We continue to investigate the properties of the earlier defined functions fm and gm, which depend on an initial arithmetic function f0. In this papers values of f0 are the Fine numbers. We investigate functions fi; gi; (i = 1; 2; 3; 4). For each function, we derive an explicit formula and give a combinatorial interpretation. It appears that g2 and g3 are well-known combinatoric object called the Catalan triangles. We finish with an identity consisting of ten items.