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\begin{document}

$\;$\hfill Due: 2011/10/12 before class

\begin{center}
{\LARGE\bf Homework 3}
\end{center}

\begin{hw}
For each of the following sub-problem, find an answer and justify. \\

(a) For any non-negative integer $n$, how many solutions $(x,y)$ are there such that $x$ and $y$ are non-negative integers and $x + 2y = n$? \\

(b) For any non-negative integer $n$, how many solutions $(x,y)$ are there such that $x$ and $y$ are non-negative integers and $2x + 3y = n$? \\
\end{hw}

\begin{hw}
For a non-negative integer $n$, let $A_n$ be the number of non-negative integer pairs $(x,y)$ where one of the following is satisfied: \\
\[x+2y = n, 2x+3y=n-1, 3x+4y=n-2, ..., nx + (n+1)y=1, (n+1)x + (n+2)y = 0.\]
Compute $A_n$.
(You may find the following approach useful: Find the generating function for $x+2y=n$,
and then find the generating function for $A_n$, and simplify it.)
\end{hw}

\begin{hw}
For each $n$, find the simple form for $S(n, n-2)$, $S(n, n-3)$, $s(n, n-2)$, and $s(n, n-3)$.
\end{hw}

\begin{hw}
Prove that the Stirling numbers of the 1st kind $s(n, k)$ is an even number whenever $2k < n$ and $n > 0$.
\end{hw}

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