Floor and Roof Function Analogs of the Bell Numbers

Define {f(n)}n=1, the floor sequence, by the linear recurrence f(n + 1) = n ∑ k=1 ⌊n k ⌋ f(k), f(1) = 1. Similarly, define {g(n)}n=1, the roof sequence, by the linear recurrence g(n + 1) = n ∑ k=1 ⌈n k ⌉ g(k), g(1) = 1. This paper studies various properties of these two sequences, including prime criteria, asymptotic approximations of { f(n+1) f(n) }∞ n=1 and { g(n+1) g(n) }∞ n=1 , and the iteration coefficients associated with f(n + r) and g(n + r), for any r ≥ 1.