I'm looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group.

For example, the simplest polynomial with non-trivial $q$-coefficient is $p_{tsut,e}(q) = 1 + q$ in type $A3.$ Where can we find the first non-trivial coefficient of $q^2$, and $q^3...$ etc.