
COMMENTS

An example of a 3place formula in predicate logic is Ex Ay Ez P(x,y,z). The number of different formulas when x, y, z have to be different is A000629(3) = 26. When variables are allowed to coincide that means that there are 20 more formulas like, e.g., Ex Ay P(x,x,y) or Ex P(x,x,x).
a(n) is the number of vertices in a cocoon concertina ncube and the sum of row n in A300695, which shows the number of vertices in that structure by rank. A000629(n) by comparison is the number of vertices in the convex concertina ncube.
The differences with A000629, i.e., the numbers of formulas with coinciding variables, are 0, 0, 2, 20, 200, 2242, 28508, 408872, 6556894, 116547952, 2277942800, ...
