Problem 951

A Game of Chance

Two players play a game using a deck of $2n$ cards: $n$ red and $n$ black. Initially the deck is shuffled into one of the $\binom{2n}{n}$ possible starting configurations. Play then proceeds, alternating turns, where a player follows two steps on each turn:

Remove the top card from the deck, taking note of its colour.
If there is a next card and it is the same colour as the previous card they toss a fair coin. If the coin lands on heads they remove that card as well; otherwise leave it on top of the deck.

The player who removes the final card from the deck wins the game.

Some starting configurations give an advantage to one of the players; while some starting configurations are fair, in which both players have exactly $50\%$ chance to win the game. For example, if $n=2$ there are four starting configurations which are fair: RRBB, BBRR, RBBR, BRRB. The remaining two, RBRB and BRBR, result in a guaranteed win for the second player.

Define $F(n)$ to be the number of starting configurations which are fair. Therefore $F(2)=4$. You are also given $F(8)=11892$.

Find $F(26)$.

机会游戏

两位玩家使用一副包含$2n$张牌的牌组进行游戏，其中包括$n$张红牌和$n$张黑牌。游戏开始前，牌组被洗成$\binom{2n}{n}$种可能的初始局面之一；之后两位玩家轮流行动，每位玩家在其回合中进行以下两个步骤：

从牌组顶部移除一张牌，并记下其颜色。
如果牌组中还有牌，且当前牌组顶部的牌与刚才被移除的牌颜色相同，则抛掷一枚公平硬币：如果硬币正面朝上，则移除当前牌组顶部的牌；否则将其留在牌组顶部。

从牌组中移除最后一张牌的玩家获胜。

有一些初始局面对其中一位玩家有利，而另一些初始配置则是公平的，即两位玩家都有恰好$50\%$的机会获胜。例如，当$n=2$时，有四种初始局面是公平的：RRBB、BBRR、RBBR、BRRB；在剩下的两种局面RBRB和BRBR下，后手玩家必胜。

定义$F(n)$为公平的初始局面数量，因此$F(2)=4$；此外已知$F(8)=11892$。

求$F(26)$。