The weighted Delannoy numbers are defined by the recurrence relation f m,n = α f m−1,n + β f m,n−1 + γ f m−1,n−1 if mn > 0, with f m,n = α m β n if nm = 0. In this work, we study a generalization of these numbers considering the same recurrence relation but with f m,n = A m B n if nm = 0. More particularly, we focus on the diagonal sequence f n,n. With some ingenuity, we are able to make use of well-established methods by Pemantle and Wilson, and by Melczer in order to determine its asymptotic behavior in the case A, B, α, β, γ ≥ 0. In addition, we also study its P-recursivity with the help of symbolic computation tools.