There is a relatively simple bijection between 0-sum sequences of 1 and -1 where the sum of all partial sequences is nonnegative and Dyck paths - this is very easy to count as a Catalan number.
However, I was trying to think about this problem with the condition that the partial sums are greater than some value, say $-5$, i.e.
Let $S$ be some $2n$ length sequence consisting of $n$ $1$s and $n$ $-1$s such that $\sum_{i=1}^{2n} S_i = 0$. Then, how many sequences $S$ exist such that the partial sum $\sum_{i=1}^{k} S_i > -5$ for all $k \in [1, 2n]$?
I tried thinking about this by 'shifting' the diagonal to $y = x + 4$, or by starting at different points, but I can't quite figure out how to count it. I also found this question, but the solutions did not really help much.