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Benedict Irwin edited section_Describing_the_Generating_Functions__.tex
about 8 years ago
Commit id: 6f11b65307f89998d959fab4147a4d53fd4608b9
deletions | additions
diff --git a/section_Describing_the_Generating_Functions__.tex b/section_Describing_the_Generating_Functions__.tex
index 7c51826..47af91b 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(2\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(p_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(3\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(p_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(4\#)} \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(p_4\#)} \chi_{11}(k)z^k}{\prod_{k=0}^3(1+z^{2^k})}\frac{1}{(1+z^{16}+z^{32})}
\end{equation}
we can then speculate that \begin{equation}
G_{13}(z) = \frac{z}{(z-1)^2}\frac{\sum_{k=0}^{\varphi(p_5\#)} \chi_{13}(k)z^k}{1-z-z^{\varphi(p_5\#)}+z^{\varphi(p_5\#)+1}}
\end{equation}
However finding the form of $\chi_{13}(n)$ is the next problem to solve.