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A sequence of

2n + 1

non-negative integers

a_1, a_2, ..., a_{2n + 1}

is given. There's also a sequence of

2n + 1

consecutive cells enumerated from

1

2n + 1

from left to right, such that initially the number

a_i

is written on the

i

-th cell, for

i = 1, 2, ..., 2n + 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 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,. . . , a_{2n+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

.
Proposed by Viktor Simjanoski, North Macedonia

Problem Statement