  Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if
$\mathcal{L}^j (a_n) \geq 0$ for all $j = 0, 1, \ldots, m$. Here, $\mathcal{L}$
is the operator defined by $\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$.
By a criterion of Craven--Csordas and McNamara--Sagan it is known that a
sequence is $\infty$-log-concave if it satisfies the stronger inequality $a_k^2
\geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent
result of Br\"and\'en shows that $\infty$-log-concave sequences include
sequences whose generating polynomial has only negative real roots. In this
paper, we investigate sequences which are fixed by a power of the operator
$\mathcal{L}$ and are therefore $\infty$-log-concave for a very different
reason. Surprisingly, we find that sequences fixed by the non-linear operators
$\mathcal{L}$ and $\mathcal{L}^2$ are, in fact, characterized by a linear
4-term recurrence. In a final conjectural part, we observe that positive
sequences appear to become $\infty$-log-concave if convoluted with themselves a
finite number of times.
