AUSTRIAN-POLISH 1992 Problem 9

Given an integer $n > 1$ , consider words composed of $n$ letters $A$ and $n$ letters $B$ . A word $X_1...X_{2n}$ is said to belong to set $R(n)$ (respectively, $S(n)$ ) if no initial segment (respectively, exactly one initial segment) $X_1...X_k$ with $1 \le k < 2n$ consists of equally many letters $A$ and $B$ . If $r(n)$ and $s(n)$ denote the cardinalities of $R(n)$ and $S(n)$ respectively, compute $s(n)/r(n)$ .