained! Jan 15 '20 at 9:51

In mathematics, the double factorial or semifactorial of a number $$n$$ (denoted by $$n!!$$) is the product of all the integers from $$1$$ up to $$n$$ that have the same parity (odd or even) as $$n$$.
Example: $$9!! = 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1$$