Let $M$ be a matroid of rank $r$ on a finite ground set $E$ with $|E|=n$, and let $I_k(M)$ be the number of independent sets of size $k$. Extend the sequence by setting $I_k(M)=0$ for $k>r$. Then $(I_0(M), I_1(M), \dots, I_n(M))$ is ultra log-concave of order $n$. In explicit form,
\begin{align*}
I_k(M)^2 \ge \left(1+\frac{1}{k}\right)\left(1+\frac{1}{n-k}\right) I_{k-1}(M)I_{k+1}(M)
\end{align*}
for $1 \le k \le r-1$.