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\documentstyle[11pt]{article}
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\textwidth=440pt
\textheight=584pt
\voffset=-36pt
\hoffset=-30pt





\newtheorem {lemma}{Lemma}

\newtheorem {theorem}[lemma]{Theorem}

\def\bbZ{Z\hspace{-2mm}Z}

\def\Erd{Erd\H{o}s}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_n(x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1482: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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{\newpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W_n(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1492: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}w_n={2n+1\choose n}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}w_n=4w_{n-1}-c_n
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}4w_{n-1}-c_n & = & 4{2n-1\choose n-1}-{1\over n+1}{2n\choose n}
\nonumber \\ 
& = & {4(2n-1)!\over n!(n-1)!}-{(2n)!\over n!(n+1)!} \nonumber \\ 
& = & {(2n-1)!\over n!(n+1)!}\{4n(n+1)-2n\} \nonumber \\ 
& = & {(2n)!(2n+1)\over n!(n+1)!}= {2n+1\choose n} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1524: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$x\geq0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1538: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{tabular*}{5.55in}{c|c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}
c@{\hspace{1mm}}cc@{\hspace{2mm}}c}
$n$&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&$k$&Total \\  \cline{2-20}
0&1&&&&&&&&&&&&&&&&&0&1 \\ 
1&&1&&&&&&&&&&&&&&&&0&1 \\ 
2&1&&1&&&&&&&&&&&&&&&1&2 \\ 
3&&2&&1&&&&&&&&&&&&&&1&3 \\ 
4&2&&3&&1&&&&&&&&&&&&&2&6 \\ 
5&&5&&4&&1&&&&&&&&&&&&2&10 \\ 
6&5&&9&&5&&1&&&&&&&&&&&3&20 \\ 
7&&14&&14&&6&&1&&&&&&&&&&3&35 \\ 
8&14&&28&&20&&7&&1&&&&&&&&&4&70 \\ 
9&&42&&48&&27&&8&&1&&&&&&&&4&126 \\ 
10&42&&90&&75&&35&&9&&1&&&&&&&5&252 \\ 
11&&132&&165&&110&&44&&10&&1&&&&&&5&462 \\ 
12&132&&297&&275&&154&&54&&11&&1&&&&&6&924 \\ 
13&&429&&572&&429&&208&&65&&12&&1&&&&6&1716 \\ 
14&429&&1001&&1001&&637&&273&&77&&13&&1&&&7&3432 \\ 
15&&1430&&2002&&1638&&910&&350&&90&&14&&1&&7&6435 \\ 
16&1430&&3432&&3640&&2548&&1260&&440&&104&&15&&1&8&12870
\end{tabular*}


\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1558: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{(2\cos\theta)}^{2n}\sin\theta = \sum_{k=0}^{n} C_k\sin(2k+1)\theta
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{k-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1580: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal S}_n^{(4)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1584: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}w(n,n)=1, w(n,n-2)=n-1, w(n,n-4)={1\over 2}n(n-3),
w(n,n-6)={1\over 6}n(n-1)(n-5)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}w(n,x)={n\choose r}-{n\choose r-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r={1\over 2}(n-x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1598: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${n\choose r}=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1600: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{n\choose r}={n-1\choose r}+{n-1\choose r-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w(n,x) & = & w(n-1,x-1)+w(n-1,x+1) \nonumber \\ 
& = & {n-1\choose r}-{n-1\choose r-1}+{n-1\choose r-1}-{n-1\choose r-2}
\nonumber \\ 
& = & {n\choose r}-{n\choose r-1} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w(2n,0)={2n\choose n}-{2n\choose n-1}={1\over n+1}{2n\choose n}=c_n\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w(n,n) & + & w(n,n-2) + w(n,n-4) + \ldots \nonumber \\ 
 & = & \left[{n\choose 0}-{n\choose -1}\right] +
\left[{n\choose 1}-{n\choose 0}\right] + \ldots +
\left[{n\choose k}-{n\choose k-1}\right] = {n\choose k} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{2k\choose k} \mbox{\hspace{5mm} or \hspace{5mm}} {2k+1\choose k}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}2{2k+1\choose k}= {2(k+1)(2k+1)!\over(k+1)k!(k+1)!}=
{2k+2\choose k+1}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w(2k+1) & = & 2w(2k)-c_k \\ 
 & = & 2{2k\choose k} - {1\over k+1}{2k\choose k} \nonumber \\ 
 & = & {2k+1\over k+1}{2k\choose k} = {2k+1\choose k} \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y\geq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1628: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{minipage}[b]{30pt}
\centerline{1}
\centerline{$n=0$}
\end{minipage}{\hfill}
\begin{minipage}[b]{1cm}
\begin{tabular*}{30pt}{c@{\hfill}c@{\hfill}c}
 & 1 & \\  1 & & 1 \end{tabular*}
\centerline{$n=1$} \end{minipage}{\hfill}
\begin{minipage}[b]{35pt}
\begin{tabular*}{35pt}{c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
 & & 1 & & \\  & 2 & & 2 & \\  1 & & 3 & & 1 \end{tabular*}
\centerline{$n=2$} \end{minipage}{\hfill}
\begin{minipage}[b]{45pt}
\begin{tabular*}{45pt}{c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
 & & & 1 & & & \\  & & 3 & & 3 & & \\  & 3 & & 8 & & 3 & \\ 
1 & & 6 & & 6 & & 1 \end{tabular*}
\centerline{$n=3$} \end{minipage}{\hfill}
\begin{minipage}[b]{90pt}
\begin{tabular*}{90pt}{c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c}
 & & & & 1 & & & & \\  & & & 4 & & 4 & & & \\  & & 6 & & 15 & & 6 & & \\ 
& 4 & & 20 & & 20 & & 4 & \\  1 & & 10 & & 20 & & 10 & & 1 \end{tabular*}
\centerline{$n=4$} \end{minipage}{\hfill}
\begin{minipage}[b]{110pt}
\begin{tabular*}{110pt}{c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
 & & & & & 1 & & & & & \\  & & & & 5 & & 5 & & & & \\ 
 & & & 10 & & 24 & & 10 & & & \\  & & 10 & & 45 & & 45 & & 10 & & \\ 
 & 5 & & 40 & & 75 & & 40 & & 5 & \\ 
1 & & 15 & & 50 & & 50 & & 15 & & 1 \end{tabular*}
\centerline{$n=5$} \end{minipage}

\begin{minipage}[b]{170pt}
\vspace{25pt}
\begin{tabular*}{170pt}{c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
 & & & & & & 1 & & & & & & \\  & & & & & 6 & & 6 & & & & & \\ 
 & & & & 15 & & 35 & & 15 & & & & \\  & & & 20 & & 84 & & 84 & & 20 & & &
\\  & & 15 & & 105 & & 189 & & 105 & & 15 & & \\ 
 & 6 & & 70 & & 210 & & 210 & & 70 & & 6 & \\ 
 1 & & 21 & & 105 & & 175 & & 105 & & 21 & & 1 \end{tabular*}
\centerline{$n=6$} \end{minipage}{\hfill}
\begin{minipage}[b]{220pt}
\begin{tabular*}{220pt}{c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
 & & & & & & & 1 & & & & & & & \\  & & & & & & 7 & & 7 & & & & & & \\ 
 & & & & & 21 & & 48 & & 21 & & & & & \\ 
 & & & & 35 & & 140 & & 140 & & 35 & & & & \\ 
 & & & 35 & & 224 & & 392 & & 224 & & 35 & & & \\ 
 & & 21 & & 210 & & 588 & & 588 & & 210 & & 21 & & \\ 
 & 7 & & 112 & & 490 & & 784 & & 490 & & 112 & & 7 & \\ 
1 & & 28 & & 196 & & 490 & & 490 & & 196 & & 28 & & 1 \end{tabular*}
\centerline{$n=7$} \end{minipage}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}w_n(0)=c_{n+1}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbt]
\begin{tabular*}{14cm}{c|c|c@{\hspace{1mm}}c@{\hspace{1mm}}c
@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c
@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{0.5mm}}c@{\hspace{0.5mm}}c}
$n$ & $w_n$ & $w_n(0)$ & $w_n(1)$ & $w_n(2)$ & $w_n(3)$ & $w_n(4)$ &
$w_n(5)$ & $w_n(6)$ & $w_n(7)$ & $w_n(8)$ & $w_n(9)$ & $w_n(10)$ \\ 
0 & 1 & 1 & & & & & & & & & & \\ 
1 & 3 & 2 & 1 & & & & & & & & & \\ 
2 & 10 & 5 & 4 & 1 & & & & & & & & \\ 
3 & 35 & 14 & 14 & 6 & 1 & & & & & & & \\ 
4 & 126 & 42 & 48 & 27 & 8 & 1 & & & & & & \\ 
5 & 462 & 132 & 165 & 110 & 44 & 10 & 1 & & & & & \\ 
6 & 1716 & 429 & 572 & 429 & 208 & 65 & 12 & 1 & & & & \\ 
7 & 6435 & 1430 & 2002 & 1638 & 910 & 350 & 90 & 14 & 1 & & & \\ 
8 & 24310 & 4862 & 7072 & 6188 & 3808 & 1700 & 544 & 119 & 16 & 1 & & \\ 
9 & 92378 & 16796 & 25194 & 23256 & 15504 & 7752 & 2907 & 798 & 152 & 18
  & 1 & \\ 
10 & 352712 & 58786 & 90440 & 87210 & 62016 & 33915 & 14364 & 4655 &
  1120 & 189 & 20 & 1
\end{tabular*}

\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1684: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}w_n={2n+1\choose n}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}w_n(x,y)={n\choose r}{n\choose s}-{n\choose r-1}{n\choose s-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r={1\over 2}(n+x-y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1688: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s={1\over 2}(n-x-y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1690: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_n(x,y)=w_n(-x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1692: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}w_n(x,y)=0.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}w_n(x,y)=w_{n-1}(x,y-1)+w_{n-1}(x,y+1)+w_{n-1}(x-1,y)+w_{n-1}(x+1,y)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w_n(x,y) & \hspace{-2.5mm} = \hspace{-2.5mm} &
{n-1\choose r}{n-1\choose s} - {n-1\choose r-1}{n-1\choose s-1} +
{n-1\choose r-1}{n-1\choose s-1} - {n-1\choose r-2}{n-1\choose s-2}
\nonumber \\ 
 & \hspace{-2.5mm} + \hspace{-2.5mm} &
{n-1\choose r-1}{n-1\choose s} - {n-1\choose r-2}{n-1\choose s-1} +
{n-1\choose r}{n-1\choose s-1} - {n-1\choose r-1}{n-1\choose s-2}
\nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w_n(y) & = & \sum_{x=-n+y}^{x=n-y}w_n(x,y)=\sum_{r=0}^{2(n-y)}w_n(x,y)
\nonumber \\ 
 & = & \left[{n\choose 0}{n\choose n-y}+{n\choose 1}{n\choose n-y-1}+
\ldots+{n\choose n-y}{n\choose 0}\right] - \nonumber \\ 
 &   & \left[{n\choose -1}{n\choose n-y-1}+{n\choose 0}{n\choose n-y-2}+
\ldots+{n\choose n-y-2}{n\choose 0}\right] \nonumber 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t^{n-y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1732: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t^{n-y-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1734: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(1+t)^n(1+t)^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1736: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}w_n(y) = {2n\choose n-y}-{2n\choose n-y-2} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}w_n(y) = {2n+1\choose n-y}-{2n+1\choose n-y-1}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w_n(y) = w(2n+1,2y+1),\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w_n(0) & = & {2n+1\choose n}-{2n+1\choose n-1}=
{(2n+1)!\over n!(n+2)!}(n+2-n) \nonumber \\ 
 & = & {2(n+1)(2n+1)!\over(n+1)!(n+2)!}={1\over n+2}{2n+2\choose n+1}=
c_{n+1}, \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{minipage}[b]{5mm}
\begin{tabular*}{5mm}{|c}
 1 \\  \hline \end{tabular*}
\end{minipage} \hspace{3mm}
\begin{minipage}[b]{1cm}
\begin{tabular*}{1cm}{|c@{\hspace{3mm}}c}
 1 & \\  & 1 \\  \hline \end{tabular*}
\end{minipage} \hspace{3mm}
\begin{minipage}[b]{15mm}
\begin{tabular*}{15mm}{|c@{\hspace{3mm}}c@{\hspace{3mm}}c}
1 & & \\  & 2 & \\  2 & & 1 \\  \hline \end{tabular*}
\end{minipage} \hspace{3mm}
\begin{minipage}[b]{2cm}
\begin{tabular*}{2cm}{|c@{\hspace{3mm}}c@{\hspace{3mm}}c@{\hspace{3mm}}c}
1 & & & \\  & 3 & & \\  5 & & 3 & \\  & 5 & & 1 \\  \hline \end{tabular*}
\end{minipage} \hspace{3mm}
\begin{minipage}[b]{25mm}
\begin{tabular*}{25mm}{|c@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{2mm}}c
@{\hspace{2mm}}c}
1 & & & & \\  & 4 & & & \\  9 & & 6 & & \\  & 16 & & 4 & \\ 
10 & & 9 & & 1 \\  \hline \end{tabular*}
\end{minipage} \hspace{3mm}
\begin{minipage}[b]{3cm}
\begin{tabular*}{3cm}{|c@{\hspace{1.4mm}}c@{\hspace{1.4mm}}c
@{\hspace{1.4mm}}c@{\hspace{1.4mm}}c@{\hspace{1.4mm}}c}
1 & & & & & \\  & 5 & & & & \\  14 & & 10 & & & \\  & 35 & & 10 & & \\ 
35 & & 35 & & 5 & \\  & 35 & & 14 & & 1 \\  \hline \end{tabular*}
\end{minipage}

\begin{minipage}[b]{35mm}
\vspace{1cm}
\begin{tabular*}{35mm}{|c@{\hspace{1mm}}c@{\hspace{1mm}}c
@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1.8mm}}c@{\hspace{1.8mm}}c}
1 & & & & & & \\  & 6 & & & & & \\  20 & & 15 & & & & \\ 
& 64 & & 20 & & & \\  84 & & 90 & & 15 & & \\  & 140 & & 64 & & 6 & \\ 
70 & & 84 & & 20 & & 1 \\  \hline
\end{tabular*} \end{minipage}{\hspace{4mm}}
\begin{minipage}[b]{4cm}
\vspace{5mm}
\begin{tabular*}{4cm}{|c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hspace{0.7mm}}c@{\hspace{0.7mm}}c}
1 & & & & & & & \\  & 7 & & & & & & \\  27 & & 21 & & & & & \\ 
& 105 & & 35 & & & & \\  168 & & 189 & & 35 & & & \\ 
& 378 & & 189 & & 21 & & \\  294 & & 378 & & 105 & & 7 & \\ 
& 294 & & 168 & & 27 & & 1 \\  \hline \end{tabular*}
\end{minipage}{\hspace{4mm}}
\begin{minipage}[b]{45mm}
\begin{tabular*}{45mm}{|c@{\hspace{-1mm}}c@{\hspace{-1mm}}c
@{\hspace{-1mm}}c@{\hspace{-1mm}}c@{\hspace{-1mm}}c@{\hfill}c
@{\hspace{0.5mm}}c@{\hspace{0.5mm}}c}
1 & & & & & & & & \\  & 8 & & & & & & & \\  35 & & 28 & & & & & & \\ 
& 160 & & 56 & & & & & \\  300 & & 350 & & 70 & & & & \\ 
& 840 & & 448 & & 56 & & & \\  840 & & 1134 & & 350 & & 28 & & \\ 
& 1344 & & 840 & & 160 & & 8 & \\  588 & & 840 & & 300 & & 35 & & 1 \\ 
\hline \end{tabular*} \end{minipage}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}w'_n(x,y)={n\choose r}{n+2\choose s}-{n+2\choose r+1}{n\choose s-1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w'_n(y,x)=w'_n(x,y)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}{n\choose r}={n\choose n-r} \mbox{\quad and \quad}{n+2\choose r+1}=
{n+2\choose n-r+1}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w'_{2k-1}(0,1)={1\over 2}c_kc_{k+1}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{tabular*}{14cm}{c|c@{\hspace{2mm}}c@{\hspace{2mm}}c
@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{1.5mm}}c@{\hspace{1.3mm}}c
@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{0.5mm}}c}
$n$ & $w'_n(0)$ & $w'_n(1)$ & $w'_n(2)$ & $w'_n(3)$ & $w'_n(4)$ &
$w'_n(5)$ & $w'_n(6)$ & $w'_n(7)$ & $w'_n(8)$ & $w'_n(9)$ \\  \hline
0 & 1 & & & & & & & & & \\ 
1 & 1 & 1 & & & & & & & & \\ 
2 & 3 & 2 & 1 & & & & & & & \\ 
3 & 6 & 8 & 3 & 1 & & & & & & \\ 
4 & 20 & 20 & 15 & 4 & 1 & & & & & \\ 
5 & 50 & 75 & 45 & 24 & 5 & 1 & & & & \\ 
6 & 175 & 210 & 189 & 84 & 35 & 6 & 1 & & & \\ 
7 & 490 & 784 & 588 & 392 & 140 & 48 & 7 & 1 & & \\ 
8 & 1764 & 2352 & 2352 & 1344 & 720 & 216 & 63 & 8 & 1 & \\ 
9 & 5292 & 8820 & 7560 & 5760 & 2700 & 1215 & 315 & 80 & 9 & 1 
\end{tabular*}

\end{figure}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w'_n(y) & = & \left[{n\choose n-y}{n+2\choose 0} -
{n+2\choose n-y+1}{n\choose -1}\right] \nonumber \\ 
 & & + \left[{n\choose n-y-1}{n+2\choose 1} -
{n+2\choose n-y}{n\choose 0}\right] \nonumber \\ 
+ \ldots & + &
\left[{n\choose {1\over 2}(n-y)}{n+2\choose {1\over 2}(n-y)}-
{n+2\choose {1\over 2}(n-y)+1}
{n\choose {1\over 2}(n-y)-1}\right] \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[{n\choose {1\over 2}(n-y+1)}{n+2\choose {1\over 2}(n-y-1)}-
{n+2\choose {1\over 2}(n-y+3)}{n\choose {1\over 2}(n-y-3)}\right]\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w'_n(y)=\left\{ \begin{array}{ll}
{n+1\choose k}{n\choose k}-{n+1\choose k}{n\choose k-1} &
\mbox{if $n-y=2k$} \\ 
{n+1\choose k}{n\choose k+1}-{n+1\choose k+1}{n\choose k-1} &
\mbox{if $n-y=2k+1$} \end{array} \right. \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w'_n(0) & = & {1\over k+1}{2k\choose k}{2k+1\choose k} \mbox{\quad or
\quad}{1\over k+2}{2k+2\choose k+1}{2k+1\choose k} \nonumber \\ 
\mbox{i.e.} & & \quad\qquad {2k+1\choose k}c_k \mbox{\qquad or \qquad}
{2k+1\choose k}c_{k+1} \nonumber \end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1828: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{tabular*}{14cm}{c|c|c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c
@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c@{\hfill}c}
$w'_n$&$n$&0&1&2&3&4&5&6&7&8&9&10&11 \\  \hline
1&0&1&&&&&&&&&&& \\ 
2&1&&2&&&&&&&&&& \\ 
6&2&2&&4&&&&&&&&& \\ 
18&3&&10&&8&&&&&&&& \\ 
60&4&10&&34&&16&&&&&&& \\ 
200&5&&70&&98&&32&&&&&& \\ 
700&6&70&&308&&258&&64&&&&& \\ 
2450&7&&588&&1092&&642&&128&&&& \\ 
8820&8&588&&3024&&3414&&1538&&256&&& \\ 
31752&9&&5544&&12276&&9834&&3586&&512&& \\ 
116424&10&5544&&31680&&43230&&26752&&8194&&1024& \\ 
426888&11&&56628&&141570&&138424&&69784&&18434&&2048 
\end{tabular*}

\end{figure}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w''_n(x+y) & = & w''_n(n-2s) = \sum_{x+y=n-2s}w'_n(x,y) \nonumber \\ 
& = & {n+2\choose s}\left[{n\choose s}+{n\choose s+1}+\ldots +
{n\choose n-s}\right] - \nonumber \\ 
& & {n\choose s-1}\left[{n+2\choose s+1}+{n+2\choose s+2}+\ldots +
{n+2\choose n-s+1}\right] \nonumber \end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}w''_n(n)\quad &\hspace{-2.8mm}=\hspace{-2.8mm}& 2^n \nonumber \\ 
w''_n(n-2) &\hspace{-2.8mm}=\hspace{-2.8mm}& (n-2)2^n+2 \nonumber \\ 
w''_n(n-4) &\hspace{-2.8mm}=\hspace{-2.8mm}&
{1\over 2!}(n^2-5n+2)2^n+n^2+3n-2 \nonumber \\ 
w''_n(n-6) &\hspace{-2.8mm}=\hspace{-2.8mm}&
{1\over 3!}[n(n^2-9n+14)2^n+n(n^3+4n^2-n-28)]
\nonumber \\ 
w''_n(n-8) &\hspace{-2.8mm}=\hspace{-2.8mm}&
{1\over4!}n(n-1)(n^2-13n+34)2^n+{1\over 72}n(n-1)(n^4+4n^3-n^2-64n-204)
\nonumber \end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w''_n(n-2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1842: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w'_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1846: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}w'_n={n\choose\lfloor{n/2}\rfloor}{n+1\choose\lfloor{(n+1)/2}\rfloor}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{n\choose r}{n\choose s}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1866: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W_n(x,y,z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1868: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}W_n(z)=W_{n-1}(z-1)+4W_{n-1}(z)+W_{n-1}(z+1)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}W_5(2)= 4 \cdot 99 + 288 + 16 = 700.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{tabular*}{14cm}{c|c|c@{\hspace{1mm}}c@{\hspace{1mm}}c
@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c
@{\hspace{1.5mm}}c@{\hspace{2mm}}c@{\hspace{2.5mm}}c}
$W_n$&$n$&0&1&2&3&4&5&6&7&8&9 \\  \hline
1&0&1&&&&&&&&& \\ 
5&1&4&1&&&&&&&& \\ 
26&2&17&8&1&&&&&&& \\ 
139&3&76&50&12&1&&&&&& \\ 
758&4&354&288&99&16&1&&&&& \\ 
4194&5&1704&1605&700&164&20&1&&&& \\ 
23460&6&8421&8824&4569&1376&245&24&1&&& \\ 
132339&7&42508&48286&28476&10318&2380&342&28&1&& \\ 
751526&8&218318&264128&172508&72128&20180&3776&455&32&1& \\ 
4290838&9&1137400&1447338&1026288&481200&156624&35739&5628&584&36&1 \\ 
\end{tabular*}

\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}W_n(n-v)=a_{n,0}{n\choose v}+a_{n,1}{n\choose v-1}+\ldots+
a_{n,t}{n\choose v-t}\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t=\lfloor(v+2)/2\rfloor$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1894: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_{n,u}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1896: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}a_{n,u}=\left[{v-u\choose u}-4^2{v-u\choose u-2}\right]4^{v-2u}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}W_n(n) \quad & = & 1 \nonumber \\ 
W_n(n-1) & = & 4n \nonumber \\ 
W_n(n-2) & = & (n-1)(8n+1) \nonumber \\ 
W_n(n-3) & = & {4\over 3}n(n-2)(8n-5) \nonumber \\ 
W_n(n-4) & = & {1\over 6}n(n-3)(64n^2-144n+83) \nonumber \\ 
W_n(n-5) & = & {2\over 15}n(n-1)(n-4)(64n^2-240n+239) \nonumber \\ 
W_n(n-6) & = & {1\over 90}n(n-1)(n-5)(512n^3-3648n^2+8872n-7233)
\nonumber \\ 
W_n(n-7) & = & {2\over 315}n(n-1)(n-2)(n-6)(512n^3-4800n^2+15496n-17007)
\nonumber \end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1902: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(2d)^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1912: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_n^{\prime}(y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1920: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1926: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1928: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${k+l\choose k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1996: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}{n\choose r}{n\choose s},\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}w_n(x,y)={n\choose r}{n\choose s}-{n\choose r-1}{n\choose s-1}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}={n\choose r}{n\choose s} - {n\choose r-1}{n\choose s-1} -
{n\choose r+1}{n\choose s-1} - {n\choose r}{n\choose s-2}\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{p}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2072: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${2n\choose n-y-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2114: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2132: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\stackrel{=}{p}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2142: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}{2n+1\choose n}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_1=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2194: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_n=\sum^{n-1}_{k=1}C_kC_{n-k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2196: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_1(q)=(1+q)/2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2202: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$((1+q^n)/2)C_n(q)=\sum^n_{k=1}C_k(q)C_{n-k}(q)q^k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2204: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal C}_n(a,q) :=
 q^{2n}(-aq^{-1};q^2)_n/(q^2,q^2)_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2206: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sum^n_{j=0}{\cal C}_j(a^{-1},q){\cal C}_{n-j}(a,q)(-aq)^j=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2208: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_n(q)=-2^{2n-1}{\cal C}_n(-1,q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2212: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal C}_n(a^{-1},q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2222: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi(k,t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2230: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}F_j=[j(m+n)]^{-1}{jm+jn\choose jm};\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sum\phi(k,t)x^k= [1- \exp(-F-1x-F_2x^2-\ldots)]^t.\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_k=\sum\phi(k,t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2238: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sum\phi(k)x^k=\exp(F_1x+F_2x^2+\ldots)-1\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_k=\phi(k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2242: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2246: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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}

{\newpage
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{\newpage
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{\newpage
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}

{\newpage
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{\newpage
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}

{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_{n,m}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2282: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}a_{n,m}=(n+1-m)(n+1)^{-1}{n+m\choose m}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_n(x)=\sum^n_{m=0}a_{n,m}x^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2286: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(1-x)^{n+1}a_n(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2290: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2294: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_n(u)=\sum^n_{m=0}a_{m,m}u^m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2298: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a(x,y)=\sum_{n=0}^{\infty}x^na_n(y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_n(x,q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2310: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}(x)_{n+1}a_n(x,q)=1-x\sum^n_{m=0}c_m(qx)^m(x)_m,\end{displaymath}
}

{\newpage
\clearpage
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{\newpage
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{\newpage
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}

{\newpage
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{\newpage
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}

{\newpage
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{\newpage
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{\newpage
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}

{\newpage
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}

{\newpage
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P(\lambda^{(k)}_{2n}=l)=P(\lambda^{(k)}_{2n+1}=l)=
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{\newpage
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{\newpage
\clearpage
\samepage \begin{displaymath}{\bf P}(\gamma_{2n}^{(2k)}=g)=
{1\over2^{2n}}{2g\choose g}{2n-2g\choose n-g+k},\quad
 g=1, 2, \ldots, n-k,\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\bf P}(\gamma_{2n}^{(2k)}=0)=
{1\over2^{2n}}\sum^k_{j=-k}{2n\choose n+j}.\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{a-kb\over d}{d\choose{d-a-b\over2}}{d\choose{d-a+b\over2}}\end{displaymath}
}

{\newpage
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{\newpage
\clearpage
\samepage \begin{displaymath}a_{n,m}={n+m\choose m}-{n+m\choose m-1}=
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}

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\samepage \setbox\sizebox=\hbox{$S_n=X_1+X_2+\ldots+X_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2498: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$C_n={1\over n+1}{2n\choose n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2524: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
\clearpage
\samepage \begin{displaymath}z=\sum_{n=1}^{\infty}q^{{n-1\choose2}}C_n(q){z^n\over
(1+z)(1+qz)\ldots(1+q^{2n-1}z)}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}z=\sum_{n=1}^{\infty}C_{n-1}(q)z^n(1-z)(1-qz)\ldots(1-q^{n-1}z).\end{displaymath}
}

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}

{\newpage
\clearpage
\samepage \begin{displaymath}c_n(\lambda;q)=\sum_{k=0}^n{1\over[n]}\left[\begin{array}{c}
n \\  k \end{array}\right]\left[\begin{array}{c}
n \\  k+1 \end{array}\right]q^{k^2+\lambda k},\end{displaymath}
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}

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{\newpage
\clearpage
\samepage \begin{displaymath}z=\sum_{n=1}^{\infty}{c_n(\lambda;q)z^n\over
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}

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\clearpage
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{\newpage
\clearpage
\samepage \begin{displaymath}z=\sum_{n=1}^{\infty}{a^{-{n\choose2}}C_n(x;a,b)z^n\over
(1+a^{-1}z)\ldots(1+a^{-n}z)(1+xbz)\ldots(1+xb^nz)}\end{displaymath}
}

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{\newpage
\clearpage
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}

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{\newpage
\clearpage
\samepage \begin{displaymath}\sum^n_{k=0}{2k\choose k}{2n-2k\choose n-k}=4^n,\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sum(W(C))^2={(p+q+1)!(2p+2q+1)!\over(p+1)!(2p+1)!(q+1)!(2q+1)!}\end{displaymath}
}

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}

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{\newpage
\clearpage
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{\newpage
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}

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\clearpage
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}

{\newpage
\clearpage
\samepage \begin{displaymath}L(n;a_1,a_2,\ldots)={\left(\sum^{\infty}_{i=0}a_i\right)!n\over
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m=n+\sum^{\infty}_{i=1}ia_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2798: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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}

{\newpage
\clearpage
\samepage \begin{displaymath}(*)\qquad A(n;t,q,p,u,v)=\sum_{\sigma\in s}t^{d(\sigma)}
q^{m(\sigma)}p^{i(\sigma)}u^{a(\sigma)}v^{b(\sigma)}.\end{displaymath}
}

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\samepage \setbox\sizebox=\hbox{$A_n(t)=A(n;t,q,p,u,v)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2812: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}(**)\quad A_{n+1}(t)=A_n(tq)+t\sum^n_{k=1}q^kp^ku^{k(n-k)}
\left[\begin{array}{c} n \\  k \end{array}\right]
A_k(t)A_{n-k}(tq^{k+1})\end{displaymath}
}

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{\newpage
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\samepage \begin{displaymath}{a+b\choose a}^{-1}\sum_k\left[{a+b\choose a-kd}-{a+b\choose a+c+kd}
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\phi\left(j-{1\over2}ab\over\sigma_{ab}\right)\right]=0\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}P\left\{{a\eta_n\over\left({4^n\over8C_{n-1}}-{n\over2}\right)}
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P_i(\delta)s_{{\bf m}}({\bf x})=
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\samepage \begin{displaymath}{x\over x+\sum^r_{k=1}(\mu_k+1)n_k}
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}

\stepcounter{section}

\end{document}
