Closed Form Representation of Recurrence Relation

Question

$$T(0) = 1$$ $$(N \gt 0) \quad T(N) = \sum_{k=0}^{n-1} (k+1)T(k)$$

"Given the recurrence relation, provide a closed-form representation for function T and provide a rigorous proof that the answer is correct."

I'm not entirely sure how this can be done through the induction method. What I have tried to do is the following:

$$1. \quad T(N) = \sum_{k=0}^{n-1} kT(k)+ \sum_{k=0}^{n-1}T(k)$$ $$2. \quad T(N) = \sum_{k=0}^{n-1} kT(k)+T(N-1) + \sum_{k=0}^{n-2}T(k)$$ $$3. \quad T(N) = \sum_{k=0}^{n-1} kT(k)+T(N-1) + (T(N-1)-1)$$ $$4. \quad T(N) = \sum_{k=0}^{n-1} kT(k)+ 2(T(N-1))-1$$

But now I'm stuck as to how I should approach $\sum_{k=0}^{n-1} kT(k)$

Any hints or on how to solve this problem with a proof?

Answer 1

Here's a hint - for n >= 3:

$$T(n) = \sum_{k=0}^{n-1}(k+1)T(k) = \sum_{k=0}^{n-2}(k+1)T(k) + nT(n-1) =T(n-1) + nT(n-1) = (n+1)T(n-1)$$