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Nim Extreme

Problem 409

Published on Saturday, 5th January 2013, 04:00 pm; Solved by 276

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.