Finding solution of given recursive equation?
The trick in this case is to consider differences:
$$
T(n)-T(n-1) = \left(1 + \sum_{j=0}^{n-1} T(j)\right) - \left(1 + \sum_{j=0}^{n-2} T(j)\right) = T(n-1).
$$
Hence $T(n) = 2T(n-1)$, and so $T(n) = 2^n T(0)$.