Subway Factorials

I was ordering a sandwich at Subway the other day when the checker said "Awesome shirt!  What are the exclamation marks for?"

I was wearing a shirt with one of Ramanujan's incredible formulas for 1/pi:



I was able to explain to him what a factorial is (6! = 6 * 5 * 4 * 3 * 2 * 1 = 720) but ran out of time before getting to double-factorials:



So, 6!! = 6 * 4 * 2 = 48, while 7!! = 7 * 5 * 3 * 1 = 105.  While double factorials look more impressive, they are always less than or equal to normal factorials.

I learned that double factorials are one case of multifactorials, where the number of exclamation points indicates the difference between factors:

8!!! = 8 * 5 * 2 = 80

10!!!! = 10 * 6 * 2 = 120

Looking at the entries for "factorial" on Wikipedia and Wolfram, I was surprised at how little I know about them.  I did find a few cool facts to take away, though:

• the factorial function growth faster than any polynomial or exponential function, but it can be approximated by an expression involving e and pi:
  


• the gamma function is just like a factorial but can take real and complex arguments (not just integers).  The gamma function does not return 0! = 1, so it must be defined as a special case.
  


• 0.5! = sqrt(pi) / 2

What other stuff is cool about factorials?