Problem 930

The Gathering

Given $n\ge 2$ bowls arranged in a circle, $m\ge 2$ balls are distributed amongst them.

Initially the balls are distributed randomly: for each ball, a bowl is chosen equiprobably and independently of the other balls. After this is done, we start the following process:

Choose one of the $m$ balls equiprobably at random.
Choose a direction to move - either clockwise or anticlockwise - again equiprobably at random.
Move the chosen ball to the neighbouring bowl in the chosen direction.
Return to step $1$.

This process stops when all the $m$ balls are located in the same bowl. Note that this may be after zero steps, if the balls happen to have been initially distributed all in the same bowl.

Let $F(n, m)$ be the expected number of times we move a ball before the process stops. For example, $F(2, 2) = \frac{1}{2}$, $F(3, 2) = \frac{4}{3}$, $F(2, 3) = \frac{9}{4}$, and $F(4, 5) = \frac{6875}{24}$.

Let $G(N, M) = \sum_{n=2}^N \sum_{m=2}^M F(n, m)$. For example, $G(3, 3) = \frac{137}{12}$ and $G(4, 5) = \frac{6277}{12}$. You are also given that $G(6, 6) \approx 1.681521567954e4$ in scientific format with $12$ significant digits after the decimal point.

Find $G(12, 12)$. Give your answer in scientific format with $12$ significant digits after the decimal point.

小球聚会

将$n\ge 2$个碗排成一圈，并向其中放入$m\ge 2$个球。

一开始，球是随机分布的：对于每个球，等概率且独立地选择一个碗，并将球放入。全部放完之后，我们进行以下操作：

等概率地从$m$个球中随机选择一个。
等概率地随机选择一个移动方向：顺时针或逆时针。
将选中的球移动到所选方向的相邻碗中。
返回步骤$1$。

当所有$m$个球都位于同一个碗中时，停止操作。如果球一开始恰好就放在同一个碗中，则不需要进行任何操作，立刻停止。

令$F(n, m)$为在停止前移动球的期望次数。例如，$F(2, 2) = \frac{1}{2}$，$F(3, 2) = \frac{4}{3}$，$F(2, 3) = \frac{9}{4}$，$F(4, 5) = \frac{6875}{24}$。

令$G(N, M) = \sum_{n=2}^N \sum_{m=2}^M F(n, m)$。例如，$G(3, 3) = \frac{137}{12}$，$G(4, 5) = \frac{6277}{12}$。已知$G(6, 6) \approx 1.681521567954e4$（以科学计数法表示，小数点后保留$12$位有效数字）。

求$G(12, 12)$，并以科学计数法表示给出你的答案，小数点后保留$12$位有效数字。