Problem 930

The Gathering

Given $n \geq 2$ bowls arranged in a circle, $m \geq 2$ 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 $G (6, 6) \approx 1.681521567954 e 4$ 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 $n \geq 2$ 个碗排成一圈，并向其中放入 $m \geq 2$ 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 $G (6, 6) \approx 1.681521567954 e 4$ （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。