\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf Generalizing Tuenter's Binomial Sums } \vskip 1cm \large Richard P. Brent\\ Mathematical Sciences Institute\\ Australian National University\\ Canberra, ACT 2614\\ Australia\\ \href{mailto:sums@rpbrent.com}{\tt sums@rpbrent.com} \end{center} \vskip .2 in \newcommand{\raisecomma}{\raisebox{2pt}{$,$}} \newcommand{\raisedot}{\raisebox{2pt}{$.$}} \newcommand{\Pbar}{{\overline P}} \newcommand{\Qbar}{{\overline Q}} \newcommand{\half}{{\mbox{$\frac{1}{2}$}}} \begin{abstract} Tuenter considered centered binomial sums of the form \[S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form \[U_r(n) = \sum_{k=0}^n \binom{n}{k}|n/2-k|^r,\] which are a generalization of Tuenter's sums and may be interpreted as moments of a symmetric Bernoulli random walk with~$n$ steps. The form of $U_r(n)$ depends on the parities of both $r$ and~$n$. In fact, $U_r(n)$ is the product of a polynomial (depending on the parities of~$r$ and $n$) times a power of two or a binomial coefficient. In all cases the polynomials can be expressed in terms of Dumont-Foata polynomials. We give recurrence relations, generating functions and explicit formulas for the functions $U_r(n)$ and/or the associated polynomials. \end{abstract} \section{Introduction} \label{sec:intro} We consider centered binomial sums of the form \begin{equation} \label{eq:U_def} U_r(n) = \sum_{k=0}^n \binom{n}{k}\left|\frac{n}{2}-k\right|^r, \end{equation} where $r,n \ge 0$. These generalize the binomial sums \begin{equation} \label{eq:S_def} S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}|n-k|^r, \end{equation} previously considered by Tuenter~\cite{Tuenter1} and other authors, since $S_r(n) = U_r(2n)$, but $U_r(n)$ is well-defined for both even and odd values of~$n$. The generalization arises naturally in the study of certain two-fold centered binomial sums of the form $\sum_{j}\sum_{k}\binom{2n}{n+j}\binom{2n}{n+k}P(j,k)$, see Brent {et al.\ }\cite[Lemma~6.4]{BOOP}. In definitions such as~\eqref{eq:U_def} and~\eqref{eq:S_def} we always interpret $0^0$ as $1$. Thus $U_0(n) = 2^n$ for all $n\ge 0$. We define $U_r(m) = 0$ if $m < 0$. For $r > 0$ we can avoid the absolute value function in~\eqref{eq:U_def} by writing \[U_r(n) = 2\sum_{k=0}^{\lfloor{n/2}\rfloor} \binom{n}{k}\left(\frac{n}{2}-k\right)^r.\] Tuenter~\cite{Tuenter1} showed in a direct manner that, for $r \ge 0$ and $n > 0$, $S_r(n)$ satisfies the recurrence \begin{equation} \label{eq:Tuenter_rec0} S_{r+2}(n) = n^2 S_r(n) - 2n(2n-1)S_r(n-1). \end{equation} Observe that this recurrence splits into two separate recurrences, one involving odd values of~$r$ and the other involving even values of~$r$. Also, $S_0(n) = 2^{2n}$ and $S_1(n) = n\binom{2n}{n}$. It follows that, for $r, n \ge 0$, \begin{equation} \label{eq:SPQ} S_{2r}(n) = Q_r(n)2^{2n-r} \;\text{ and }\; S_{2r+1}(n) = P_r(n)n\binom{2n}{n}, \end{equation} where $P_r(n)$ and $Q_r(n)$ are polynomials of degree~$r$ with integer coefficients, satisfying the recurrences \begin{align} \label{eq:Prec} P_{r+1}(n) &= n^2P_r(n) - n(n-1)P_r(n-1),\\ \label{eq:Qrec} Q_{r+1}(n) &= 2n^2Q_r(n) - n(2n-1)Q_r(n-1) \end{align} for $r \ge 0$, with initial conditions $P_0(n) = Q_0(n) = 1$. The Dumont-Foata polynomials $F_r(x,y,z)$ are $3$-variable polynomials satisfying the recurrence relation \begin{equation} \label{eq:DF-recurrence} F_{r+1}(x,y,z) = (x+z)(y+z)F_r(x,y,z+1) - z^2F_r(x,y,z) \end{equation} for $r \ge 1$, with $F_1(x,y,z) = 1$. Dumont and Foata~\cite{DF} gave a combinatorial interpretation for the coefficients of $F_r(x,y,z)$ and showed that $F_r(x,y,z)$ is symmetric in the three variables $x,y,z$. Tuenter~\cite{Tuenter1} showed that $P_r(n)$ and $Q_r(n)$ may be expressed in terms of Dumont-Foata polynomials. In fact, for $r \ge 1$, \begin{equation} \label{eq:PQF} P_r(n) = (-1)^{r-1}nF_r(1,1,-n) \;\text{ and }\; Q_r(n) = (-2)^{r-1}nF_r(\half,1,-n). \end{equation} Thus, we can obtain explicit formulas and generating functions for the polynomials $P_r(n)$ and $Q_r(n)$ as special cases of the results of Carlitz~\cite{Carlitz} on Dumont-Foata polynomials. We show that all the above results for $S_r(n)$ can be generalized to cover $U_r(n)$. In particular, Theorem~\ref{thm:recurrence} shows that $U_r(n)$ satisfies a recurrence~\eqref{eq:Urec} similar to the recurrence~\eqref{eq:Tuenter_rec0} satisfied by $S_r(n)$. Also, $U_r(n)$ is the product of a polynomial (depending on the parity of~$r$) times a power of two or a binomial coefficient, and Theorem~\ref{thm:polynomials} shows that these polynomials satisfy three-term recurrence relations analogous to~\eqref{eq:Prec}--\eqref{eq:Qrec}. Using the recurrences, the polynomials can be expressed in terms of Dumont-Foata polynomials, so the results of Carlitz allow us to obtain explicit formulas (in~\S\ref{sec:explicit}) and exponential generating functions (in~\S\ref{sec:egfs}). \section{Recurrence relations} \label{sec:main} Theorems~\ref{thm:recurrence}--\ref{thm:polynomials} give recurrence relations for $U_r(n)$ and associated polynomials. The recurrence~\eqref{eq:Urec} in Theorem~\ref{thm:recurrence} implies the recurrence~\eqref{eq:Tuenter_rec0} satisfied by $S_r(n)$. \begin{theorem} \label{thm:recurrence} For all $r, n \ge 0$, $U_r(n)$ satisfies the recurrence \begin{equation} \label{eq:Urec} 4U_{r+2}(n) = n^2U_r(n) - 4n(n-1)U_r(n-2), \end{equation} and may be computed from the recurrence using the initial conditions \[ U_0(n) = 2^n,\; U_1(2n) = n\binom{2n}{n},\; U_1(2n+1) = (2n+1)\binom{2n}{n} \text{ for all } n \ge 0. \] \end{theorem} \begin{proof} We have \begin{align} 4U_{r+2}(n) &= \sum_{k=0}^n 4\binom{n}{k}\left|\frac{n}{2}-k\right|^{r+2} = \sum_{k=0}^n \binom{n}{k}\left|\frac{n}{2}-k\right|^{r}(n-2k)^2, \label{eq:rec1} \end{align} \begin{align} n^2U_r(n) &= \sum_{k=0}^n\binom{n}{k}\left|\frac{n}{2}-k\right|^{r}n^2, \label{eq:rec2} \end{align} and \begin{align} 4n(n-1)U_r(n-2) &= \sum_{k=0}^{n-2} 4n(n-1)\binom{n-2}{k}\left|\frac{n-2}{2}-k\right|^r \nonumber\\ &= \sum_{k=1}^{n-1} 4n(n-1)\binom{n-2}{k-1}\left|\frac{n}{2}-k\right|^{r} \nonumber\\ &= \sum_{k=1}^{n-1} 4k(n-k)\binom{n}{k}\left|\frac{n}{2}-k\right|^{r}. \label{eq:rec3} \end{align} Since $(n-2k)^2 - n^2 - 4k(n-k) = 0$, the recurrence~\eqref{eq:Urec} follows from~\eqref{eq:rec1}--\eqref{eq:rec3}. For the initial conditions, it is easily verified that $U_0(n) = 2^n$. The solution by Hillman~\cite{Hillman} to the Putnam problem 35-A4 gives $U_1(n) = n\binom{n-1}{\lfloor{n/2}\rfloor}$, and taking account of the parity of $n$ gives the remaining conditions. \end{proof} It is now clear that the structure of $U_r(n)$ depends upon the parities of both $r$ and $n$, and one can elucidate the recursion~\eqref{eq:Urec} by the following substitutions: \begin{align} U_{2r+1}(2n-1) &= 2^{-(2r+1)}n\Pbar_r(n)\binom{2n}{n}, \label{eq:U_odd_odd_Pbar}\\ U_{2r}(2n+1) &= 2^{2n+1-2r}\Qbar_r(n). \label{eq:U_even_odd_Qbar} \end{align} It is easily verified that $\Pbar_r(n)$ and $\Qbar_r(n)$ are polynomials in~$n$ of degree~$r$. By substitution into~\eqref{eq:Urec}, we obtain the following theorem. \begin{theorem} \label{thm:polynomials} For $r \ge 0$, the polynomials $\Pbar_r(n)$ and $\Qbar_r(n)$ defined by equations \eqref{eq:U_odd_odd_Pbar}--\eqref{eq:U_even_odd_Qbar} satisfy the following recurrence relations: \begin{align} \Pbar_{r+1}(n) &= (2n-1)^2\,\Pbar_r(n) - 4(n-1)^2\,\Pbar_r(n-1), \label{eq:Pbar_rec}\\ \Qbar_{r+1}(n) &= (2n+1)^2\,\Qbar_r(n) - 2n(2n+1)\Qbar_r(n-1), \label{eq:Qbar_rec} \end{align} with initial conditions $\Pbar_0(n) = \Qbar_0(n) = 1$. \end{theorem} \section{Explicit formulas} \label{sec:explicit} Using the recurrence~\eqref{eq:DF-recurrence} and matching the initial conditions, we can verify that the polynomials $\Pbar_r(n)$ and $\Qbar_r(n)$ can be expressed in terms of Dumont-Foata polynomials, as follows: \begin{align} \Pbar_r(n) &= (-4)^r F_{r+1}\left(\half,\half,\half-n\right), \label{eq:Pbar_F}\\ \Qbar_r(n) &= (-1)^{r-1}2^{2r-1} \left(n+\half\right)F_r\left(\half,1,-n-\half\right). \label{eq:Qbar_and_Q} \end{align} Also, replacing $n$ by $n+\half$ in the second half of~\eqref{eq:PQF} shows that \begin{equation} \label{eq:QandQbar} \Qbar_r(n) = 2^r Q_r\!\left(n+\half\right). \end{equation} Carlitz~\cite[{eqns.\ }(1.13) and (1.16)]{Carlitz} gives an explicit formula for the Dumont-Foata polynomials. We follow Carlitz and let $(z)_k$ denote the Pochhammer symbol or ``rising factorial''. Note that, although $F_{r+1}(x,y,z)$ is symmetric in $x,y$ and $z$, the representation given in Proposition~\ref{prop:DF} is not symmetric. Thus, it is sometimes necessary to permute variables before applying Proposition~\ref{prop:DF}. \begin{proposition}[Carlitz] \label{prop:DF} For $r \ge 0$ and $z>0$, \begin{equation*} F_{r+1}(x,y,z) = \sum_{k=0}^r (-1)^{r-k}{(x+z)_k(y+z)_k}\,A_{r,k}(z), \end{equation*} \vspace*{-10pt} where \vspace*{-12pt} \begin{equation*} \;\;\; A_{r,k}(z) = \frac{2}{k!}\, \sum_{j=0}^k(-1)^{k-j}\binom{k}{j}\frac{(z+j)^{2r+1}}{(2z+j)_{k+1}} \,\raisedot \end{equation*} \end{proposition} Application of Proposition~\ref{prop:DF} gives, after some simplification, the following formulas for $U_r(n)$, valid for $r\ge 1$ and $n \ge 0$: \begin{align} \label{eq:U_even_a2} U_{2r}(n) &= 2^{n+1} \sum_{1 \le j \le k \le r} (-1)^j \frac{\left(-\frac{n}{2}\right)_k \left(\half\right)_k} {(k-j)!(k+j)!}\,j^{2r},\\ \label{eq:U_odd_even2} U_{2r+1}(2n) &= 2n\binom{2n}{n}\sum_{1 \le j \le k \le r} (-1)^{j}\frac{(-n)_{k}}{(k-j)!\,(k+1)_j}\,j^{2r},\\ \label{eq:U_odd_odd2} U_{2r-1}(2n-1) &= \binom{2n}{n}\sum_{1 \le j \le k \le r} (-1)^{j}\frac{(-n)_{k}}{(k-j)!\,(k)_j}\, \left(j-\half\right)^{2r-1}. \end{align} \section{Generating functions} \label{sec:egfs} The exponential generating function (egf) \begin{equation} \label{eq:U_even_gf} \sum_{r\ge 0} U_{2r}(n)\frac{z^{2r}}{(2r)!} = 2^n \cosh^n (z/2). \end{equation} generalizes the egf \begin{equation} \label{eq:S_even_gf} \sum_{r\ge 0} S_{2r}(n)\frac{z^{2r}}{(2r)!} = 2^{2n} \cosh^{2n} (z/2) \end{equation} given by Tuenter~\cite[\S5]{Tuenter1}, since replacing $n$ by $2n$ in~\eqref{eq:U_even_gf} gives~\eqref{eq:S_even_gf}. The proof of~\eqref{eq:U_even_gf} is straightforward, and does not require the results of Carlitz. We can obtain other egfs from the results of Carlitz. First, we note that Carlitz~\cite[{eqn.\ }(4.2)]{Carlitz} gives the egf \begin{equation} \label{eq:Carlitz42} \sum_{r \ge 1} (-1)^rF_r(x,y,1) \frac{z^{2r}}{(2r)!} = \frac{1}{xy}\sum_{k\ge 1}(-1)^k\frac{(x)_k(y)_k}{(2k)!} \left(2\sinh\frac{z}{2}\right)^{2k}. \end{equation} In view of~\eqref{eq:SPQ} and~\eqref{eq:PQF}, this allows us to obtain an egf for $U_{2r+1}(2n)$: \begin{equation} \label{eq:U_odd_even_egf} \sum_{r\ge 0}U_{2r+1}(2n)\frac{z^{2r}}{(2r)!} = n\binom{2n}{n}\sum_{k=0}^n 2^{2k}\binom{n}{k}\binom{2k}{k}^{-1} \sinh^{2k}\left(\frac{z}{2}\right)\,\raisedot \end{equation} In order to calculate $U_{2r+1}(2n)$ from~\eqref{eq:U_odd_even_egf}, it is only necessary to sum the terms on the right-hand side for $k \le \min(r,n)$. The remaining case $U_{2r+1}(2n-1)$ is more difficult because \eqref{eq:Carlitz42} does not apply. We can use the egf \begin{align} \sum_{r\ge 0}&(-1)^r F_{r+1}(x,y,z)\frac{u^{2r+1}}{(2r+1)!} \nonumber\\ &=\, 2\,\sum_{k=0}^\infty \sum_{j=0}^k \frac{(-1)^j (x+z)_k (y+z)_k (2z)_j}{j! (k-j)! (2z)_{k+1} (2z+k+1)_j} \sinh((z+j)u), \label{eq:Carlitz222} \end{align} which follows from the discussion in Carlitz~\cite[{pp.\ }221--222]{Carlitz}. Using~\eqref{eq:U_odd_odd_Pbar}, \eqref{eq:Pbar_F} and~\eqref{eq:Carlitz222}, after some simplification followed by a change of variables ($u \mapsto z$), we obtain the egf: \begin{align} \sum_{r\ge 0}&\,U_{2r+1}(2n-1)\frac{z^{2r+1}}{(2r+1)!} \nonumber \\ &=\, n\binom{2n}{n}\sum_{0 \le j \le k < n} \frac{(-1)^{k-j}\binom{n-1}{k}\binom{2k}{k-j}}{\binom{2k}{k}}\, \frac{\sinh\left(\left(j+\half\right)z\right)} {j+k+1}\,\raisecomma \label{eq:U_odd_odd_egf} \end{align} which is valid for $n\ge 1$. \section{Acknowledgment} Thanks are due to an anonymous referee for detailed and helpful comments on a preliminary version of this paper. This research was supported by Australian Research Council grant DP140101417. \subsection*{Appendix 1: The polynomials $\Pbar_r, \Qbar_r$ for $r \le 5$} \begin{align*} \Pbar_0(n) &= 1,\\ \Pbar_1(n) &= 4n-3,\\ \Pbar_2(n) &= 32n^2 - 56n + 25,\\ \Pbar_3(n) &= 384n^3 - 1184n^2 + 1228n - 427,\\ \Pbar_4(n) &= 6144n^4 - 29184n^3 + 52416n^2 - 41840n + 12465,\\ \Pbar_5(n) &= 122880n^5 - 829440n^4 +\\ & \phantom{= xx} 2258688n^3 - 3076288n^2 + 2079892n - 555731,\\ \Qbar_0(n) &= 1,\\ \Qbar_1(n) &= 2n+1,\\ \Qbar_2(n) &= 12n^2 + 8n + 1,\\ \Qbar_3(n) &= 120n^3 + 60n^2 + 2n + 1,\\ \Qbar_4(n) &= 1680n^4 - 168n^2 + 128n + 1,\\ \Qbar_5(n) &= 30240n^5 - 25200n^4 + 5040n^3 + 7320n^2 - 2638n + 1. \end{align*} A similar table for Tuenter's polynomials $P_r(n), Q_r(n)$ may be found in Brent~\cite{rpb259-arxiv}. The triangles of coefficients of $-P_r(-n)/n, -Q_r(-n)/n$, and $\Qbar_r(-n)$ for $r \ge 1$ are OEIS~\cite{OEIS} sequences \seqnum{A036970}, \seqnum{A083061}, and \seqnum{A160485} respectively. We have contributed the coefficients of $\Pbar_r(n)$ as sequence \seqnum{A245244}. The values $(-1)^r\Pbar_r(0)$ are sequence \seqnum{A009843} (see Appendix~$2$ for details). The bijection~\eqref{eq:QandQbar} between \seqnum{A083061} and \seqnum{A160485} (by a shift of $\pm\half$ and scaling by a power of~$2$) was not mentioned in the relevant OEIS entries as at July 14, 2014; we have now contributed comments to this effect. \subsection*{Appendix 2: Special values of $P_r, Q_r, \Pbar_r, \Qbar_r$} In Table~\ref{tab:one}, we give values of the polynomials $P_r, Q_r, \Pbar_r$, and $\Qbar_r$ at $0, 1, \infty$. Here $P(\infty)$ denotes the leading coefficient of the polynomial $P(z)$, $\delta_{i,j}$ is the Kronecker delta, and ${\cal S}_r$ is the $r$-th Secant number. \begin{table}[ht] \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline &&&&\\[-10pt] $z$ & $P_r(z)$ & $Q_r(z)$ & $\Pbar_r(z)$ & $\Qbar_r(z)$\\[1pt] \hline &&&&\\[-10pt] $0$ & $\delta_{0,r}$ & $\delta_{0,r}$ & $(-1)^r(2r+1){\cal S}_r$ & $1$\\[1pt] $1$ & $1$ & $\max(1,2^{r-1})$ & $1$ & $(3^{2r} + 3)/4$\\[1pt] $\infty$ & $r!$ & $(2r)!/(2^r r!)$ & $2^{2r}r!$ & $(2r)!/r!$\\[1pt] \hline \end{tabular} \caption{Special values of the polynomials} \label{tab:one} \end{center} \end{table} The values $(-1)^r\Pbar_r(0)$ are OEIS sequence \seqnum{A009843}, and are given by the egf \begin{equation} \label{eq:Pbar_gf} \sum_{r=0}^\infty \Pbar_r(0)\frac{x^{2r+1}}{(2r+1)!} = \frac{x}{\cosh x}\,\raisedot \end{equation} They may be expressed in terms of the Secant numbers ${\cal S}_r$, which comprise OEIS sequence \seqnum{A000364}. In view of~\eqref{eq:Pbar_F}, we obtain a special value of the Dumont-Foata polynomials: \begin{equation} \label{eq:F_half_half_half} F_{r+1}\left(\half,\half,\half\right) = 2^{-2r}(2r+1){\cal S}_r. \end{equation} The values $\Qbar_r(1)$ comprise OEIS sequence \seqnum{A054879}. The values in the last row of Table~\ref{tab:one} may also be found in OEIS: they are sequences \seqnum{A000142}, \seqnum{A001147}, \seqnum{A047053}, and \seqnum{A001813}. Tuenter~\cite{Tuenter1} observed that, for $r \ge 1$, the constant terms of $-P_r(n)/n$ are the Genocchi numbers (\seqnum{A001469}), and the constant terms of $(-1)^{r-1}Q_r(n)/n$ are the reduced tangent numbers (\seqnum{A002105}). \begin{thebibliography}{9} \bibitem{rpb259-arxiv} R. P. Brent, {Generalising Tuenter's binomial sums}, preprint, \newline \url{http://arxiv.org/abs/1407.3533v4/}, January 19 2015. \bibitem{BOOP} R. P. Brent, H. Ohtsuka, J. H. Osborn, and H. Prodinger, {Some binomial sums involving absolute values}, preprint, \url{http://arxiv.org/abs/1411.1477v1/}, \newline November 6 2014. \bibitem{Carlitz} L. Carlitz, Explicit formulas for the Dumont-Foata polynomial, \emph{Discrete Math.\ }\textbf{30} (1980), 211--225. \bibitem{DF} D. Dumont and D. Foata, Une propri\'et\'e de sym\'etrie des nombres de Genocchi, \emph{Bull.\ Soc.\ Math.\ France} \textbf{104} (1976), 433--451. \bibitem{Hillman} A. P. Hillman, The William Lowell Putnam mathematical competition, \emph{Amer.\ Math.\ Monthly} \textbf{82} (1975), 905--912. \bibitem{OEIS} OEIS Foundation Inc., \emph{The On-Line Encyclopedia of Integer Sequences}, published electronically at \url{http://oeis.org}. \bibitem{Tuenter1} H. J. H. Tuenter, Walking into an absolute sum, \emph{Fibonacci Quart.\ }\textbf{40} (2002), 175--180. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 05A10; Secondary 11B65, 05A15, 05A19, 44A60, 60G50. \noindent \emph{Keywords: } Bernoulli random walk, binomial sum identity, Catalan number, Dumont-Foata polynomial, explicit formula, generating function, Genocchi number, moment, polynomial interpolation, secant number, tangent number. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000142}, \seqnum{A000364}, \seqnum{A001147}, \seqnum{A001469}, \seqnum{A001813}, \seqnum{A002105}, \seqnum{A009843}, \seqnum{A036970}, \seqnum{A047053}, \seqnum{A054879}, \seqnum{A083061}, \seqnum{A160485}, and \seqnum{A245244}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received July 16 2014; revised versions received January 18 2015; January 25 2015. Published in {\it Journal of Integer Sequences}, January 26 2015. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}