Eccoci.

A group of $k(k-1) / 2 + 1$ children play a game of $k$ rounds.
At the beginning, they are all seated on chairs arranged in a circle.

During the $i$-th round:

1. The music starts playing and $i$ children are randomly selected, with all combinations being equally likely. The selected children stand up and dance around.
2. When the music stops, these $i$ children sit back down randomly in the $i$ available chairs, with all permutations being equally likely.

Let $P(k)$ be the probability that every child ends up sitting exactly one chair to the right of their original chair when the game ends (at the end of the $k$-th round).

You are given $P(3) \approx 1.3888888889 \mathrm {e}{-2}$.

Find $P(7)$. Give your answer in scientific notation rounded to ten significant digits after the decimal point. Use a lowercase e to separate the mantissa and the exponent.