A group of children play a game of rounds. At the beginning, they are all seated on chairs arranged in a circle.
During the -th round:
The music starts playing and children are randomly selected, with all combinations being equally likely. The selected children stand up and dance around.
When the music stops, these children sit back down randomly in the available chairs, with all permutations being equally likely.
Let 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 -th round).
You are given .
Find . 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.
964. 重访抢椅子游戏
一组 个孩子将玩 轮抢椅子游戏。开始时，他们都坐在排列成一圈的椅子上。
在第 轮中：
乐声响起的同时，随机选出 个孩子（所有组合被选中的概率均相同）。被选中的 个孩子须起身并围圈起舞。
乐声停止时，这 个孩子须随机坐回空出的 把椅子上（每种入座方式出现的概率相同）。
记 为：第 轮结束时，每个孩子都恰好坐在其初始座位右侧相邻椅子上的概率。
已知 。
求 ，将你的答案用科学记数法表示，用小写字母 e 分开尾数和指数，并将尾数四舍五入至小数点后第 10 位。