The weighted Delannoy numbers are defined by the recurrence relation $f_{m,n}=αf_{m-1,n}+ βf_{m,n-1}+ γf_{m-1,n-1}$ if $m n>0 $, with $f_{m,n}=α^m β^n$ if $n m=0$. 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 $n m=0$. 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 $A,B,α,β,γ\geq 0$. In addition, we also study its P-recursivity with the help of symbolic computation tools.