KöMaL 2013 A582

Let $p$ be a fixed positive prime. For any polynomial $f$($x$) with integer coefficients, let $f^*$($x$)=$f$ ( (1+$x$)$^p$-1 ) . Find all polynomials $g$ with integer coefficients for which the following holds: for every positive integer $n$there is a polynomial $f$ with integer coefficients such that the coefficients of $g$-($f$-$f^*$) with degree less than $n$ are all divisible by $p^n$.