We continue to investigate the properties of the earlier defined functions f_m and g_m, which depend on an initial arithmetic function f₀. In this paper, values of f₀ are the Fine numbers. We investigate functions f_i, g_i, (i = 1, 2, 3, 4), and show that these functions count Dyck paths having hills in different colors. For each function, we also derive an explicit formula. We also prove several results which mutually connect these functions. It appears that g₂ and g₃ are well-known objects called the Catalan triangles.

We finish with two identities relating different kind of combinatorial objects.