  The growth of Hilbert coefficients for powers of ideals are studied. For a
graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely
generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are
polynomial functions. Given two families of graded ideals $(I_k)_{k\geq 0}$ and
$(J_k)_{k\geq 0}$ with $J_k\subset I_k$ for all $k$ with the property that
$J_kJ_\ell\subset J_{k+\ell}$ and $I_kI_\ell\subset I_{k+\ell}$ for all $k$ and
$\ell$, and such that the algebras $A=\Dirsum_{k\geq 0}J_k$ and
$B=\Dirsum_{k\geq 0}I_k$ are finitely generated, we show the function $k
\mapsto_0(I_k/J_k)$ is of quasi-polynomial type, say given by the polynomials
$P_0,..., P_{g-1}$. If $J_k = J^k$ for all $k$ then we show that all the $P_i$
have the same degree and the same leading coefficient. As one of the
applications it is shown that $\lim_{k\to \infty}\length(\Gamma_\mm(S/I^k))/k^n
\in \mathbb{Q}.$ We also study analogous statements in the local case.
