Let $wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$$ Then we have an integer sequence given by $$a(n)=\sum\limits_{j=0}^{2^{wt(n)}-1}m^{wt(n)-wt(j)}\prod\limits_{k=0}^{wt(n)-1}(1+wt(\left\lfloor\frac{j}{2^k}\right\rfloor))^{t_{k+1}+1}, a(0)=1$$ Let $$s(n,m)=\sum\limits_{k=0}^{2^n-1}a(k)$$ then I conjecture that for any $m$ $$s(n,m)=\sum\limits_{k=0}^{n+1}k!{n+1\brace k}(m+1)^{n-k+1}$$ Is there a way to prove it?