In the present article, we give the explicit formulation of the linear recurrence sequence satisfied by the sum of the elements lying over any finite ray of the generalized Delannoy matrix, and find the generating function of the cited sums. We also establish the unimodality of sequences located along the infinite transversal of direction (1,−1) of the generalized Delannoy matrix. We also discuss the corresponding Morgan-Voyce phenomenon. Finally, we show the log-convexity of the central sequence in the inverse of the tribonacci matrix from a computational point of view.