\batchmode \documentstyle[11pt]{article} \makeatletter \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 } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$w_n(y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1484: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$w'_n(x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1486: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$w'_n(y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1488: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$w''_n(x+y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1490: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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 } {\newpage \clearpage \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 \clearpage \samepage \setbox\sizebox=\hbox{${\cal C}_n(a,q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2220: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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 \samepage \setbox\sizebox=\hbox{$x_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2248: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$x_i=\pm1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2250: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i=x_0+ \ldots +x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2252: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2254: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_{i-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2258: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2260: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$0\leq m\leq n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2266: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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 \samepage \setbox\sizebox=\hbox{$1-xP_n(x(1-x))$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2292: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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 \samepage \setbox\sizebox=\hbox{$(x)_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2316: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(1-x)(1-qx)\ldots(1-q^{k-1}x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2318: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$a_{n,m}(q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2320: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$q^{k_1+\ldots+k_m}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2324: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$k_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2326: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2328: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$1\leq k_1\leq \ldots \leq k_m\leq n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2330: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$k_1\geq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2332: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$k_m\geq m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(1/((r-1)n+1)){rn\choose n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2338: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$t_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2352: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\sum t_i=n-1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2356: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(1/n){n\choose t_1}{n\choose t_2} \ldots {n\choose t_r}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2358: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(1/n){(r-1)n\choose k}{n\choose n-1-k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2366: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$p_{i,i+1}=p_{i,i-1}={1\over2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2368: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\lambda_j^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2370: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\lambda_j^{(0)}=\lambda_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2376: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$P(\lambda_{2n-1}=l)=P(\lambda_{2n}=l)=2^{-2n+2}{2n-1\choose n+l}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2378: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$l\neq0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$2^{-2n+1}{2n\choose n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2382: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$P(\lambda^{(k)}_{2n-1}=l)=P(\lambda^{(k)}_{2n}=l)= 2^{-2n+1}{2n\choose n+l+k/2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2388: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$P(\lambda^{(k)}_{2n}=l)=P(\lambda^{(k)}_{2n+1}=l)= 2^{-2n}{2n+1\choose n+l+{1\over2}(k+1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2392: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$S_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2394: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$n\rightarrow\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2408: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\pm1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_0=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2418: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$2\gamma_{2n}^{(2k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2420: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(i=1, \ldots, 2n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2424: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i>2k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2426: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i=2k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2428: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_{i-1}=2k+1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2430: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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 \clearpage \samepage \setbox\sizebox=\hbox{$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2464: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$n\geq m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2474: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$c(x;n,m)=\sum^m_{k=0}a_{n+k,m-k}x^k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2478: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$t(x;n,m)=\sum^m_{k=0}a_{n-1,m-k}(2x)^k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2484: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{displaymath}a_{n,m}={n+m\choose m}-{n+m\choose m-1}= {n+1-m\over n+1}{n+m\choose m}.\end{displaymath} } {\newpage \clearpage \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 } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$X_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2500: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{${1\over2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2504: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{${2k\choose k}{2n-2k\choose n-k}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2506: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$C_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2522: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \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 } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$z=\sum_{n=1}^{\infty}C_nz^n/(1+z)^{2n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2526: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$z=\sum_{n=1}^{\infty}C_nz^n(1-z)^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2528: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$C_n(q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2538: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$[n]=(q^n-1)/(q-1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2546: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\left[\begin{array}{c} n \\ k \end{array}\right]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2548: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\lambda=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2552: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$c_n(1;q)=C_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2554: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{displaymath}z=\sum_{n=1}^{\infty}{c_n(\lambda;q)z^n\over q^{{n\choose2}}(-q^{-n}z)_n(-q^{\lambda}z)^n},\end{displaymath} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(a)_n=(1-a)(1-qa)\ldots(1-q^{n-1}a)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2560: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$3^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2590: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{${1\over2}((1+t)/\sqrt{1-2t-3t^2}-1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2594: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$0\leq x\leq y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2598: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$m_{n-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2600: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{displaymath}p_k=\sum F_1^{k_1}F_2^{k_2}\ldots[k_1!k_2!\ldots]^{-1}\end{displaymath} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$k_1+2k_2+\ldots=k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2618: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$F_j=j^{-1}(m+n)^{-1}{jm+jn\choose jn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2620: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$X_n=x_1x_2\ldots x_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2680: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$Y_n=y_1y_2\ldots y_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2682: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$x_1x_2\ldots x_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2698: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$y_1y_2\ldots y_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2700: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(X_n,Y_n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2706: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(Y_n,X_n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2708: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$N_n(p,q)={n-1\over pq}{n-2\choose p-1}{n-2\choose q-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2720: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2726: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$m-\mu n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2728: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\alpha\geq0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2746: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$a_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2758: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$b_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2762: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$-45^{\circ}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2764: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$c_{mn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2768: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\sum_mc_{mn}=b_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2776: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\sum_1^{\infty}a_mx^m=x+2x^2+6x^3+19x^4+61x^5+ \ldots = x(1-x)^3/(1-5x+7x^2-4x^3)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2784: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$b_n=(1/(4n-2)){2n\choose n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2786: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$a_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2788: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$a_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2790: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$\sum a_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2792: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$L=L(n;a_1,a_2,\ldots)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2794: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \begin{displaymath}L(n;a_1,a_2,\ldots)={\left(\sum^{\infty}_{i=0}a_i\right)!n\over \prod^{\infty}_{i=0}(a_i!)m}\end{displaymath} } {\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 } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$a_0=n+\sum^{\infty}_{i=1}(i-1)a_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2800: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\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} } {\newpage \clearpage \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} } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(\vartheta_1, \vartheta_2, \ldots \vartheta_{2n})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2838: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$(i,s_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2848: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i=\vartheta_1+\vartheta_2+\ldots+\vartheta_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2850: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$s_i-r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2856: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox } {\newpage \clearpage \samepage \setbox\sizebox=\hbox{$1\leq a_1