The sequence A080416, proposed by Barry, is a Stirling-like number triangle defined by a paired decomposition of C(n+3,3). The recurrence formula for this sequence can be derived from an article by Corcino et al. However, explicit formulas have not been discovered. In this paper, we derive two formulas for calculating the members of this sequence. One of them involves matrix multiplication, while the other one does not require it. We also generalize the approach for sequences defined by counting North-East lattice paths with weights.