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Problem 409

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Nim Extreme

Let n be a positive integer. Consider nim positions where:

• There are n non-empty piles.
• Each pile has size less than 2ⁿ.
• No two piles have the same size.

Let W(n) be the number of winning nim positions satisfying the above conditions (a position is winning if the first player has a winning strategy). For example, W(1) = 1, W(2) = 6, W(3) = 168, W(5) = 19764360 and W(100) mod 1 000 000 007 = 384777056.

Find W(10 000 000) mod 1 000 000 007.

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极限取石子游戏

记 n为正整数。考虑如下的取石子游戏设计：

• 有n个非空的堆。
• 每个堆的石子数小于2ⁿ。
• 没有两堆的石子数相同。

记W(n)是所有设计中必胜态的数目（必胜态是指先手玩家有必胜策略的状态）。例如，W(1) = 1, W(2) = 6, W(3) = 168, W(5) = 19764360，以及W(100) mod 1 000 000 007 = 384777056。

求W(10 000 000) mod 1 000 000 007。

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