Problem 409: Nim Extreme

FCC link

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.