Putnam 1984 B5

For each nonnegative integer $k$ , let $d(k)$ denote the number of $1$ 's in the binary expansion of $k$ . Let $m$ be a positive integer. Express $$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$ in the form $(-1)^ma^{f(m)}g(m)!$ , where $a$ is an integer and $f$ and $g$ are polynomials.