A question about making a function for this sequence?

Please excuse my weak math terminology, it's been a while...

I got bored at work today and was playing around with exponents. I would raise the numbers 1 to 9 to the first power, then the second power, then the third power, and so on, all the way up to the tenth power. Then I took the answers and looked for patterns. I found that, if you take the differences between the numbers, and then the differences between those numbers, eventually you reach a constant number. i.e. 1 squared is 1, 2 squared is 4, 3 squared is 9, 5 squared is 25, and 6 squared in 36. 4-1=3, 9-3=5, 16-9=7, 25-16=9, 36-25=11. The difference between all those numbers is 2. So that was the constant that I found for the second power. Each time I went up a power, I had to do one more 'tier' of calculations to get to a constant. Here were the constants that I found: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880. I looked for a pattern in this, and noticed that 1*2=2, 2*3=6, 6*4=24, 24*5=120, and so on. If you increase the number that you multiply by by 1 each time you end up with that sequence. So my question is twofold. Firstly, what is this called? What are other ways to end up with this sequence? Secondly, how would I derive a function from this list of numbers? I want to play with derivatives now, but I've completely forgotten how to get the function!