First, consider the differences between consecutive values: We have $\begin{align*}T_{N+1}-T_N &= \left(\sum_{k=0}^{N-1}(N-k)T_k\right)-\left(\sum_{k=0}^{N-2}(N-1-k)T_k\right) \\ &= \bigl(N-(N-1)\bigr)T_{N-1} + \sum_{k=0}^{N-2}\biggl((N-k)-(N-1-k)\biggr)T_k \\ &= T_{N-1}+\sum_{k=0}^{N-2}T_k \\ &= \sum_{k=0}^{N-1}T_k\end{align*}$ Or in other words $T_{N+1} = \sum_{k=0}^N T_k$. Can you figure out the rest from there?
(Incidentally, one way of arriving at a good closed-form for your recurrence, or at least starting a guess, is just to start plugging in values! Quick calculation shows that $T_2=1+1\cdot T_0 = 2$, $T_3 = 1+2\cdot T_0 + 1\cdot T_1 = 4$, and $T_4 = 1+3\cdot T_0+2\cdot T_1+1\cdot T_2 = 8$, and that should lead pretty easily to a hypothesis about $T_N$; from there it's just a matter of proving your guess.)