A game is transitive if any position reached by any number of consecutive moves by one player can be reached in a single move by that player. It is hereditarily transitive if it and all its followers are transitive.
The hereditarily transitive games born by day n form a distributive lattice whose Hasse diagram is planar. It is conjectured (known for n<=3) that the number of antichains in this lattice is 2^A000372(n)-2.
Aaron Siegel attributes the values up to a(3) to Angela Siegel, and a(4) to Neil McKay.