for $c$ in the equation $\lambda_{p-1}-1\leq c$, we get a new lower bound for ${\mathsf{w}}_{2p+1}$, namely $$\frac{\mu+p-1}{2p}\leq {\mathsf{w}}_{2p+1}.$$

We therefore have:

\[prop:TrivialBundleOddCase\] Let $k=2p+1\geq 9$ and consider $\mu\geq 1$. The $(2p+1)^{\text{th}}$ generalized Gromov width of $M_{\mu}^{0}$ is $${\mathsf{w}}_{2p+1}(M_{\mu}^{0})=
  \begin{cases}
    c_{\operatorname{vol}}            & \text{~if~}\mu\in
                           \left[ 1,~ p+1-\sqrt{2p+1} \right) \\
    \frac{\mu+p}{2p+1} & \text{~if~}\mu\in
                           \left[ p+1-\sqrt{2p+1},~ p+1 \right) \\
    1                  & \text{~if~}\mu\in  \left[ p+1,~ \infty \right)
  \end{cases}$$

The previous discussion shows that ${\mathsf{w}}_{2p+1}(M_{\mu}^{0}) =
\min\left\{c_{\operatorname{vol}}, 1, \frac{\mu+p}{2p+1}\right\}$.

An immediate consequence is the following

\[cor:MainPackingNumbersTrivialOdd\] Let $k=2p+1\geq 9$ and consider $\mu\geq 1$. The $(2p+1)^{\text{th}}$ packing number of $M_{\mu}^{0}$ is $$p_{2p+1}(M_{\mu}^{0}) =
  \begin{cases}
    1                             & \text{~if~}\mu\in
                                      \left[ 1,~ p+1-\sqrt{2p+1} \right) \\
    \frac{(\mu+p)^{2}}{2\mu(2p+1)} & \text{~if~}\mu\in
                                      \left[ p+1-\sqrt{2p+1},~ p+1 \right) \\
    \frac{2p+1}{2\mu}             & \text{~if~}\mu\in
                                      \left[ p+1,~ \infty \right)
  \end{cases}$$

\[cor:OddStabilityNumbersTrivial\] The odd stability number of $M_{\mu}^{0}$ is $$N_{{\rm odd}}(M_{\mu}^{0})=
\begin{cases}
7 &\text{~if~}\mu=\frac{8}{7}\\
9 &\text{~if~}\mu\in\left[1,~\frac{8}{7}\right)\cup\left(\frac{8}{7},~2\right]\\
2\left\lceil\mu+\sqrt{2\mu}\right\rceil + 1 
                    &\text{~if~}\mu\in\left(2,~\infty\right]
\end{cases}$$

By Proposition \[prop:PackingNumbersProduct&lt;8\], the only pair $(\mu,k)$ for which we have full packing by $k=2p+1\leq 7$ balls is $(8/7,~7)$. On the other hand, the largest root of the polynomial in $p$ $$(2p+1)(\mu+p)^{2}-2\mu(2p+1)^{2}$$ obtained by setting $$c_{\operatorname{vol}}^{2}=\frac{\mu+p}{2p+1}$$ is $$r(\mu) =  \mu+\sqrt{2\mu}$$ The integer $J(\mu) := \max \{9,~ 2\lceil r(\mu) \rceil + 1\}$ gives the odd stability number in the range 