Representation of a 2-power as sum of k 2-powers: the asymptotic behavior

A $k$-representation of an integer $\l$ is a representation of $\l$ as sum of $k$ powers of $2$, where representations differing by the order are considered as distinct. Let $\W(\sigma,k)$ be the maximum number of such representations for integers $\l$ whose binary representation has exactly $\sigma$ non-zero digits. $\W(\sigma,k)$ can be recovered from $\W(1,k)$ via an explicit formula, thus in some sense $\W(1,k)$ is the fundamental object. In this paper we prove that $(\W(1,k)/k!)^{1/k}$ tends to a computable limit as $k$ diverges. This result improves previous bounds which were obtained with purely combinatorial tools.