Given the recursive function $T: \mathbb{N}_0 \to \mathbb{R}_+$
$$T(n) ≤ max_{1 ≤ k ≤ n - 1}\{T(k-1) + T(n - k) + c n^2\}$$ with c > 0 constant, I want to determine an absolute formula to quickly determine a supremum or an upper limit as low as possible for what the n-th value of T could be, and prove that this formula is indeed valid.
I was a little irritated myself that there's no base case given for $T(1)$ and $T(0)$, so let's simply assume $T(1) = T(0) = c$.
Now my idea was that once I've figured out such a formula, I could probably d an induction proof to show that they are correct indeed. But I don't know how to find a formula in the first place. Thanks in advance.