deletions | additions       diff --git a/section_Describing_the_Generating_Functions__.tex b/section_Describing_the_Generating_Functions__.tex  index 7a958fb..7c51826 100644  --- a/section_Describing_the_Generating_Functions__.tex  +++ b/section_Describing_the_Generating_Functions__.tex 
...
\end{cases}  \end{equation}  This allows us to write \begin{equation}  G_5(z) = \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(3\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(2\#)}  \chi_5(k)z^k}{\prod_{k=0}^0(1+z^{2^k})} \end{equation}  We may define a function \begin{equation} 
...
\end{cases}  \end{equation}  This allows us to write \begin{equation}  G_7(z) = \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(5\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(3\#)}  \chi_7(k)z^k}{\prod_{k=0}^2(1+z^{2^k})} \end{equation}  Similarly we may write \begin{equation} 
...
\end{cases}  \end{equation}  Which allows us to write \begin{equation}  G_{11}(z) = \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(7\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(4\#)}  \chi_{11}(k)z^k}{\prod_{k=0}^3(1+z^{2^k})}\frac{1}{(1+z^{16}+z^{32})} \end{equation}