\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{dos05}
\citation{asai}
\citation{bren}
\citation{dos10}
\citation{dos04}
\citation{dos08}
\citation{mil04}
\citation{pre04}
\citation{stan89}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\citation{dos05}
\citation{dos05}
\citation{zhao15}
\citation{zhao14}
\citation{zhao15}
\citation{zhangtt16}
\citation{zhangtt16}
\citation{zhangtt16}
\@writefile{toc}{\contentsline {section}{\numberline {2}Sufficient conditions for log-balancedness}{2}{section.2}}
\newlabel{thm:1a}{{1}{2}{}{theorem.1}{}}
\newlabel{thm:2a}{{2}{3}{}{theorem.2}{}}
\newlabel{thm:3a}{{3}{3}{}{theorem.3}{}}
\citation{liu07}
\@writefile{toc}{\contentsline {section}{\numberline {3}Log-balancedness of some sequences}{4}{section.3}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}The derangement numbers}{4}{subsection.3.1}}
\newlabel{zf-3.1}{{1}{4}{The derangement numbers}{equation.3.1}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Some initial values of $\{d_n\}_{n\ge 0}$}}{4}{table.1}}
\newlabel{tab:table1}{{1}{4}{Some initial values of $\{d_n\}_{n\ge 0}$}{table.1}{}}
\newlabel{thm:4a}{{4}{4}{}{theorem.4}{}}
\newlabel{zf-3.2}{{2}{4}{The derangement numbers}{equation.3.2}{}}
\citation{hara70}
\citation{liu07}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Numbers counting tree-like polyhexes}{5}{subsection.3.2}}
\newlabel{zf-3.3}{{3}{5}{Numbers counting tree-like polyhexes}{equation.3.3}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Some initial values of $\{h_n\}_{n\ge 0}$}}{5}{table.2}}
\newlabel{tab:table2}{{2}{5}{Some initial values of $\{h_n\}_{n\ge 0}$}{table.2}{}}
\newlabel{thm:5a}{{5}{5}{}{theorem.5}{}}
\newlabel{zf-3.4}{{4}{6}{Numbers counting tree-like polyhexes}{equation.3.4}{}}
\citation{guy}
\citation{liu07}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Numbers counting walks on the cubic lattice}{7}{subsection.3.3}}
\newlabel{zf-3.5}{{5}{7}{Numbers counting walks on the cubic lattice}{equation.3.5}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Some initial values of $\{w_n\}_{n\ge 0}$}}{7}{table.3}}
\newlabel{tab:table3}{{3}{7}{Some initial values of $\{w_n\}_{n\ge 0}$}{table.3}{}}
\newlabel{thm:6a}{{6}{7}{}{theorem.6}{}}
\newlabel{zf-3.6}{{6}{7}{Numbers counting walks on the cubic lattice}{equation.3.6}{}}
\newlabel{zf-3.7}{{7}{7}{Numbers counting walks on the cubic lattice}{equation.3.7}{}}
\citation{zhao15J}
\citation{comt74}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Numbers counting a class of arrays}{9}{subsection.3.4}}
\newlabel{zf-3.8}{{8}{9}{Numbers counting a class of arrays}{equation.3.8}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces Some initial values of $\{A_n\}_{n\ge 0}$}}{9}{table.4}}
\newlabel{tab:table4}{{4}{9}{Some initial values of $\{A_n\}_{n\ge 0}$}{table.4}{}}
\newlabel{thm:7a}{{7}{9}{}{theorem.7}{}}
\newlabel{zf-3.9}{{9}{9}{Numbers counting a class of arrays}{equation.3.9}{}}
\newlabel{zf-3.10}{{10}{9}{Numbers counting a class of arrays}{equation.3.10}{}}
\citation{Odly95}
\citation{fish89}
\citation{zhao15J}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Numbers satisfying a three-term recurrence}{10}{subsection.3.5}}
\newlabel{zf-3.11}{{11}{10}{Numbers satisfying a three-term recurrence}{equation.3.11}{}}
\newlabel{thm:8a}{{8}{10}{}{theorem.8}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5}{\ignorespaces Some initial values of $\{t_n\}_{n\ge 0}$}}{11}{table.5}}
\newlabel{tab:table5}{{5}{11}{Some initial values of $\{t_n\}_{n\ge 0}$}{table.5}{}}
\newlabel{zf-3.12}{{12}{11}{Numbers satisfying a three-term recurrence}{equation.3.12}{}}
\newlabel{zf-3.13}{{13}{11}{Numbers satisfying a three-term recurrence}{equation.3.13}{}}
\citation{zhao15J}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.6}Numbers counting bipermutations}{12}{subsection.3.6}}
\newlabel{zf-3.14}{{14}{12}{Numbers counting bipermutations}{equation.3.14}{}}
\@writefile{lot}{\contentsline {table}{\numberline {6}{\ignorespaces Some initial values of $\{P_n\}_{n\ge 0}$}}{12}{table.6}}
\newlabel{tab:table6}{{6}{12}{Some initial values of $\{P_n\}_{n\ge 0}$}{table.6}{}}
\newlabel{thm:9a}{{9}{12}{}{theorem.9}{}}
\newlabel{zf-3.15}{{15}{12}{Numbers counting bipermutations}{equation.3.15}{}}
\newlabel{zf-3.16}{{16}{12}{Numbers counting bipermutations}{equation.3.16}{}}
\citation{stan99}
\citation{stan99}
\citation{dos08}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.7}Numbers satisfying a four-term recurrence (``minus" case)}{13}{subsection.3.7}}
\newlabel{zf-3.17}{{17}{13}{Numbers satisfying a four-term recurrence (``minus" case)}{equation.3.17}{}}
\newlabel{thm:10a}{{10}{13}{}{theorem.10}{}}
\@writefile{lot}{\contentsline {table}{\numberline {7}{\ignorespaces Some initial values of $\{G_n\}_{n\ge 0}$}}{14}{table.7}}
\newlabel{tab:table7}{{7}{14}{Some initial values of $\{G_n\}_{n\ge 0}$}{table.7}{}}
\newlabel{zf-3.18}{{18}{14}{Numbers satisfying a four-term recurrence (``minus" case)}{equation.3.18}{}}
\newlabel{zf-3.19}{{19}{14}{Numbers satisfying a four-term recurrence (``minus" case)}{equation.3.19}{}}
\citation{zhao15J}
\citation{comt74}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.8}Numbers satisfying a four-term recurrence (``plus" case)}{15}{subsection.3.8}}
\newlabel{zf-3.20}{{20}{15}{Numbers satisfying a four-term recurrence (``plus" case)}{equation.3.20}{}}
\@writefile{lot}{\contentsline {table}{\numberline {8}{\ignorespaces Some initial values of $\{g_n\}_{n\ge 0}$}}{15}{table.8}}
\newlabel{tab:table8}{{8}{15}{Some initial values of $\{g_n\}_{n\ge 0}$}{table.8}{}}
\newlabel{thm:11a}{{11}{15}{}{theorem.11}{}}
\newlabel{zf-3.21}{{21}{15}{Numbers satisfying a four-term recurrence (``plus" case)}{equation.3.21}{}}
\newlabel{zf-3.22}{{22}{15}{Numbers satisfying a four-term recurrence (``plus" case)}{equation.3.22}{}}
\citation{zhao15J}
\citation{comt74}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.9}Numbers counting permutation with ordered orbits}{16}{subsection.3.9}}
\newlabel{zf-3.23}{{23}{16}{Numbers counting permutation with ordered orbits}{equation.3.23}{}}
\@writefile{lot}{\contentsline {table}{\numberline {9}{\ignorespaces Some initial values of $\{T_n\}_{n\ge 0}$}}{16}{table.9}}
\newlabel{tab:table9}{{9}{16}{Some initial values of $\{T_n\}_{n\ge 0}$}{table.9}{}}
\newlabel{thm:12a}{{12}{16}{}{theorem.12}{}}
\newlabel{zf-3.24}{{24}{16}{Numbers counting permutation with ordered orbits}{equation.3.24}{}}
\newlabel{zf-3.25}{{25}{16}{Numbers counting permutation with ordered orbits}{equation.3.25}{}}
\citation{wang14}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.10}The Domb numbers}{17}{subsection.3.10}}
\newlabel{zf-3.26}{{26}{17}{The Domb numbers}{equation.3.26}{}}
\newlabel{thm:13a}{{13}{17}{}{theorem.13}{}}
\@writefile{lot}{\contentsline {table}{\numberline {10}{\ignorespaces Some initial values of $\{D_n\}_{n\ge 0}$}}{18}{table.10}}
\newlabel{tab:table10}{{10}{18}{Some initial values of $\{D_n\}_{n\ge 0}$}{table.10}{}}
\newlabel{zf-3.27}{{27}{18}{The Domb numbers}{equation.3.27}{}}
\newlabel{zf-3.28}{{28}{18}{The Domb numbers}{equation.3.28}{}}
\citation{dos08}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.11}Numbers counting a class of $n\times n$ symmetric matrices}{19}{subsection.3.11}}
\newlabel{zf-3.29}{{29}{19}{Numbers counting a class of $n\times n$ symmetric matrices}{equation.3.29}{}}
\@writefile{lot}{\contentsline {table}{\numberline {11}{\ignorespaces Some initial values of $\{\tau _n\}_{n\ge 0}$}}{19}{table.11}}
\newlabel{tab:table11}{{11}{19}{Some initial values of $\{\tau _n\}_{n\ge 0}$}{table.11}{}}
\newlabel{thm:14a}{{14}{19}{}{theorem.14}{}}
\newlabel{zf-3.30}{{30}{19}{Numbers counting a class of $n\times n$ symmetric matrices}{equation.3.30}{}}
\newlabel{zf-3.31}{{31}{19}{Numbers counting a class of $n\times n$ symmetric matrices}{equation.3.31}{}}
\citation{zhao15}
\citation{zhao15}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.12}The harmonic numbers}{20}{subsection.3.12}}
\newlabel{thm:15a}{{15}{20}{}{theorem.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.13}The Fibonacci and Lucas numbers}{20}{subsection.3.13}}
\newlabel{thm:16a}{{16}{20}{}{theorem.16}{}}
\bibstyle{plain}
\bibcite{asai}{1}
\bibcite{bren}{2}
\bibcite{comt74}{3}
\bibcite{dos05}{4}
\bibcite{dos10}{5}
\bibcite{dos04}{6}
\bibcite{dos08}{7}
\bibcite{fish89}{8}
\bibcite{guy}{9}
\bibcite{hara70}{10}
\bibcite{liu07}{11}
\@writefile{toc}{\contentsline {section}{\numberline {4}Conclusions}{21}{section.4}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Acknowledgment}{21}{section.5}}
\bibcite{mil04}{12}
\bibcite{Odly95}{13}
\bibcite{pre04}{14}
\bibcite{stan99}{15}
\bibcite{stan89}{16}
\bibcite{wang14}{17}
\bibcite{zhangtt16}{18}
\bibcite{zhao15J}{19}
\bibcite{zhao15}{20}
\bibcite{zhao14}{21}
