I haven't quite worked this out, but I can give you some hints.
First of all, I'm pretty sure you mean Stirling Numbers of the 2nd Kind, not the first. You're using the Knuth notation, where
stands for the Stirling Numbers first kind (I think it's pronounced "p bracket p-1") and
stands for the Stirling Numbers of the second kind ("p brace p-1.")
If you look at a table of Stirling Numbers (2nd kind), you'll see a diagonal of 1's. All numbers of the form
are equal to one. So the numbers you're interested in are in the "second diagonal" just below the first, that is, each one has a 1 above it in the table. Think of the recurrence relation. What does that mean?
Now consider the numbers in that second diagonal as a sequence. Construct a table of differences. What do you observe?