  For an ideal $I$ in a regular local ring or a graded ideal $I$ in the
polynomial ring we study the limiting behavior of the Betti numbers of S/I^k as
k goes to infinity. By Kodiyalam's result it is known that in each homological
degree the Betti number is a polynomial for large k. We call these polynomials
the Kodiyalam polynomials and encode the limiting behavior in their generating
polynomial. It is shown that the limiting behavior depends only on the
coefficients on the Kodiyalam polynomials in the highest possible degree. For
these we exhibit lower bounds in special cases and conjecture that the bounds
are valid in general. We also show that the Kodiyalam polynomials have weakly
descending degrees and identify a situation where the polynomials have all
highest possible degree.
