We consider the sequence whose n^th term is the number F(n) of anti-chains in the partially ordered set whose elements are 0, 1, ..., n–1 and the order relation is coordinate-wise on the binary representation of each integer. This sequence is a sort of "background" sequence to its more prominent subsequence of Dedekind numbers, that is, the sequence whose terms are F(2^k). We also consider the sequence of first differences with terms F(n) – F(n–1). We discuss, state, and prove some (recursive) relations between the terms of these three sequences.