The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras and are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.