  Negative type inequalities arise in the study of embedding properties of
metric spaces, but they often reduce to intractable combinatorial problems. In
this paper we study more quantitative versions of these inequalities involving
the so-called $p$-negative type gap. In particular, we focus our attention on
the class of finite ultrametric spaces which are important in areas such as
phylogenetics and data mining.
  Let $(X,d)$ be a given finite ultrametric space with minimum non-zero
distance $\alpha$. Then the $p$-negative type gap $\Gamma_{X}(p)$ of $(X,d)$ is
positive for all $p \geq 0$. In this paper we compute the value of the limit
\begin{eqnarray*} \Gamma_{X}(\infty) & = & \lim\limits_{p \rightarrow \infty}
\frac{\Gamma_{X}(p)}{\alpha^{p}}. \end{eqnarray*} It turns out that this value
is positive and it may be given explicitly by an elegant combinatorial formula.
On the basis of our calculations we are then able to characterize when
$\Gamma_{X}(p)/ \alpha^{p}$ is constant on $[0, \infty)$.
  The determination of $\Gamma_{X}(\infty)$ also leads to new, asymptotically
sharp, families of enhanced $p$-negative type inequalities for $(X,d)$. Indeed,
suppose that $G \in (0, \Gamma_{X}(\infty))$. Then, for all sufficiently large
$p$, we have \begin{eqnarray*} \frac{G \cdot \alpha^{p}}{2} \left(
\sum\limits_{k=1}^{n} |\zeta_{k}| \right)^{2} + \sum\limits_{j,i =1}^{n}
d(z_{j},z_{i})^{p} \zeta_{j} \zeta_{i} & \leq & 0 \end{eqnarray*} for each
finite subset $\{ z_{1}, \ldots, z_{n} \} \subseteq X$ and each choice of real
numbers $\zeta_{1}, \ldots, \zeta_{n}$ with $\zeta_{1} + \cdots + \zeta_{n} =
0$. We note that these results do not extend to general finite metric spaces.
