\documentclass[12pt,reqno]{article} %\sloppy \usepackage [colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \usepackage{fullpage} \usepackage{amsmath} \usepackage{amssymb} \usepackage{color} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\topmargin}{-.2in} \setlength{\textheight}{8.0in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \makeatletter \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \def\endofproofmark{$\Box$} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\endofproofmark{$\Box$} \begin{center} \vskip 1cm {\LARGE\bf On an Integer Sequence Related to a Product of Trigonometric Functions, and its Combinatorial Relevance} \vskip 1cm \large Dorin Andrica \\ \vskip .5cm ``Babe\c s--Bolyai'' University\\ Faculty of Mathematics and Computer Science\\ Str. M. Kog\v alniceanu nr. 1\\ 3400 Cluj--Napoca, Romania \\ \href{mailto:dandrica@math.ubbcluj.ro}{\tt dandrica@math.ubbcluj.ro} \\ \vskip 1cm \large Ioan Tomescu\\ \vskip .5cm University of Bucharest\\ Faculty of Mathematics and Computer Science\\ Str. Academiei, 14\\ R-70109 Bucharest, Romania\\ \href{mailto:ioan@math.math.unibuc.ro}{\tt ioan@math.math.unibuc.ro}\\ \end{center} \date{} %\maketitle \pagestyle{myheadings} %%%%%%%%%%%%%%%%%%%%%%%%%MACROS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theo}{Theorem}[section] \newtheorem{prop}[theo]{Proposition} \newtheorem{lemma}[theo]{Lemma} \newtheorem{cor}[theo]{Corollary} \newtheorem{problem}{Problem} \def\frameqed{\framebox(5.2,6.2){}} \def\deshqed{\dashbox{2.71}(3.5,9.00){}} \def\ruleqed{\rule{5.25\unitlength}{9.75\unitlength}} \def\myqed{\rule{8.00\unitlength}{12.00\unitlength}} \def\qed{\hbox{\hskip 6pt\vrule width 7pt height11pt depth1pt\hskip 3pt} \bigskip} \newenvironment{proof}{\trivlist\item[\hskip\labelsep{\bf Proof}:]}{\hfill $\frameqed$ \endtrivlist} \newcommand{\COM}[2]{{#1\choose#2}} %%%%%%%%%%%%%%%%%%%END%OF%MACROS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \thispagestyle{empty} \null \addtolength{\textheight}{2cm} \begin{abstract} In this paper it is shown that for $n\equiv 0$ or 3 (mod 4), the middle term $S(n)$ in the expansion of the polynomial $(1+x)(1+x^{2})\cdots (1+x^{n})$ occurs naturally when one analyzes when a discontinuous product of trigonometric functions is a derivative of a function. This number also represents the number of partitions of $T_{n}/2=n(n+1)/4$, (where $T_{n}$ is the $n$th triangular number) into distinct parts less than or equal to $n$. It is proved in a constructive way that $S(n)\geq 6S(n-4)$ for every $n\geq 8$, and an asymptotic evaluation of $S(n)^{1/n}$ is obtained as a consequence of the unimodality of the coefficients of this polynomial. Also an integral expression of $S(n)$ is deduced. \end{abstract} \section{Notation and preliminary results} In a paper of Andrica \cite{andr} the following necessary and sufficient condition that some product of derivatives is also a derivative is deduced: \begin{theo} Let $n_{1},\ldots ,n_{k}\geq 0$ be integers with $n_{1}+\ldots +n_{k}\geq 1$ and let $\alpha _{1},\ldots ,\alpha _{k}$ be real numbers different from zero. The function $f_{n_{1},\ldots ,n_{k}}^{\alpha _{1},\ldots ,\alpha _{k}}:\RR \rightarrow \RR $, defined by $$f_{n_{1},\ldots ,n_{k}}^{\alpha _{1},\ldots ,\alpha _{k}}(x)=\left\{ \begin{array}{ll} \cos^{n_{1}}(\alpha _{1}/x)\cdots \cos^{n_{k}}(\alpha _{k}/x), & \mbox{if $ x\neq 0 $;} \\ \alpha, & \mbox{if $ x=0 $;} \end{array} \right. $$ is a derivative if and only if $$\alpha =\frac{1}{2^{n_{1}+\ldots +n_{k}}}S(n_{1},\ldots ,n_{k};\alpha _{1}, \ldots ,\alpha _{k}),$$ where $S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})$ is the number of all choices of signs $+$ and $-$ such that \begin{equation} \underbrace{\pm \alpha _{1} \pm \ldots \pm \alpha _{1}}_{n_{1}\, times}\, \, \underbrace{\pm \alpha _{2}\pm \ldots \pm \alpha _{2}}_{n_{2} \, times}\, \pm \ldots \, \underbrace{\pm \alpha _{k}\pm \ldots \pm \alpha _{k}}_{n_{k}\, times} =0. \end{equation} \end{theo} Note that this theorem extends one previously published in \cite{andr-buz}.\\ We shall present another combinatorial interpretations of the numbers $$S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})$$ and an integral representation, while the last section is devoted to the sequence $S(n)= S(\underbrace{1,\ldots ,1}_{n\, {\rm times}};1,2,3,\ldots ,n)$ for $n\geq 1$. \medskip Let $M$ be a multiset of type $\alpha _{1}^{n_{1}}\alpha _{2}^{n_{2}}\ldots \alpha _{k}^{n_{k}}$, i.e., a multiset containing $\alpha _{i}$ with multiplicity $n_{i}$ for every $1\leq i\leq k$. It is clear that $S(n_{1},\ldots ,n_{k}; \alpha _{1},\ldots \alpha _{k})$ is the number of ordered partitions having equal sums of $M$, i.e., of ordered pairs $(C_{1},C_{2})$ such that $C_{1}\cup C_{2}=M$, $C_{1}\cap C_{2}=\emptyset $ and $\sum_{x\in C_{1}}x =\sum_{y\in C_{2}} y =\frac{1}{2}\sum_{i=1}^{k}n_{i}\alpha _{i}$. Indeed, there exists a bijection between the set of all choices of $+$ or $-$ signs in (1) and the set of all ordered partitions with equal sums of $M$ defined as follows: We put $\alpha _{i}$ from (1) in $C_{1}$ if its sign is $+$ and in $C_{2}$ otherwise. \par It is also clear that $S(n_{1},\ldots ,n_{k};\alpha _{1}\ldots ,\alpha _{k})$ is the term not depending on $z$ in the expansion \begin{equation} F(z)=\left(z^{\alpha _{1}}+\frac{1}{z^{\alpha _{1}}}\right)^{n_{1}}\left(z^{\alpha _{2}}+ \frac{1}{z^{\alpha _{2}}}\right)^{n_{2}}\ldots \left(z^{\alpha _{k}}+\frac{1}{z^{\alpha _{k}}}\right)^{n_{k}}. \end{equation} Wilf \cite{wilf} outlines a proof that for $n_{1}=n_{2}=\ldots =n_{k}=1$, the coefficient of $z^{n}$ in $F(z)$ represents the number of ways of choosing $+$ or $-$ signs such that $\pm \alpha _{1}\pm \alpha _{2}\pm \ldots \pm \alpha _{k}=n$. If $\alpha _{1},\ldots ,\alpha _{k}$ are positive integers, from (2) one gets \begin{equation} F(z)=S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})+ \sum_{\alpha \neq 0} a_{\alpha}z^{\alpha}, \end{equation} where the sum has only a finite number of terms and $\alpha $ and $a_{\alpha}$ are integers. By substituting $z=\cos t+ i\sin t$, $t\in \RR $ in (3) one deduces $$2^{n_{1}+\ldots +n_{k}}\prod _{j=1}^{k}(\cos\alpha _{j}t)^{n_{j}}= S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})+\sum_{\alpha \neq 0} a_{\alpha}(\cos\alpha t+i\sin\alpha t)$$ By integration on $[0,2\pi ]$ we find the following integral expression of $S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})$: $$S(n_{1},\ldots ,n_{k};\alpha _{1},\ldots ,\alpha _{k})=\frac {2^{n_{1}+ \ldots +n_{k}}}{2\pi }\int _{0}^{2\pi }(\cos\alpha _{1}t)^{n_{1}}\cdots (\cos \alpha _{k}t)^{n_{k}} dt. $$ \section{A particular case and its connection with polynomial unimodality} An interesting particular case is obtained for $n_{1}=n_{2}=\ldots =n_{k}=1$ and $\alpha _{i}=i$ for every $1\leq i\leq k$. In this case $S(n)$ is the number of ways of choosing $+$ and $-$ signs such that $\pm 1\pm 2\pm \ldots \pm n=0$. Since now $M=\{1,2,\ldots ,n\}$ has sum $T_{n}=n(n+1)/2$ and every class of an ordered bipartition of $M$ must have sum $T_{n}/2$, it follows that $S(n)=0$ for $n\equiv 1$ or 2 (mod 4) and $S(n)\neq 0$ for $n\equiv 0$ or 3 (mod 4). The following theorem proposes several equivalent definitions of the sequence $S(n)$ for $n\geq 1$. \begin{theo} For every $n\geq 1$ the following properties are equivalent:\\ \hspace*{0.5cm}(i) $S(n)$ is the number of choices of $+$ and $-$ signs such that $\pm 1\pm 2\pm \ldots \pm n=0;$\\ \hspace*{0.4cm}(ii) $S(n)$ is the number of ordered bipartitions into classes having equal sums of $\{1,2,\ldots ,n\}$;\\ \hspace*{0.3cm}(iii) $S(n)$ is the term not depending on $x$ in the expansion of $$\left(x+\frac{1}{x}\right)\left(x^{2}+\frac{1}{x^{2}}\right)\ldots \left(x^{n}+\frac{1}{x^{n}}\right);$$ \hspace*{0.4cm}(iv) $S(n)$ is the number of partitions of $T_{n}/2$ into distinct parts, less than or equal to $n$, if $n\equiv 0$ or $3\ ({\rm mod}\, 4)$, and $S(n)=0$ otherwise;\\ \hspace*{0.5cm}(v) $S(n)$ is the number of distinct subsets of $\{1,\ldots ,n\}$ whose elements sum to $T_{n}/2 $ if $n\equiv 0\, or \, 3\ ({\rm mod}\, 4)$, and $S(n)=0$ if $n\equiv 1\, or\, 2 \ ({\rm mod}\, 4)$;\\ \hspace*{0.4cm}(vi) $S(n)$ is the coefficient of $x^{T_{n}/2}$ in the polynomial $G_{n}(x)= (1+x)(1+x^{2})\ldots (1+x^{n})$ when $n\equiv 0\, or\, 3\ ({\rm mod}\, 4)$, and $S(n)=0$ otherwise;\\ \hspace*{0.3cm}(vii) $$S(n)=\frac{2^{n-1}}{\pi }\int_{0}^{2\pi}\cos t \cos 2t \cdots \cos nt\, dt;$$ \hspace*{0.2cm}(viii) $S(n)/2^{n}$ is the unique real number $\alpha $ having the property that the function $f:\RR \rightarrow \RR $, defined by $$ f(x)=\left\{ \begin{array}{ll} \cos (1/x)\cos (2/x)\cdots \cos (n/x), & \mbox{if $ x\neq 0$;}\\ \alpha, & \mbox{if $ x=0$;} \end{array} \right. $$ is a derivative. \end{theo} \begin{proof} Some equivalences are obvious or were shown in the general case. For example, the equivalence between (ii) and (v) is given by the bijection $\varphi$ defined for every bipartition $M=C_{1}\cup C_{2}$ such that $\sum_{x\in C_{1}}x=\sum _{y\in C_{2}}y $ by $\varphi (C_{1}\cup C_{2})= C_{1}\subset M$. \end{proof} Let us denote \begin{equation} G_{n}(x)=(1+x)(1+x^{2})\ldots (1+x^{n})=\sum_{i=0}^{T_{n}}G(n,i)x^{i}. \end{equation} Note that the property that the coefficient of $x^{i}$ in $G_{n}(x)$ is the number of distinct subsets of $\{1,\ldots ,n\}$ whose elements sum to $i$ was used by Friedman and Keith \cite{friedm-keith} to deduce a necessary and sufficient condition for the existence of a basic ($n$,$k$) magic carpet. Stanley \cite{stan}, using the ``hard Lefschetz theorem'' from algebraic geometry, proved that the posets $M(n)$ of all partitions of integers into distinct parts less than or equal to $n$ are rank unimodal, by showing the existence of a chain decomposition for $M(n)$. This fact is equivalent to the unimodality of the polynomial $G_{n}(x)$, which implies that $S(n)$ is the maximum coefficient in the expansion of $G_{n}(x)$ for $n\equiv 0$ or 3 ({\rm mod} 4). Stanley's proof was subsequently simplified by Proctor \cite{pro}. The property of symmetry of the coefficients in (4), namely $G(n,i)= G(n,T_{n}-i)$ for every $0\leq i\leq T_{n}$ was pointed out by Friedman and Keith\cite{friedm-keith}; they also found the recurrence $G(n,i)=G(n-1,i)+G(n-1,i-n)$. This latter recurrence, which is a consequence of the identity $G_{n}(x)=G_{n-1}(x)(1+x^{n})$, allows us to compute any finite submatrix of the numbers $G(n,i)$ and thus the numbers $S(n)=G(n,T_{n}/2)$. Some values of $S(n)$, starting with $n=3$, are given in the following table: \bigskip \begin{center} \begin{tabular}{|r|r||r|r||r|r||r|r|} \hline $n$ & $S(n)$ & $n$ & $S(n)$ & $n$ & $S(n)$ & $n$ & $S(n)$ \\ \hline 3 & 2 & 13 & 0 & 23 & 99,820 & 33 & 0 \\ \hline 4 & 2 & 14 & 0 & 24 & 187,692 & 34 & 0 \\ \hline 5 & 0 & 15 & 722 & 25 & 0 & 35 & 221,653,776 \\ \hline 6 & 0 & 16 & 1,314 & 26 & 0 & 36 & 425,363,952 \\ \hline 7 & 8 & 17 & 0 & 27 & 1,265,204 & 37 & 0 \\ \hline 8 & 14 & 18 & 0 & 28 & 2,399,784 & 38 & 0 \\ \hline 9 & 0 & 19 & 8,220 & 29 & 0 & 39 & 3,025,553,180 \\ \hline 10 & 0 & 20 & 15,272 & 30 & 0 & 40 & 5,830,034,720\\ \hline 11 & 70 & 21 & 0 & 31 & 16,547,220 & 41 & 0 \\ \hline 12 & 124 & 22 & 0 & 32 & 31,592,878 & 42 & 0 \\ \hline \end{tabular} \\ \end{center} \bigskip \noindent and thus the terms different from zero form a subsequence of the sequence A025591 in Sloane \cite{sloane}. Another recurrence satisfied by the numbers $G(n,i)$ is the following: \begin{lemma} We have $G(n,i)=\sum_{j\geq 0}G(n-1-j,i-n+j).$ \end{lemma} \begin{proof} Let ${\cal P}(k,i)$ denote the set of partitions of $i$ into distinct parts such that the maximum part is equal to $k$. It is clear that $$G(n,i)=\left|\bigcup_{j\geq 0}{\cal P}(n-j,i)\right|=\sum_{j\geq 0}|{\cal P}(n-j,i)|= \sum_{j\geq 0}G(n-1-j,i-n+j). $$ Indeed, there is a bijection between the set of partitions of $i$ into distinct parts such that the maximum part equals $n-j$ and the set of partitions of $i-n+j$ into distinct parts less than or equal to $n-1-j$, defined by deleting the maximum part, equal to $n-j$, in any partition in ${\cal P}(n-j,i)$. Hence $|{\cal P}(n-j,i)|=G(n-1-j,i-n+j)$. \end{proof} \begin{theo} For any $n\geq 8$ we have $S(n)\geq 6S(n-4)$. \end{theo} \begin{proof} For $n\leq 11$ this inequality is verified by inspection. For $n\geq 12$ we shall propose a constructive proof yielding for any ordered partition of $\{1,\ldots ,n-4\}$ in two classes $C_{1}$ and $C_{2}$ with equal sums six ordered partitions of $\{1,\ldots ,n\}$ in two classes ${\tt C_{1}'}$ and ${\tt C_{2}'}$ having equal sums and all partitions generated will be distinct. Indeed, for any ordered bipartition with equal sums $\{1,\ldots ,n-4\}=C_{1} \cup C_{2}$ we can generate six ordered bipartitions with equal sums $\{1,\ldots ,n\}={\tt C_{1}'} \cup {\tt C_{2}'}$ as follows: \medskip (a) ${\tt C_{1}'}=C_{1}\cup \{n-3,n\}$ and ${\tt C_{2}'}=C_{2}\cup \{n-2,n-1\}$; \medskip (b) ${\tt C_{1}'}=C_{1}\cup \{n-2,n-1\}$ and ${\tt C_{2}'}=C_{2}\cup \{n-3,n\}$; \medskip (c) Without loss of generality suppose $1\in C_{1}$. We define ${\tt C_{1}''}= C_{1}\backslash \{1\}$, ${\tt C_{2}''}=C_{2}\cup \{1\}$, ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-2,n\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n-3,n-1\}$; \medskip (d) Without loss of generality suppose $2\in C_{1}$. Now ${\tt C_{1}''}=C_{1} \backslash \{2\}$, ${\tt C_{2}''}=C_{2}\cup \{2\}$, ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-1,n\}$, ${\tt C_{2}'}= {\tt C_{2}''}\cup \{n-3,n-2\}$. \medskip Case (e) is a little more complicated, but we will be able to do it by combining two simple transformations. \medskip (e) Suppose $1\in C_{1}$. If $n-4$ belongs to the same class, we define ${\tt C_{1}''}=C_{1}\backslash \{1,n-4\}$, ${\tt C_{2}''}=C_{2}\cup \{1,n-4\}$, ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-3,n-2,n-1\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n\}$. This transformation resolves the imbalance of $2n-6$ between ${\tt C_{1}''}$ and ${\tt C_{2}''}$ and will be called of type A. Otherwise $1\in C_{1}$ and $n-4\in C_{2}$. If $2\in C_{2}$ one defines ${\tt C_{2}''}=C_{2}\backslash \{2,n-4\}$, ${\tt C_{1}''}=C_{1}\cup \{2,n-4\}$, ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-1\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n,n-2,n-3\}$. This transformation balances classes ${\tt C_{1}''}$ and ${\tt C_{2}''}$ by $2n-4$ and will be called of type B. Otherwise $2\in C_{1}$, hence $C_{1}=\{1,2,\ldots \}$ and $C_{2}=\{n-4,\ldots \}$. If $n-5\in C_{1}$ then ${\tt C_{1}''}=C_{1}\backslash \{2,n-5\}$, ${\tt C_{2}''}= C_{2}\cup \{2,n-5\}$, ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-3,n-2,n-1\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n\}$. Otherwise $n-5\in C_{2}$, hence $C_{1}=\{1,2,\ldots \}$, $C_{2}=\{n-4,n-5,\ldots \}$. Now if $3\in C_{2}$ we move 3 and $n-5$ into $C_{1}$ and apply a type B transformation. Otherwise $3\in C_{1}$ and if $n-6\in C_{1}$, we add $n-6$ and 3 to $C_{2}$ and apply a type A transformation; otherwise $C_{1}=\{1,2,3,\ldots \}$ and $C_{2}=\{n-4,n-5,n-6,\ldots \}$ and so on. Note that a transformation of type A or B can be applied to every partition $\pi =C_{1}\cup C_{2}$ of $\{1,\ldots ,n-4\}$ since otherwise $\pi$ must have classes $C_{1}=\{1,2,3,\ldots \}$ and $C_{2}=\{n-4,n-5,n-6,\ldots \}$ such that for every $k\in C_{1}$ verifying $1\leq k\leq (n-4)/2$, the number $n-k-3$ belongs to $C_{2}$. But this contradicts the property that $C_{1}$ and $C_{2}$ have the same sum for every $n\geq 8$. If $1\in C_{2}$ this algorithm runs similarly and all partitions generated in this way are pairwise distinct. \medskip (f) Suppose $3\in C_{1}$. If $n-4\in C_{1}$, we move 3 and $n-4$ into $C_{2}$ and annihilate the imbalance equal to $2n-2$ by defining ${\tt C_{1}'}={\tt C_{1}''}\cup \{n,n-1,n-3\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n-2\}$ (a type C transformation). Otherwise $C_{1}=\{3,\ldots \}$, $C_{2}=\{n-4,\ldots \}$. If $4\in C_{2}$ we move 4 and $n-4$ into $C_{1}$ which produces an imbalance equal to $2n$; then define ${\tt C_{1}'}={\tt C_{1}''}\cup \{n-3\}$ and ${\tt C_{2}'}={\tt C_{2}''}\cup \{n,n-1,n-2\}$ (a type D transformation). Otherwise $C_{1}=\{3,4,\ldots \}$ and $C_{2}= \{n-4,\ldots \}$. If $n-5\in C_{1}$ we move 4 and $n-5$ into $C_{2}$ and apply a type C transformation; otherwise $C_{1}=\{3,4,\ldots \}$ and $C_{2}=\{ n-4,n-5,\ldots \}$. In this way we can apply a transformation of type C or D to every partition $\pi $ of $\{1,\ldots ,n-4\}$ since otherwise $C_{1}=\{3,4,5,\ldots \}$, $C_{2}=\{n-4,n-5,n-6,\ldots\}$ such that for every $k\in C_{1}$, $3\leq k\leq (n-2)/2$, we have $n-k-1\in C_{2}$. This is a contradiction, since in this case $C_{1}$ and $C_{2}$ cannot have the same sum for every $n\geq 12$. As in the previous cases all partitions produced in this way are distinct. \end{proof} This theorem has the following consequence: \begin{cor} We have \begin{equation} S(n)>6^{n/4}\approx 1.56508^{n} \end{equation} for every $n\equiv 0$ $ or$ $3$ $ (mod\, 4)$ and $n\geq 16$. \end{cor} \begin{proof} If $n=4k$ one gets $S(4k)\geq 6^{n/4-4}S(16)>6^{n/4}$ since $S(16)=1,314$. Similarly, $S(4k+3)\geq 6^{k-3}S(15)=6^{(n-15)/4}S(15)>6^{n/4}$ because $S(15)=722$. \end{proof} Note that in \cite{friedm-keith} the maximum coefficient in the polynomial $G_{n}(x)$, which coincides with $S(n)$ for $n\equiv $ 0 or 3 (mod 4), is bounded below by $2(n+1)$ for every $n\geq 10$. Although the lower bound (5) is exponential, its order of magnitude is far from being exact, as can be seen below. \begin{lemma} \begin{equation} \lim_{n\rightarrow \infty}S(4n)^{1/(4n)}=\lim_{n\rightarrow \infty} S(4n+3)^{1/(4n+3)}=2. \end{equation} \end{lemma} \begin{proof} Since the sequence of coefficients $(G(n,i))_{i=0,\ldots ,T_{n}}$ in $G_{n}(x)$ is unimodal (\cite{pro,sloane}) and symmetric, and the first and last coefficient are equal to 1, it follows that for every $n\geq 5$, $n\equiv 0$ or 3 (mod 4), $$S(n)>\frac{2^{n}-2}{T_{n}-1}>\frac{2^{n}}{T_{n}}= \frac{2^{n+1}}{n^{2}+n}.$$ Indeed, $\sum_{i=0}^{T_{n}}G(n,i)=G_{n}(1)=2^{n}$ and $T_{n}<2^{n-1}$ for every $n\geq 5$. On the other hand, $S(n)<2^{n}-2$, the number of ordered partitions having two classes of $\{1,\ldots ,n\}$, and these two inequalities imply (6). \end{proof} A better upper bound for $S(n)$ is $\COM{n}{\lfloor n/2\rfloor }\leq C_{1}\frac{2^{n}}{\sqrt{n}}$ for some constant $C_{1}>0$. This follows from the following particular case of a result of Erd\H os (see \cite{aigner} or \cite{erdos}): Fix an interval of length $2$ and consider the set of combinations $\sum_{i=1}^{n}\varepsilon _{i}i$, that lie within the interval, where $\varepsilon _{i}\in \{1,-1\}$ for every $1\leq i\leq n$. The sets $\{i: \varepsilon _{i}=1\}$ that correspond to these combinations form an antichain in the poset of subsets of $\{1,\ldots ,n\}$ ordered by inclusion. By Sperner's theorem \cite{sper} the maximum number of elements in such an antichain is $\COM{n}{\lfloor n/2\rfloor}$, which is an upper bound for the number of combinations $\sum _{i=1}^{n}\varepsilon _{i}i$ that sum to 0. \bigskip \noindent {\bf Conjecture 2.6} {\em For} $n\equiv 0$ or 3 (mod 4) {\em we have} $$S(n)\sim \sqrt{6/\pi}\cdot \frac{2^{n}}{n\sqrt{n}}\, , $$ where $f(n)\sim g(n)$ means that $\lim_{n\rightarrow \infty}f(n)/g(n)=1$.\\ \bigskip This behavior was verified by computer experiments up to $n=100$. \section{Acknowledgements} The authors are grateful to J. Radcliffe from University of Nebraska (Lincoln) for useful discussions related to the upper bound for $S(n)$. Also, the authors are indebted to the referee of the paper for his/her very useful remarks, including the present form of Conjecture 2.6. \begin{thebibliography}{99} \bibitem{aigner} %1 M. Aigner and G.-M. Ziegler, {\em Proofs from THE BOOK}, Springer Verlag, Berlin, Heidelberg, 1998. \bibitem{andr-buz} %2 D. Andrica and \c S. Buze\c teanu, On the product of two or more derivatives, {\em Revue Roumaine Math.\ Pures Appl.}, {\bf 30} (1985), 703--710. \bibitem{andr} %3 D. Andrica, A combinatorial result concerning the product of more derivatives, {\em Bull.\ Calcutta Math.\ Soc.}, {\bf 92} (4) (2000), 299--304. \bibitem{erdos} %4 P. Erd\H os, On a lemma of Littlewood and Offord, {\em Bull.\ Amer.\ Math.\ Soc.}, {\bf 5} (1945), 898--902. \bibitem{friedm-keith} %5 E. Friedman and M. Keith, \htmladdnormallink{Magic carpets}{http://www.math.uwaterloo.ca/JIS/VOL3/KEITH/carpet.html}, \htmladdnormallink{\em Journal of Integer Sequences}{http://www.math.uwaterloo.ca/JIS}, {\bf 3} (2000), Article 00.2.5. \bibitem{pro} %6 R. Proctor, Solution of two difficult combinatorial problems with linear algebra, {\em Amer.\ Math.\ Monthly}, {\bf 89} (1982), 721--734. \bibitem{sloane} %7 N. J. A. Sloane \newblock{\em\htmladdnormallink{The On-Line Encyclopedia of Integer Sequences}{http://www.research.att.com/~njas/sequences/}}, {\tt {http://www.research.att.com/\char'176njas/sequences/}}. \bibitem{sper} %8 E. Sperner, Ein Satz \"uber Untermengen einer endlichen Menge, {\em Math.\ Zeitschrift}, {\bf 27} (1928), 544--548. \bibitem{stan} %9 R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, {\em SIAM J. Alg.\ Discr.\ Math.}, {\bf 1} (1980), 164--184. \bibitem{wilf} %10 H. Wilf, \htmladdnormallink{\em Generatingfunctionology}{http://www.math.upenn.edu/~wilf/DownldGF.html}, Academic Press, New York, 1994. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: 05A15, 05A16, 05A17, 05A18, 06A07, 11B75 .\ \ \noindent \emph{Keywords: unimodal polynomial, triangular number, derivative, partition, Sperner's theorem, generating function} \bigskip \hrule \bigskip \noindent (Concerned with sequence \seqnum{A025591}.) \vspace*{+.1in} \noindent Received September 25, 2002; revised version received November 3, 2002. Published in {\it Journal of Integer Sequences} November 14, 2002. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \end{document}