Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead to consider the zero distribution of the sequence of polynomials $$P(z,x)=\sum_{j=0}^{m}\left(\sum_{k=j}^{m}{m+1\choose k-j}(-1)^{m-k}D_{m+1-k}(x)\right)z^j$$ of degree $m=m(x)=\max \{\Omega(n):n\leq x\}\sim \log_2 x$.

I am interested in the question of whether the zeros of the sequence of polynomials $P(z,x)$ have strictly negative real parts-the truth of which depends, among other things, on the non-negativity of the coefficients $$c(j,x)=\sum_{k=j}^{m}{m+1\choose k-j}(-1)^{m-k}D_{m+1-k}(x).$$

I have checked this up to $x=50$ and for each such $x$ the $c(j,x)$ are positive and unimodal. For instance, for $1\leq x \leq 20$, the coefficients are: [1], [1, 2], [1, 3], [1, 4, 4], [1, 5, 5], [1, 4, 6], [1, 5, 7], [1, 6, 12, 8], [1, 6, 13, 9], [1, 5, 13, 10], [1, 6, 15, 11], [1, 6, 13, 12], [1, 7, 15, 13], [1, 6, 15, 14], [1, 5, 15, 15], [1, 6, 20, 30, 16], [1, 7, 23, 33, 17], [1, 7, 21, 32, 18], [1, 8, 24, 35, 19], [1, 8, 22, 34, 20].

Is there a good reason why these numbers should be positive? I know this would follow if it could be shown that the sequence $${m(x)+1\choose k}D_{m(x)-j+1-k}(x)$$ is unimodal for $k\in\{0,m(x)-j\}$, but I don't see how this could be proved.