BALKAN-SL 2021 C4

A sequence of $2 n+1$ non-negative integers $a_{1}, a_{2}, \ldots, a_{2 n+1}$ is given. There's also a sequence of $2 n+1$ consecutive cells enumerated from 1 to $2 n+1$ from left to right, such that initially the number $a_{i}$ is written on the $i$-th cell, for $i=1,2, \ldots 2 n+1$. Starting from this initial position, we repeat the following sequence of steps, as long as it's possible:

Step 1: Add up the numbers written on all the cells, denote the sum as $s$. Step 2: If $s$ is equal to 0 or if it is larger than the current number of cells, the process terminates. Otherwise, remove the $s$-th cell, and shift all cells that are to the right of it one position to the left. Then go to Step 1.

Example: $(1,0,1, \underline{2}, 0) \rightarrow(1, \underline{0}, 1,0) \rightarrow(1, \underline{1}, 0) \rightarrow(\underline{1}, 0) \rightarrow(0)$. A sequence $a_{1}, a_{2}, \ldots, a_{2 n+1}$ of non-negative integers is called balanced, if at the end of this process there's exactly one cell left, and it's the cell that was initially enumerated by $(n+1)$, i.e. the cell that was initially in the middle.

Find the total number of balanced sequences as a function of $n$.