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Button-up Some More

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

You might like to have a look at Button-up before trying this problem.
 
I have a jacket which has four buttons.
Sometimes, I do the buttons up starting with the top button. Sometimes, I start somewhere else.  
How many different ways of buttoning it up can you find?
 
Look back at the number of different ways you found for buttoning up three buttons and four buttons.
Can you predict the number of ways of buttoning up a coat with five buttons?
Six buttons ...?  
 

This problem was inspired by an idea from Bernard Murphy.