## Nim Extreme

### Problem 409

Published on Saturday, 5th January 2013, 04:00 pm; Solved by 275; Difficulty rating: 55%Let `n` be a positive integer. Consider **nim** positions where:

- There are
`n`non-empty piles. - Each pile has size less than 2
^{n}. - 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.