Nim Extreme (noch nicht übersetzt)

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.