I'm trying to count the number of ways of writing a general natural number $n\geq 2$ as the sum of $r$ smaller numbers where each of these numbers is at least $2$ - that is, I want to count the number of $\left\{2,\dots , n\right\}$-restricted $r$-compositions of $n$. This is not an area I expected to find myself when attempting this problem (which started out as a hard group theory exercise), but I've done some research into working with these things via generating functions (see http://www.mathcs.emory.edu/~rg/ordgenfct.pdf for a nice intro).
Essentially I need to calculate the coefficient of $t^n$ in the sum $\left(\sum_{k=1}^{n-1} t^k\right)^r$ for general $r \leq \left\lfloor n/2 \right\rfloor$ and my brain has completely shut off. Induction on $r$ would be the obvious thing to try but I haven't thought of any hypotheses which would work nicely. Some things which might help you spot a pattern:
$\left(\sum_{k=1}^{n-1}t^k\right)^2 = \sum_{k=1}^{2 n - 3} k t^{k+1}$
$\left(\sum_{k=1}^{n-1}t^k\right)^3 = \sum_{k=1}^{3 n - 5} \Delta_k t^{k+2}$ where $\Delta_k$ is the $k$th triangular number
Thanks in advance!