This is the number of ways a game of chess ends in checkmate after n-ply moves. (NB a ply is a move by one side only, and the sequence starts at 0 ply)
As you can see, the numbers grow quite rapidly, as they do in a number of other chess related integer sequences. While the size of the numbers is impressive, the fact that someone made the effort to work these things out is just as impressive (in a nerdy way).
A more obvious sequence begins with 1,20,400, 8902 which is the number of different games that can be played in n plys, while 1,20,400, 5362 is a companion sequence, as it is the number of positions that can be reached after n plys (the difference in the 4th term is due to transpositions)
The source for this (and other sequences) is the Online Encyclopedia of Integer Sequences. If you do a search for chess you will find around 450 chess related sequences. Included in those are at least 2 with connections to former World Champions, Lasker and Euwe, who of course were both Mathematicians as well.