Asymmetric extension of Pascal-Dellanoy triangles

Abstract: We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L = (1, 0), L1 = (1, 1), L2 = (2, 1), . . . , Ls = (s, 1)}; an explicit formula is given. Thereby linking the elements of the quasi s-Pascal triangle with the bisnomial coefficients. We establish the recurrence relation for the sum of elements lying over any finite ray of the quasi s-Pascal triangle. The generating function of the cited sums is produced. We also give identities among which one equivalent to the de Moivre sum and establish a q-analogue of the coefficient of the quasi s-Pascal triangle.