We all know that 5! equals 5x4x3x2x1, and when we encounter something less intuitive, such as -5.98!, we are all capable of plugging the necessary notation into Wolfram Alpha, or some similar calculator, to obtain the correct answer. However, error starts to creep in when dealing with multiple exclamation points, such as 5!!. Too often these multifactorials are incorrectly interpreted by calculators and their fleshy operators as nested factorials ((5!)!), when in fact they are completely different. In order to calculate a multifactorial, such as 17!!!!, you must first count the number of exclamation points, in this case 4. Since there are 4 exclamation points here, we will multiply every 4th number until we can no longer get smaller. So, 17!!!!=17×13×9×5×1, a much smaller number than ((((17!)!)!)!). Likewise, 14!!! has three exclamation points, and thus we descend by threes to get 14×11×8×5×2. As a final example, 5!! has two exclamation points and is therefore equal to 5×3×1.

Wolfram alpha is able to calculate double factorials correctly (the case with just two exclamation points), but cannot interpret any of the higher order multi factorials. As a result, when calculating multifactorials of less workable numbers, this alternative but esentially equivalent definition must be used in order to obtain the correct result, however note this function has some unique domain restrictions, as well as some divergence from the initial definition.

TLDR: n!!! ≠ ((n!)!)! and Wolfram Alpha and other calculators don't always realize this. Be careful when calculating!

Sources: https://en.m.wikipedia.org/wiki/Factorial#Alternative_extension_of_the_multifactorial

http://mathworld.wolfram.com/Multifactorial.html