10. Observe the following equations:
$$
\begin{array}{l}
\mathrm{C}_{5}^{1}+\mathrm{C}_{5}^{5}=2^{3}-2, \\
\mathrm{C}_{9}^{1}+\mathrm{C}_{9}^{5}+\mathrm{C}_{9}^{9}=2^{7}+2^{3}, \\
\mathrm{C}_{13}^{1}+\mathrm{C}_{13}^{5}+\mathrm{C}_{13}^{9}+\mathrm{C}_{13}^{13}=2^{11}-2^{5}, \\
\mathrm{C}_{17}^{1}+\mathrm{C}_{17}^{5}+\mathrm{C}_{17}^{9}+\mathrm{C}_{17}^{13}+\mathrm{C}_{17}^{17}=2^{15}+2^{7}, \\
\ldots \ldots
\end{array}
$$

From the above equations, deduce the general conclusion: For
$$
n \in \mathbf{Z}_{+}, \sum_{i=0}^{n} \mathrm{C}_{4 n+1}^{4 i+1}=
$$
$\qquad$
10. $2^{4 n-1}+(-1)^{n} 2^{2 n-1}$