\section{Coins}


Consider the following problem:

\say{How many different ways can you make change for \$0.50 \par
using pennies, nickels, dimes, quarters and half-dollars?}

\vspace{2mm}

Most ways of solving this involve awkward brute-force
approaches that don't reveal anything interesting about the problem:
how can we change our answer if we want to make change for
\$0.51, or \$1.05, or some other quantity?

\vspace{2mm}

We can use generating functions to solve this problem in a general way.

\definition{}
Let $p_0, p_1, p_2, ...$ be such that $p_k$ is the number
of ways to make change for $k$ cents with only pennies.

Similarly, let...
\begin{itemize}
	\item $n_k$ be the number of ways to make change for $k$ cents with only nickels;
	\item $d_k$ be the number of ways using only dimes;
	\item $q_k$ be the number of ways using only quarters;
	\item and $h_k$ be the number of ways using only half-dollars.
\end{itemize}

\problem{}<pcoins>
Let $p(x)$ be the generating function that corresponds to $p_n$. \par
Express $p(x)$ as a rational function.

\vfill

\problem{}
Modify \ref{pcoins} to find expressions for $n(x)$, $d(x)$, $q(x)$, and $h(x)$.

\vfill
\pagebreak

\definition{}
Now, let $N(x)$ be the generating function for the sequence
$n_0, n_1, ...$, where $n_k$ is the number of ways to make
change for $k$ cents using pennies and nickels.

Similarly, let...
\begin{itemize}
	\item let $D(x)$ be the generating function for the sequence using pennies, nickels, and dimes;
	\item let $Q(x)$ use pennies, nickels, dimes, and quarters;
	\item and let $H(x)$ use all coins.
\end{itemize}

\problem{}
Express $N(x)$ as a rational function.

\vfill

\problem{}
Using the previous problem, write $D(x)$, then $Q(x)$, then $H(x)$
as rational functions.


\vfill

\problem{}
Using these generating functions, find recurrence relations for
the sequences $N_k$, $D_k$, $Q_k$, and $H_k$.

\hint{
	Your recurrence relation for $N_k$ should refer to the
	previous values of itself and some values of $p_k$.
	Your recurrence for $D_k$ should refer to itself and $N_k$;
	the one for $Q_k$ should refer to itself $D_k$;
	and the one for $H_k$ should refer to itself and $Q_k$.
}

\vfill


\problem{}
Using these recurrence relations, fill following table
and solve the original problem.

\begin{center}
	\begin{tabular}{ c|cccc|cccc|ccc }
		$n$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\
		\hline
		$p_k$ &&&&&&&&&& \\
		$N_k$ &&&&&&&&&& \\
		\hline
		$D_k$ &&&&&&&&&& \\
		\hline
		$Q_k$ &&&&&&&&&& \\
		$H_k$ &&&&&&&&&&
	\end{tabular}
\end{center}

\vspace{1cm}
\pagebreak