On Dedekind Numbers and Two Sequences of Knuth

We consider the sequence whose nth 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.