Swapping parentheses P37366

Let ${\cal P}_n$ be the set of words with exactly $n$ opening parentheses and $n$ closing parentheses, such that every ‘)’ matches a ‘(’. For instance, $${\cal P}_3 \enspace = \enspace \{ \enspace \mbox{``\texttt{\small ((()))}''} \enspace, \enspace \mbox{``\texttt{\small (()())}''} \enspace, \enspace \mbox{``\texttt{\small (())()}''} \enspace, \enspace \mbox{``\texttt{\small ()(())}''} \enspace, \enspace \mbox{``\texttt{\small ()()()}''} \enspace \} \enspace .$$

Consider the following experiment: Choose one word $w$ from ${\cal P}_n$ at random. Then, pick one ‘(’ and one ‘)’ of $w$, independently at random, and swap them. What is the probability that the result is also a word in ${\cal P}_n$?

For example, let $n = 3$. If we choose $w =$ “((()))”, then there are exactly four swaps that produce a word in ${\cal P}_3$, namely 2-4, 2-5, 3-4, 3-5. The rest of swaps (1-4, 1-5, 1-6, 2-6, 3-6) are incorrect. Each of the other words in ${\cal P}_3$ has three correct swaps. Therefore, the probability for $n = 3$ is $$\frac{1}{5} \left( \frac{4}{9} + \frac{3}{9} + \frac{3}{9} + \frac{3}{9} + \frac{3}{9} \right) = \frac{16}{45} \simeq 0.355556 \enspace .$$

Input

Input consists of several integer numbers $n$ between 1 and 30.

Output

For every given $n$, print with six digits after the decimal point the probability that swapping a random ‘(’ with a random ‘)’ of a random word in ${\cal P}_n$ produces a word also in ${\cal P}_n$.