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

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    $\begingroup$ That looks like being the double-factorial. It's smaller than the factorial: every other number multiplied together, not every number multiplied together. $\endgroup$ – Patrick Stevens Jan 12 at 17:25
  • $\begingroup$ @Patrick Stevens thanks I'll look it up and try to apply. $\endgroup$ – onepound Jan 12 at 17:25
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    $\begingroup$ This is not quite a duplicate of The double factorial notation $\endgroup$ – Mark S. Jan 12 at 17:56
  • $\begingroup$ 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! $\endgroup$ – onepound Jan 15 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: $9!! = 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1$


to supplement N3buchadnezzar, some ways to work out the double factorial or semifactorial are as follows:


following this definition, these handy relations can be used

for even !!


and odd !!

enter image description here


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