The state complexity of L2 and Lk

We show that if M is a DFA with n states over an alphabet with at least two letters and L=L(M), then the worst-case state complexity of L2 is nn2−2n−1. If, however, M is a DFA over a unary alphabet, then the worst-case state complexity of Lk is kn−k+1 for all k⩾2.