I just into this sequence for a problem I'm working on and wondered if it had a name and a mathematical notation. It's the result of creating permutations with replacement where there's also a null case but they can't all be null. There's also no distinction between choosing "null" and then an object and the reverse order. Here's an example if you're selecting from two objects:
object1
object1-object1
object1-object2
object2
object2-object1
object2-object2
So for two objects there are six possible cases. I was curious what the number of cases there would be for each number of objects and it looks like it is:
1 = 1
2 + 2*2 = 6
3 + 3*3 + 3*3*3 = 39
4 + 4*4 + 4*4*4 + 4*4*4*4 = 340
Is there a mathematical symbol for this calculation? Does this sequence have a name?
$n+n^2+n^3+\cdots+n^n$ is a geometric progression with first term $n$, $n$ terms, and common ratio $n$. By the formula for the sum of a geometric progression, we have $$n+n^2+n^3+\cdots+n^n={n(n^n-1)\over n-1}$$ The sequence is tabulated, with more information, at the Online Encyclopedia of Integer Sequences.