  We give necessary and sufficient conditions of the existence of a
left-invariant metric of strictly negative Ricci curvature on a solvable Lie
group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie
algebra $\mathfrak{n}$. It turns out that such a metric always exists, except
for in the two cases, when $\mathfrak{n}$ is one of the algebras of rank two,
$L_n$ or $Q_n$, and $\mathfrak{g}$ is a one-dimensional extension of
$\mathfrak{n}$, in which cases the conditions are given in terms of certain
linear inequalities for the eigenvalues of the extension derivation.
