On certain plane curves with many integral points

We define a sequence of polynomials P-d is an element of Z [x, Y], such that P-d is absolutely irreducible, of degree d, has low height, and has at least d(2) + 2d + 3 integral solutions to P-d(x, y) = 0. We know of no other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree.