Representation of a 2-power as sum of k 2-powers: a recursive formula

For every integer k, a k-representation of $2^{k-1}$ is a string $\n=(n_1, \ldots,n_k)$ of nonnegative integers such that $\sum_{j=1}^k 2^{n_j} = 2^{k-1}$, and W(1,k) is their number. We present an efficient recursive formula for W(1,k); this formula allows also to prove the congruence $W(1,k) = 4+(-1)^k\pmod{8}$ for $k\geq 3$.