Chapman's Problem

Back in my mega-nerd days (which, I assure you, are long gone [who am I kidding, no they're not]), my friend John and I used to make up goofy math problems for each other. Every once in a while, we'd stumble on something that looked simple, but wasn't as simple as we thought. One problem in particular remains unsolved today, despite its deceiving simplicity.

The premise is this: How many ways can you arrange n circles?

With 1 circle, there's one way to arrange it:

With 2 circles, there are two ways to arrange them:

With 3 circles, there are four ways:

 

Once we got this far, we assumed the pattern was simple: it just doubles each time. 1 makes 1, 2 makes 2, 3 makes 4, so 4 will make 8. Right? 

As you can see, there are not 8 but 9 ways to arrange 4 circles. 

With 5 circles, we get 20 possible arrangements.

Can anybody out there give me an equation to predict how many arrangements (excluding size and overlap) there will be for n circles?

The challenge to solve Chapman's Problem begins!