In this case I would say your initial formulation is correct, you should just define your terms so that everyone knows exactly what you mean. I haven't seen that notation used in proofs, but my experience is limited and it seems like a reasonable enough use of notation.

So you'll end up having $\displaystyle S_0 = 0$, $\displaystyle S_1 = A[0]$, $\displaystyle S_2 = A[0]+A[1]$, etc., which it seems you're going for.

Whether or not a sum from 0 to -1 is conventionally considered the empty sum, I think you can rest easy because mathematicians make up and adapt notation for their purposes all the time.