Problem 964

Musical Chairs Revisited

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:

The music starts playing and $i$ children are randomly selected, with all combinations being equally likely. The selected children stand up and dance around.
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 $\mathrm{e}$ to separate the mantissa and the exponent.

变种音乐椅游戏

一群$k(k-1) / 2 + 1$个孩子正在玩游戏，游戏需要进行$k$轮。

游戏开始时，孩子们坐在围成一圈的椅子上。

在第$i$轮游戏中：

当音乐开始时，随机选择$i$个孩子（所有组合的概率相等）；被选中的孩子站起来，随着音乐跳舞。
当音乐停止时，这$i$个孩子随机坐回$i$个空椅子上（所有排列的概率相等）。

当游戏进行完$k$轮时，记$P(k)$为每个孩子恰好都坐在他在游戏开始时的座位右侧那个椅子上的概率。

已知$P(3) \approx 1.3888888889 \mathrm{e}{-2}$。

求$P(7)$，并以科学计数法给出你的答案：四舍五入到小数点后十位有效数字，并使用小写字母$\mathrm{e}$来分隔尾数和指数。