Let
![$ S=K[x_1,\ldots,x_n]$](img19.png)
be the polynomial ring over the field
![$ K$](img1.png)
, and let
![$ I\subset S$](img20.png)
be a graded ideal. It is shown that for
![$ k \gg0$](img21.png)
the postulation number of
![$ I^k$](img22.png)
is bounded by a linear function of
![$ k$](img23.png)
, and it is a linear function of
![$ k$](img23.png)
, if
![$ I$](img24.png)
is generated in a single degree. By using the relationship of the
![$ h$](img18.png)
-vector with the higher iterated Hilbert coefficients of
![$ I^k$](img22.png)
it is shown that the Hilbert coefficients
![$ e_i(I^k)$](img25.png)
of
![$ I^k$](img22.png)
are polynomials for
![$ k \gg0$](img21.png)
, whenever
![$ I$](img24.png)
is generated in a single degree.