Counting characters of small degree in upper unitriangular groups

Let U_n denote the group of upper n×n unitriangular matrices over a fixed finite field of order q. That is, U_n consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U_n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U_n of degree q^e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases.