Nim Extreme

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 $W (10 000 000) mod 1 000 000 007$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.