Jack and Jill

There are $n$ males and $n$ females who take part in a dance competition. The competition runs under these rules:

1. At first, the males and females are randomly matched into $n$ couples, and all the couples are arranged in a circle.
2. The judge flips a coin to choose $k$, which is either $1$ or $2$ with equal probability. Then another coin flip chooses "clockwise" or "counter-clockwise", also with equal probability.
3. Following the coin flips from the previous step, each female moves $k$ positions around the circle in the chosen direction and takes the partner she lands on. The males stay in place.
4. If, after moving, a female is matched with a male she already danced with in an earlier round, the competition ends and the judges determine the winners. Otherwise, the current couples dance a round, the judges evaluate them, and the process returns to step 2.

Find the expected number of rounds danced during the competition.

A single line containing an integer $n$ ($2 \le n \le 50$).

Print the answer with an accuracy of $10^{-9}$.