I have a sequence of random variables, $$X_0, X_1, \ldots, X_n, \ldots$$ defined in a probability space where $$X_0 = 0$$ by definition, and $$X_n = r_1(x) + r_2(x) + \cdots + r_n(x), \quad n \ge 1,$$ where $r(x)$ is the rademacher function, with $x$ any real number in the interval $[0,1)$. The sequence of random variables given describes a random walk in the set of integers.
I want to find the distribution of the $X$ in the space given by $A = \{-n, -n + 2, \ldots , n - 2, n\}$ is
$$P(X_n=k) = \frac 1 {2^n}\cdot \binom n {(n - k)/2} \text{ when } k \in A.$$
and that the probability is zero when $$ k \in \mathbb{Z}\smallsetminus A$$
I know how to calculate the distribution of a sequence of i.i.d. random variables, like they are in this problem, but the restriction to the set $A$ is troubling me out. Any help?