Vectors Sums

$\text{V is a vector }\in \{-1,1{\}}^n$

We define the following $n$ sums: $$S_j=\sum _{i=1}^j{V}_i-\sum _{i=j+1}^n{V}_i\text{ for }1\le j\le n$$ For example if $V=(-1,-1,1,1,1)\text{ then }S=(-3,-5,-3,-1,1)$

We define then $T_j=1\text{ if }{S}_j>0\text{ and }0\text{ otherwise. Thus }T=(0,0,0,0,1)$

There are $2^n\text{ possible vectors }V.$

How many different vectors $T$ are there?

Give your answer for n = 50

Hint : there are 2728 possibilities for n = 15.