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.
solutions/s930.md.