Let S=K[x_1,...,x_n] be the polynomial ring over the field $K$, and let $I\subset S$ be a graded ideal. It is shown that for k>0 the postulation number of I^k is bounded by a linear function of k, and it is a linear function of k, if I is generated in a single degree. By using the relationship of the $h$-vector with the higher iterated Hilbert coefficients of I^k it is shown that the Hilbert coefficients e_i(I^k) of I^k are polynomials for k>0, whenever I is generated in a single degree.
