\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \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} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand\HH{\mathcal H} \newcommand\card{\mathrm{card}} \newcommand\tr{\mathrm{tr}} \newcommand\Id{\mathrm{Id}} \newcommand\ZZ{\mathbb{Z}} \newcommand\CC{\mathbb{C}} \newcommand\FF{\mathbb{F}} \newcommand\Pg{\overline{P}} \newcommand\TT{\overline{T}} \newcommand\IS{\mathrm{IS}} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\rm \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 Pairs of Intertwined Integer Sequences } \vskip 1cm \large Christian Kassel\\ Institut de Recherche Math\'e\-ma\-tique Avanc\'ee\\ Universit\'e de Strasbourg \& CNRS \\ 7~rue Ren\'e-Descartes\\ 67084 Strasbourg\\ France\\ \href{mailto:kassel@math.unistra.fr}{\tt kassel@math.unistra.fr} \\ \ \\ Christophe Reutenauer\\ Math\'ematiques \\ Universit\'e du Qu\'ebec \`a Mont\-r\'eal\\ CP 8888, succ.\ centre-ville\\ Montr\'eal, PQ H3C 3P8 \\ Canada \\ \href{mailto:reutenauer.christophe@uqam.ca}{\tt reutenauer.christophe@uqam.ca} \\ \end{center} \vskip .2 in \begin{abstract} In previous work we computed the number~$C_n(q)$ of ideals of codimension~$n$ of the algebra $\FF_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field~$\FF_q$: it turned out that $C_n(q)$ is a palindromic polynomial of degree~$2n$ in~$q$, divisible by~$(q-1)^2$. The quotient $P_n(q) = C_n(q)/ (q-1)^2$ is a palindromic polynomial of degree~$2n-2$. For each $n\geq 1$ let~$\Pg_n(X) \in \ZZ[X]$ be the degree~$n-1$ polynomial such that $\Pg_n(q+q^{-1}) = P_n(q)/q^{n-1}$. In this article we show that for every integer~$N$ the integer value $\Pg_n(N)$ is close to the value at~$N$ of the degree~$n-1$ polynomial $F_{n-1}(X) = 1 + \sum_{k=1}^{n-1} \, \TT_k(X)$, which is a sum of monic versions $\TT_k(X)$ of Chebyshev polynomials of the first kind. We give a precise formula for~$\Pg_n(X)$ as a linear combination of the $F_k(X)$, each appearance of the latter being indexed by an odd divisor of~$n$. As a consequence, $\Pg_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of~$2$. We exhibit similar formulas for~$C_n(q)$. \end{abstract} \section{Introduction}\label{sec-intro} In this article, for each integer~$N$ we produce a pair $(\Pg_n(N), (F_{n-1}(N))_{n\geq 1}$ of integer sequences. Both sequences, though of differing origin, consist of amazingly close elements. Some of them do even coincide: we have $\Pg_n(N) = F_{n-1}(N)$ whenever $n$ is a power of~$2$. The reader is invited to examine Table~\ref{table-values} which exhibits such pairs for $N = 3,4,5$. In this table coinciding values have been set in boldface and values differing by~$1$ in italics. \begin{table}[ht] \renewcommand\arraystretch{1.30} \noindent\[ \centering \begin{array}{|c||c|c||c|c||c|c||} \hline n & \Pg_n(3) & F_{n-1}(3) & \Pg_n(4) & F_{n-1}(4) & \Pg_n(5) & F_{n-1}(5) \\ \hline\hline {1} & \textbf{1} & \textbf{1} &\textbf{1} & \textbf{1} & \textbf{1} & \textbf{1}\\ \hline {2} & \textbf{4} & \textbf{4} & \textbf{5} & \textbf{5} & \textbf{6} & \textbf{6} \\ \hline 3 & \textit{10} & \textit{11} & \textit{18} & \textit{19} & \textit{28} & \textit{29} \\ \hline {4} & \textbf{29} & \textbf{29} & \textbf{71} & \textbf{71} & \textbf{139} & \textbf{139} \\ \hline 5 & 72 & 76 & 260 & 265 & 660 & 666\\ \hline 6 & \textit{200} & \textit{199} & \textit{990} & \textit{989} & \textit{3192} & \textit{3191} \\ \hline 7 & 510 & 521 & 3672 & 3691 & 15260 & 15289 \\ \hline {8} & \textbf{1364} & \textbf{1364} & \textbf{13775} & \textbf{13775} & \textbf{73254} & \textbf{73254} \\ \hline 9 & 3546 & 3571 & 51343 & 51409 & 350848 & 350981 \\ \hline 10 & \textit{9348} & \textit{9349} & \textit{191860} & \textit{191861} & \textit{1681650} & \textit{1681651} \\ \hline 11 & 24400 & 24476 & 715770 & 716035 & 8056608 & 8057274 \\ \hline 12 & 64090 & 64079 & 2672298 & 2672279 & 38604748 & 38604719 \\ \hline 13 & 167562 & 167761 & 9972092 & 9973081 & 184963130 & 184966321 \\ \hline 14 & 439200 & 439204 & 37220040 & 37220045 &886226880 & 886226886 \\ \hline 15 & 1149360 &1149851 & 138903480 & 138907099 &4246152960 & 4246168109 \\ \hline {16} & \textbf{3010349} & \textbf{3010349} & \textbf{518408351} & \textbf{518408351} & \textbf{20344613659} & \textbf{20344613659}\\ \hline \end{array} \] \caption{Values of $\Pg_n(N)$ and $F_{n-1}(N)$ for $N = 3,4,5$.}\label{table-values} \end{table} Let us now describe the components of such pairs. Starting with~$\Pg_n(N)$, let~$\FF_q$ be a finite field of cardinality~$q$ and $\FF_q[x,y,x^{-1}, y^{-1}]$ be the algebra of two-variable Laurent polynomials with coefficients in~$\FF_q$. For every integer $n\geq 1$, let $C_n(q)$ denote the number of ideals~$I$ of codimension~$n$ of~$\FF_q[x,y,x^{-1}, y^{-1}]$, i.e., such that the quotient $\FF_q[x,y,x^{-1}, y^{-1}]/I$ is a $\FF_q$-vector space of dimension~$n$. (For algebraic geometers, $C_n(q)$ counts the $\FF_q$-points of the Hil\-bert scheme of $n$~points on the two-dimensional torus.) The authors\ \cite{KR2} showed that $C_n(q)$ is a monic polynomial of degree~$2n$ with integer coefficients in the variable~$q$; moreover, $C_n(q)$ is palindromic and divisible by~$(q-1)^2$. They~\cite[Th.\ 1.1]{KR3} gave explicit formulas for the coefficients of~$C_n(q)$; also see \S~\ref{ssec-Cnq} below. The polynomial $C_n(q)$ being divisible by~$(q-1)^2$, we may consider the degree $2n-2$ polynomial~$P_n(q) = C_n(q)/(q-1)^2$: it is monic, palindromic and has integer coefficients. One of the most striking features of~$P_n(q)$ is that its coefficients are all \emph{nonnegative}; actually, each coefficient of~$P_n(q)$ can be expressed as the number of divisors of~$n$ in a certain real interval; see~\cite[Thm.\ 1.3]{KR3} and \S~\ref{ssec-approx} below. The authors~\cite[Thm.\ 1.6]{KR3} also computed the values taken by $P_n(q)$ when $q$ is a complex root of unity of order $1,2,3,4$ or~$6$, equivalently when $q + q^{-1}$ takes one of the respective values $2, -2, -1, 0, 1$, thus recovering well-known arithmetical functions. For instance, $P_n(1) = \sigma(n)$ is equal to the sum of divisors of~$n$ and $P_n(-1) = r(n)/4$, where $r(n)$ is the number of representations of~$n$ as a sum of squares of two integers. More generally, for every integer $n \geq 1$, the Laurent polynomial ${P_n(q)}/{q^{n-1}}$ takes integer values whenever $q + q^{-1} = N$ is an integer. Computing such values is the main objective of this paper. For simplicity we replace the Laurent polynomial $P_n(q)/q^{n-1}$ in the variable~$q$ by the degree $n-1$ polynomial~$\Pg_n(X)$ in the variable~$X$, defined by \begin{equation}\label{Pg-P} \Pg_n(q + q^{-1}) = P_n(q)/q^{n-1} . \end{equation} (This is possible since $P_n(q)$ is palindromic of degree~$2n-2$.) We have thus reduced our problem to the computation of the values of~$\Pg_n(X)$ at integers~$N$. Searching the \emph{On-Line Encyclopedia of Integer Sequences} (OEIS)~\cite{OEIS} for the sequence \[ (\Pg_1(5), \Pg_2(5),\Pg_3(5), \Pg_4(5), \Pg_5(5), \Pg_6(5)) = (1,6,28, 139,660, 3192) \] yielded, as of January 7 2025, what the OEIS called an ``approximate match'', namely the sequence of integers consisting of the first few coefficients of the power series expansion of the rational function \[ \frac{1+t}{1 - 5t + t^2} \, . \] Replacing $5$ by~$3$ and by~$4$ in this rational function has produced similar approximate matches for $\Pg_n(3)$ and $\Pg_n(4)$ respectively. In view of this, it was natural for us to replace $3,4,5$ by an indeterminate~$X$ as follows. The second integer sequence mentioned above consists of integer values at~$N$ of the polynomials~$F_k(X)$ defined by the following equality of formal power series: \begin{equation}\label{gf-F} \sum_{k\geq 0} \, F_k(X) \, t^k = \frac{1+t}{1 - Xt + t^2} \, . \end{equation} It turns out that each polynomial $F_k(X)$ is of degree~$k-1$, has integer coefficients, and we have \begin{equation}\label{eq-FFTT} F_k(X) = 1 + \sum_{m=1}^{k} \, \TT_m(X) , \end{equation} which is the sum of the monic versions~$\TT_m(X)$ of the Chebyshev polynomials of the first kind defined in \S~\ref{ssec-Tchb} below. With this notation we establish that each~$\Pg_n(X)$ can be approximated by~$F_{n-1}(X)$ in the sense that the difference between these two monic polynomials, both of degree~$n-1$, is a polynomial of much lower degree (see Theorem~\ref{th-approx} for a precise statement). More interestingly, $\Pg_n(X)$ can be expressed as the following linear combination of~$F_k(X)$, whose terms are indexed by the odd divisors $d$ of~$n$: \begin{equation}\label{formula-PFodd} \Pg_n(X) = \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) \geq 0} \, F_{r_n(d)}(X) - \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) <0} \, F_{-r_n(d) -1}(X) , \end{equation} where $r_n(d) = {n/d} - {(d+1)/2}$; see Theorem~\ref{th-PFodd}. For $d=1$ we have $r_n(1) = n-1$, which implies the presence of~$F_{n-1}(X)$ in~\eqref{formula-PFodd} for all $n\geq 1$. Since the powers of~$2$ are the only integers with a unique odd divisor, namely $d=1$, we have $\Pg_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of~$2$. This explains the coincidences observed in Table~\ref{table-values}. We deduce Formula~\eqref{formula-PFodd} from the following new expression for~$C_n(q)$: \begin{equation}\label{formula-Cnodd} C_n(q) = q^n \, \sum_{d | n , \, d \, \mathrm{odd}} \, \left( q^{\frac{n}{d} - \frac{d-1}{2}} + q^{-\frac{n}{d} + \frac{d-1}{2}} - q^{\frac{n}{d} - \frac{d+1}{2}} - q^{-\frac{n}{d} + \frac{d+1}{2}} \right) . \end{equation} See Theorem~\ref{th-Cn}. Formula~\eqref{formula-Cnodd} allows us to express the local zeta function and the Hasse--Weil zeta function of the Hilbert scheme of $n$~points on the two-dimensional torus as products whose factors are indexed by the odd divisors of~$n$ (see \S~\ref{ssec-zeta}). Let us detail the content of each section. In Section~\ref{sec-Pn}, for each $n\geq 1$ we define the degree~$n-1$ polynomial~$\Pg_n(X)$ by~\eqref{Pg-P}. As mentioned above, the integers of interest to us are the values~$\Pg_n(N)$ at arbitrary integers~$N$. Building on previous work~\cite{KR3}, we show that the function $n \mapsto |\Pg_n(N)|$ is multiplicative if $N= -2, -1, 0$ or~$2$; for all remaining integer values of~$N$ it is not multiplicative. Section~\ref{sec-aux} is devoted to the monic versions $\TT_k(X)$ of the Chebyshev polynomials of the first kind and to the polynomials~$F_k(X)$ defined above. In Section~\ref{sec-resultsP} we state the results on~$\Pg_n(X)$ mentioned above. Theorem~\ref{th-approx} and Theorem~\ref{th-PFodd} with Formula~\eqref{formula-PFodd} are the main results. As special cases of~\eqref{formula-PFodd} we obtain formulas for~$P_n(X)$ in terms of the polynomials~$F_k(X)$ for various families of integers~$n$. Section~\ref{sec-Cnq} is devoted to the polynomials~$C_n(q)$: we prove~\eqref{formula-Cnodd} and state the formulas for the zeta functions mentioned above. In Section~\ref{pf-th-PFodd} we use the results of Section~\ref{sec-Cnq} to prove Theorem~\ref{th-PFodd}. \section{The polynomials $\Pg_n(X)$}\label{sec-Pn} \subsection{Definition and basic properties}\label{ssec-defPn} For every positive integer~$n$ we define the degree $n-1$ polynomial $\Pg_n(X)$ by \begin{equation}\label{def-Pgn} \Pg_n(q + q^{-1}) = P_n(q)/q^{n-1} , \end{equation} where $P_n(q)$ was introduced in Section~\ref{sec-intro}. This makes sense since $P_n(q)$ is a palindromic polynomial of degree~$2n-2$. As $P_n(q)$ has integer coefficients, so does~$\Pg_n(X)$. As a consequence of~\cite[Cor.\ 1.4]{KR2}, we have the equality of formal power series \begin{equation*}\label{gf-Pnq} 1 + (q + q^{-1} - 2) \sum_{n\geq 1} \, \frac{P_n(q)}{q^{n-1}} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{equation*} This translates into the following equality for the polynomials~$\Pg_n (X)$: \begin{equation}\label{gf-Pgn} 1 + (X - 2) \sum_{n\geq 1} \, \Pg_n (X) \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1- X t^i + t^{2i}} \, . \end{equation} The authors~\cite[Thm.\ 1.3]{KR3} established that each polynomial~$P_n(q)$ is of the form \begin{equation}\label{Pn-coeff} P_n(q) = a_{n,0}\, q^{n-1} + \sum_{i=1}^{n-1} \, a_{n,i} \, \left( q^{n+i-1} + q^{n-i-1} \right) , \end{equation} where the coefficients~$a_{n,i}$ are all \emph{nonnegative integers} (to be expressed explicitly in the proof of Lemma~\ref{lem-ani}). It follows that for each positive integer~$n$, \begin{equation}\label{PPTT} \Pg_n(X) = a_{n,0} + \sum_{i=1}^{n-1} \, a_{n,i} \, \TT_i(X) , \end{equation} where $\TT_i(X)$ is the monic Chebyshev polynomial of degree~$i$ defined by~\eqref{defTT} below. Since the coefficients~$a_{n,i}$ are nonnegative integers, each~$\Pg_n(X)$ is a \emph{sum} of the polynomials~$\TT_k(X)$. A list of the polynomials $\Pg_n(X)$ for $1 \leq n \leq 12$ is given in Table~\ref{tablePP}. It is based on the corresponding list of $P_n(q)$ in Table~2 of either~\cite{KR2} or~\cite{KR3}. \begin{table}[H] \renewcommand\arraystretch{1.25} \noindent\[ \centering \begin{array}{|c||c|} \hline n & \Pg_n(X) \\ \hline\hline 1 & 1 \\ \hline 2 & X + 1 \\ \hline 3 & X^2 + X - 2 \\ \hline 4 & X^3 + X^2 - 2X -1 \\ \hline 5 & X^4 + X^3 - 3 X^2 - 3X \\ \hline 6 & X^5 + X^4 - 4X^3 - 3 X^2 + 3X + 2 \\ \hline 7 & X^6 + X^5 - 5 X^4 -4 X^3 + 5X^2 + 2X \\ \hline 8 & X^7 +X^6 - 6X^5 - 5 X^4 + 10 X^3 + 6 X^2 - 4X -1 \\ \hline 9 & X^8 + X^7 - 7 X^6 - 6 X^5 +15 X^4 + 9 X^3 - 11 X^2 - X +3 \\ \hline 10 & X^9 + X^8 -8 X^7 -7 X^6 + 21 X^5 + 15 X^4 - 20 X^3 - 10 X^2 + 5X \\ \hline & X^{10} + X^9 - 9 X^8 - 8 X^7 + 28 X^6 + 21 X^5 \\ 11 & - 36 X^4 - 21 X^3 + 18 X^2 + 7X - 2 \\ \hline & X^{11} + X^{10} - 10 X^9 - 9 X^8 + 36 X^7 + 28 X^6 \\ 12 & - 56 X^5 -35 X^4 + 35 X^3 + 16 X^2 - 5 X -2 \\ \hline \end{array} \] \caption{The polynomials $\Pg_n (X)$.}\label{tablePP} \end{table} The following result explains why all entries in Table~\ref{table-values} are positive. \begin{proposition} Let $N$ be an integer $\geq 2$. Then $\Pg_n(N) > 0$ for all $n \geq 1$. \end{proposition} \begin{proof} This is a consequence of Proposition~\ref{TTpos} below and of the fact mentioned above that each $\Pg_n(X)$ is a sum of the $\TT_k(X)$. \end{proof} \subsection{Multiplicative values of $|P_n(q)|$ at roots of unity}\label{ssec-mult} The content of this subsection is an addendum to \cite[Th.\ 1.6]{KR3}. It concerns the cases when $q + q^{-1} = N$ is an integer such that $-2 \leq N \leq 2$, equivalently when $q$ is a root of unity of order $d = 1, 2, 3, 4$ or~$6$. \begin{proposition}\label{prop-mult} Let $q$ be a complex root of unity of order $d = 1, 2, 3$ or $4$. Then the sequence of nonnegative integers \begin{equation*} \left |P_n(q) \right | = \left | P_n(q)/q^{n-1} \right | = \left |\Pg_n(q + q^{-1}) \right | \end{equation*} is multiplicative as a function of~$n$. \end{proposition} Recall that a function $f(n)$ defined on positive integers is \emph{multiplicative} if $f(mn) = f(m) f(n)$ whenever $m$ and $n$ are coprime. \begin{proof} When $q = 1$, by \cite[Cor.\ 1.2]{KR2} we have $\Pg_n(2) = P_n(1) = \sigma(n)$, the sum of divisors of~$n$, which is well-known to be multiplicative. See Apostol \cite[Sect.\ 2.13]{Ap} and the OEIS~\cite{OEIS} at \seqnum{A000203}. For $q = -1$ we have $\Pg_n(-2) = P_n(-1) = r(n)/4$, where $r(n)$ is the number of representations of~$n$ as a sum of squares of two integers. By the OEIS~\cite[\seqnum{A002654}]{OEIS} the sequence $r(n)/4$ is multiplicative. Then so is~$P_n(-1)$. For $q = j$, a root of unity of order~$3$, we have $\Pg_n(-1) = P_n(j)/j^{n-1} = \lambda(n)$, where $\lambda(n)$ is the sequence \seqnum{A113063} in the OEIS~\cite{OEIS}. Fine \cite[Sect.\ 32, p.\ 79]{Fi} states that $\lambda(n)$ is multiplicative. See also Caballero~\cite{RC2}. Let $i$ be a square root of~$-1$. It follows again from~\cite[Th.\ 1.6]{KR3} that we have \[ |\Pg_n(0)| = |P_n(i)/i^{n-1}| = r'(n)/2, \] where $r'(n)$ is the number of representations of~$n$ as a sum of a square and twice another square. The sequence of integers $r'(n)/2$ listed as \seqnum{A002325} in the OEIS~\cite{OEIS} is known to be multiplicative. \end{proof} The case when $q = \omega$ is a root of unity of order~$6$ is slightly more involved. \begin{proposition}\label{prop-mult1} Let $\omega$ be a root of unity of order~$6$. Then for all pairs $(m,n)$ of coprime positive integers we have \begin{equation*} |P_m(\omega)| |P_n(\omega)| = |P_{mn}(\omega)| \; \text{when} \; (m,n) \equiv (0,1), (1,1) \; \text{or} \; (1,2) \hskip -10pt \pmod{3}, \end{equation*} \begin{equation*} |P_m(\omega)| |P_n(\omega)| = 2 \, |P_{mn}(\omega)| \quad \text{when} \;\; (m,n) \equiv (0,2) \pmod{3}, \end{equation*} \begin{equation*} |P_m(\omega)| |P_n(\omega)| = 4 \, |P_{mn}(\omega)| \quad \text{when} \;\; (m,n) \equiv (2,2) \pmod{3}. \end{equation*} \end{proposition} \begin{proof} This is a consequence of the multiplicativity of the sequence~$r(n)/4$ observed above and of the following reformulation of a result due to the authors~\cite[Th.\ 1.1]{KR1} and~\cite[Th.\ 1.6\,(d)]{KR3}: \begin{equation*} |P_n(\omega)| = \begin{cases} r(n), & \text{if $n \equiv 0 \pmod{3}$;} \\ r(n)/4, & \text{if $n \equiv 1 \pmod{3}$;} \\ r(n)/2, & \text{if $n \equiv 2 \pmod{3}$.} \end{cases} \end{equation*} \end{proof} We have the following consequence of Propositions\,\ref{prop-mult} and\,\ref{prop-mult1}. \begin{corollary} Given a pair $(m,n)$ of coprime positive integers, the polynomial $\Pg_{mn}(X)^2 - \Pg_m(X)^2 \Pg_n(X)^2$ is divisible by $X(X+1)(X^2-4)$, and moreover by $X-1$ if $(m,n) \equiv (0,1), (1,1)$ or $(1,2) \pmod{3}$. \end{corollary} We may wonder whether there are other integers $N$ such that $n \mapsto |\Pg_n(N)|$ is a multiplicative function. The answer is no. \begin{proposition} The function $n \mapsto |\Pg_n(N)|$ is multiplicative if and only if $N= -2, -1, 0$ or~$2$. \end{proposition} \begin{proof} The ``if direction'' is a reformulation of Proposition~\ref{prop-mult}. Conversely, if $n \mapsto |\Pg_n(N)|$ is multiplicative, then we have $|\Pg_6(N)| = |\Pg_2(N)| |\Pg_3(N)|$ as a special case. Now the polynomial $\Pg_6(X)^2 - \Pg_2(X)^2 \, \Pg_3(X)^2$ can be factored as \begin{equation*} \Pg_6(X)^2 - \Pg_2(X)^2 \, \Pg_3(X)^2 = X(X-1)^3 (X+1)^3 (X-2)(X+2)^2 . \end{equation*} This implies that multiplicativity holds only for integers~$N$ such that $|N| \leq 2$. We rule out the case $N = 1$ in view of Proposition~\ref{prop-mult1}. \end{proof} \section{The polynomials $F_n(X)$}\label{sec-aux} Let us define the monic Chebyshev polynomials and the polynomials~$F_n(X)$ mentioned in the introduction. \subsection{Monic Chebyshev polynomials}\label{ssec-Tchb} Given an integer $k\geq 0$, the \emph{standard Chebyshev polynomial of the first kind}~$T_k(X)$ is the degree~$k$ polynomial defined by \begin{equation*} \cos(k \theta) = T_k(\cos (\theta)) , \end{equation*} equivalently by \begin{equation*} \frac{q^k + q^{-k}}{2} = T_k \left( \frac{q + q^{-1}}{2} \right) . \end{equation*} Rivlin~\cite{Ri} gave a useful overview of the important properties of Chebyshev polynomials. We are interested in the following variant of~$T_k(X)$: let $\TT_k(X)$ be the degree~$k$ polynomial defined by \begin{equation}\label{defTT} q^k + q^{-k} = \TT_k (q + q^{-1}). \end{equation} Both $T_k (X)$ and~$\TT_k(X)$ have integer coefficients. They are clearly related by \begin{equation}\label{eq-TTT} \TT_k (X) = 2 T_k(X/2) . \end{equation} The polynomials $\TT_k (X)$ are monic for all $k\geq 1$. That is why we call them \emph{monic Chebyshev polynomials}. Abramowitz et al.\ \cite[p.\ 778, Eq.~22.5.11]{AS} denote the polynomials~$\TT_k (X)$ by~$C_k$; Horadam~\cite{Ho} calls them \emph{Vieta--Lucas polynomials} and denotes them by~$v_n(x)$. Table\,\ref{tableTT} lists the polynomials~$\TT_k (X)$ for $1 \leq k \leq 12$. \begin{table}[H] \renewcommand\arraystretch{1.25} \noindent\[ \centering \begin{array}{|c||c|} \hline k & \TT_k(X) \\ \hline\hline 0 & 2 \\ \hline 1 & X \\ \hline 2 & X^2 - 2 \\ \hline 3 & X^3 - 3X \\ \hline 4 & X^4 - 4X^2 +2 \\ \hline 5 & X^5 - 5X^3 + 5X \\ \hline 6 & X^6 - 6X^4 + 9 X^2 - 2 \\ \hline 7 & X^7 - 7 X^5 + 14 X^3 - 7X \\ \hline 8 & X^8 - 8 X^6 + 20 X^4 -16 X^2 +2 \\ \hline 9 & X^9 - 9 X^7 + 27 X^5 - 30 X^3 + 9 X \\ \hline 10 & X^{10} - 10 X^8 + 35 X^6 - 50 X^4 + 25 X^2 - 2 \\ \hline 11 & X^{11} - 11 X^9 + 44 X^7 - 77 X^5 + 55 X^3 - 11X \\ \hline 12 & X^{12} - 12 X^{10} + 54 X^8 - 112 X^6 + 105 X^4 - 36 X^2 +2 \\ \hline \end{array} \] \caption{The monic Chebyshev polynomials $\TT_k (X)$.}\label{tableTT} \end{table} Let us state a few properties of the polynomials~$\TT_k (X)$ which can be easily derived from well-known properties of~$T_k (X)$. First, the polynomials~$\TT_k (X)$ can be defined inductively by $\TT_0 (X) = 2$, $\TT_1 (X) = X$, and for $k\geq 1$ by \begin{equation}\label{rec-TT} \TT_{k+1}(X) = X\TT_k(X) - \TT_{k-1}(X) . \end{equation} The previous linear recurrence relation can be written in matrix form as \begin{equation}\label{matrix-TT} \begin{pmatrix} \TT_{k+1}(X) \\ \TT_k(X) \end{pmatrix} = M \begin{pmatrix} \TT_k(X) \\ \TT_{k-1}(X) \end{pmatrix}, \;\; \text{where} \; M = \begin{pmatrix} X & - 1\\ 1 & 0 \end{pmatrix} . \end{equation} Next, the generating function of the polynomials~$\TT_k (X)$ is the power series expansion of the following rational function: \begin{equation}\label{genfun-TT} \sum_{k \geq 0} \, \TT_k (X) \, t^k = \frac{2 - Xt}{1 - Xt + t^2} . \end{equation} We have the following expression of~$\TT_n(X)$ in terms of powers of~$X$: \begin{equation*}\label{eq-TX} \TT_n(X) = \sum_{m=0}^{[n/2]} \, (-1)^m \left( \binom{n-m}{m} - \binom{n-m-1}{m-1} \right) X^{n-2m} . \end{equation*} We also have a positivity result for the polynomials $\TT_k(X)$. \begin{proposition}\label{TTpos} For each $k \geq 0$, we have $\TT_k (x) > 0$ if $x$ is a real number $\geq 2$. \end{proposition} \begin{proof} On one hand, as is well known, the zeros of the standard Chebyshev polynomials~$T_k(X)$ all lie in the open interval $(-1,1)$. Hence, in view of~\eqref{eq-TTT} the zeros of~$\TT_k(X)$ lie in $(-2,2)$. On the other, since $\TT_k(X)$ is monic, $\lim_{x \to + \infty} \, \TT_k(x) = + \infty$. Combining these two facts implies the desired positivity. \end{proof} We close our summary on~$\TT_k (X)$ with the following result, which may be of independent interest. \begin{proposition}\label{prop-recTT} For all $k\geq 0$ we have $\TT_k (X) = \tr(M^k)$, where $M$ is the $2 \times 2$-matrix defined in \eqref{matrix-TT}. \end{proposition} \begin{proof} First, observe that $\tr(M) = X = \TT_1(X)$ and $\tr(M^0) = \tr(\Id) = 2 = \TT_0(X)$, where $\Id$ is the identity matrix. In order to complete the proof of the desired equality, it suffices to show that the sequence $(\tr(M^k))_{k\geq 0}$ satisfies the linear recurrence relation~\eqref{rec-TT}. By the Cayley--Hamilton theorem applied to the matrix~$M$, we have \[ M^2 = \tr(M) M - \det(M) \Id = XM - \Id . \] Now multiply both sides by~$M^{k-1}$ and take traces. \end{proof} \subsection{Definition and basic properties of $F_n(X)$}\label{ssec-F} We define the sequence of polynomials $(F_n(X))_{n\geq 0}$ inductively by $F_0(X) = 1$ and by \begin{equation}\label{def-F} F_n(X) = F_{n-1}(X) + \TT_n(X) \end{equation} for $n\geq 1$. In other words, \begin{equation}\label{def-F2} F_n(X) = 1 + \sum_{k=1}^n \, \TT_k(X) . \end{equation} It is clear that each $F_n(X) $ has integer coefficients and is monic of degree~$n$. We give a list of the polynomials $F_n(X)$ for $n \leq 11$ in Table~\ref{table-F}. \begin{table}[ht] \renewcommand\arraystretch{1.25} \noindent\[ \centering \begin{array}{|c||c|} \hline n & F_n(X) \\ \hline\hline 0 & 1 \\ \hline 1 & X + 1 \\ \hline 2 & X^2 + X - 1 \\ \hline 3 & X^3 + X^2 - 2X -1 \\ \hline 4 & X^4 + X^3 - 3 X^2 - 2X +1 \\ \hline 5 & X^5 + X^4 - 4X^3 - 3 X^2 + 3X + 1 \\ \hline 6 & X^6 + X^5 - 5 X^4 -4 X^3 + 6X^2 + 3X -1 \\ \hline 7 & X^7 +X^6 - 6X^5 - 5 X^4 + 10 X^3 + 6 X^2 - 4X -1 \\ \hline 8 & X^8 + X^7 - 7 X^6 - 6 X^5 + 15 X^4 + 10 X^3 -10 X^2 -4 X +1 \\ \hline & X^9 + X^8 -8 X^7 -7 X^6 + 21 X^5 + 15 X^4 \\ 9 & - 20 X^3 - 10 X^2 + 5X + 1 \\ \hline & X^{10} + X^9 - 9 X^8 - 8 X^7 + 28 X^6 + 21 X^5 \\ 10 & - 35 X^4 - 20 X^3 + 15 X^2 + 5X -1 \\ \hline & X^{11} + X^{10} - 10 X^9 - 9 X^8 + 36 X^7 + 28 X^6 \\ 11 & - 56 X^5 -35 X^4 + 35 X^3 + 15 X^2 - 6 X - 1 \\ \hline \end{array} \] \caption{The polynomials $F_n (X)$.}\label{table-F} \end{table} Our first result is the following. \begin{proposition}\label{prop-gfF} The generating function of the polynomials $F_n(X)$ is the power series expansion of the following rational function: \[ \sum_{n \geq 0} \, F_n(X) \, t^n = \frac{1+t}{1-Xt +t^2} \, . \] \end{proposition} \begin{proof} Using~\eqref{genfun-TT}, we obtain \begin{eqnarray*} 1 + \sum_{k \geq 1} \, \TT_k (X) \, t^k & = & \frac{2 - Xt}{1 - Xt + t^2} + 1 - \TT_0(X) \\ & = & \frac{2 - Xt}{1 - Xt + t^2} - 1 = \frac{1 - t^2}{1 - Xt + t^2} \, . \end{eqnarray*} Therefore, \begin{eqnarray*} \sum_{n \geq 0} \, F_n(X) \, t^n & = & \sum_{n \geq 0} \, \left( 1 + \sum_{m=1}^n \, \TT_m(X) \right) t^n \\ & = & \left( \sum_{\ell \geq 0} \, t^{\ell} \right) \left( 1 + \sum_{k \geq 1} \, \TT_k (X) \, t^k \right) \\ & = & \frac{1}{1-t} \cdot \frac{1 - t^2}{1 - Xt + t^2} = \frac{1 + t}{1 - Xt + t^2} \, . \end{eqnarray*} \end{proof} Setting $X= 0$ in the formula of Proposition~\ref{prop-gfF} yields the \emph{constant term} of~$F_n(X)$: for all $m \geq 0$ we have \begin{equation*}\label{constant-F} F_{2m}(0) = F_{2m+1}(0) = (-1)^m . \end{equation*} For the remaining coefficients of~$F_n(X)$ we have the following expression. \begin{proposition}\label{prop-Fn} For $n\geq 1$ we have \[ F_n(X) = \sum_{0 \leq m \leq n/2} \, (-1)^m \binom{n - m}{m} \, X^{n-2m} + \sum_{0 \leq m \leq (n-1)/2} \, (-1)^m \binom{n - m - 1}{m} \, X^{n-2m-1} . \] \end{proposition} \begin{proof} Expanding the following fraction, we obtain \begin{eqnarray*} \frac{1}{1-Xt +t^2} & = & 1 + \sum_{r \geq 1} \, t^r (X - t)^r \\ & = & 1 + \sum_{r \geq 1} \sum_{s = 0}^r \, (-1)^s \binom{r}{s} X^{r-s} t^{r+s} \\ & = & 1 + \sum_{n \geq 1} \, \sum_{0 \leq m \leq n/2} \, (-1)^m \binom{n-m}{m} X^{n-2m} t^n . \end{eqnarray*} In view of Proposition~\ref{prop-gfF}, multiplying by $1+t$ yields the desired formula for~$F_n(X)$. \end{proof} \begin{corollary}\label{coro-Fn} For $n \geq 5$ we have \begin{eqnarray*} F_n(X) & = & X^n + X^{n-1} - (n-1) X^{n-2} - (n-2) X^{n-3} \\ && {} + \frac{(n-2)(n-3)}{2} X^{n-4} + \frac{(n-3)(n-4)}{2} X^{n-5} \\ && {} + {\rm \text{\rm monomials of degree}} \leq n-6 . \end{eqnarray*} \end{corollary} In Section~\ref{pf-th-PFodd} we need the following linear recurrence relation satisfied by the polynomials~$F_k(X)$. \begin{proposition}\label{prop-recF} For $k\geq 1$ we have $F_{k+1}(X) = X F_k(X) - F_{k-1}(X)$. \end{proposition} \begin{proof} The rational function expressing the generating function in Proposition~\ref{prop-gfF} has the same denominator as the rational function in~\eqref{genfun-TT}. Therefore the polynomials $F_k(X)$ satisfy the same linear recurrence relation as the polynomials $\TT_k(X)$ in~\eqref{rec-TT}; see, e.g., Berstel et~al.\ \cite[Chap.\ 6, Prop.\ 1.2 and Cor.\ 1.3]{BR}. \end{proof} \section{Results for the polynomials $\Pg_n(X)$}\label{sec-resultsP} Now we establish a connection between the polynomials $\Pg_n(X)$ and the polynomials~$F_k(X)$. We start with an approximation result. \subsection{Approximating $\Pg_n(X)$ by $F_{n-1}(X)$}\label{ssec-approx} The aim of this subsection is the following result. \begin{theorem}\label{th-approx} For all $n\geq 2$, the difference $\Pg_n(X) - F_{n-1}(X)$ is a polynomial whose degree is smaller than $n/2 - 1$. \end{theorem} Recall that $\Pg_n(X)$ and $F_{n-1}(X)$ are both of degree~$n-1$. The theorem states that they coincide in every degree $\geq n/2-1$. The bound on degrees is sharp as can be seen from Table~\ref{table-PPTTF}: indeed, $\Pg_9(X) - F_8(X) = - F_3(X) + F_1(X)$ is of degree~$3 < 9/2 - 1$; similarly, $\Pg_{11}(X) - F_{10}(X) = - F_4(X)$ is of degree~$4 < 11/2 - 1$, and $\Pg_{15}(X) - F_{14}(X)$ is of degree~$6 < 15/2 - 1$. Together with Corollary~\ref{coro-Fn} we derive the following. \begin{corollary} For all $n \geq 10$ we have \begin{eqnarray*} \Pg_n(X) & = & X^{n-1} + X^{n-2} - (n-2) X^{n-3} - (n-3) X^{n-4} \\ && {} + \frac{(n-3)(n-4)}{2} X^{n-5} + \frac{(n-4)(n-5)}{2} X^{n-6} \\ && {} + {\rm \text{\rm monomials of degree}} \leq n-7 . \end{eqnarray*} \end{corollary} Recall the (nonnegative) coefficients $a_{n,i}$ of~$\Pg_n(X)$ appearing in \eqref{Pn-coeff} and~\eqref{PPTT}. For the proof of Theorem~\ref{th-approx} we need the following result. \begin{lemma}\label{lem-ani} We have $a_{n,i} = 1$ for all $i$ such that $n/2 - 1 \leq i \leq n-1$. \end{lemma} \begin{proof} Given $0 \leq i \leq n-1$, the authors~\cite[Thm.\ 1.3]{KR3} showed that the coefficient~$a_{n,i}$ is equal to the number of divisors of~$n$ belonging to the half-open interval \[ I_{n,i} = \left( \frac{i + \sqrt{2n+ i^2}}{2}, i + \sqrt{2n+ i^2} \right] . \] It is enough to show that $n$ is the only divisor of~$n$ belonging to~$I_{n,i}$. Since all other divisor of~$n$ are $\leq n/2$, it suffices to check that $n \leq i + \sqrt{2n+ i^2}$ and $n/2 \leq (i + \sqrt{2n+ i^2})/2$; the latter inequality is equivalent to the former. They are both equivalent to $n(n-2i) = (n - i)^2 - i^2 \leq 2n$, which simplifies into $n-2i \leq 2$ hence to $i \geq (n-2)/2= n/2 - 1$. \end{proof} \begin{proof}[Proof of Theorem~\ref{th-approx}] By \eqref{def-F2} and~\eqref{PPTT} we have \begin{eqnarray*} \Pg_n(X) - F_{n-1}(X) & = & (a_{n,0} - 1) + \sum_{1 \leq i \leq n - 1} \, (a_{n,i} -1) \, \TT_i(X) \\ & = & (a_{n,0} - 1) + \sum_{1 \leq i < n/2 - 1} \, (a_{n,i} -1) \, \TT_i(X) . \end{eqnarray*} The last equality is a consequence of Lemma~\ref{lem-ani}. Since $\TT_i(X)$ is of degree~$i$, we obtain the desired result. \end{proof} \begin{remark} The sequence of integers $\Pg_n(3)$ is Sequence \seqnum{A329156} in the OEIS \cite{OEIS}. The sequence $(F_{n-1}(3))_{n\geq 1}$ is \seqnum{A002878}: we have $F_{n-1}(3) = L(2n+1)$, where $L(n)$ is the Lucas sequence defined by $L(n) = L(n-1) + L(n-2)$, $L(0) = 2$, and $L(1) = 1$. The sequence~$(\Pg_n(4))_{n\geq 1}$ is now referenced as \seqnum{A386706} in the OEIS~\cite{OEIS}. According to Kimberling's remark in the OEIS~\cite[\seqnum{A001834}]{OEIS}, the sequence $(F_{n-1}(4))_{n\geq 1}$ consist of the numerators~$p_n$ of the even-rank convergents $p_n/q_n$ to~$\sqrt{3}$; these are the convergents that are smaller than~$\sqrt{3}$. The sequence~$(\Pg_n(5))_{n\geq 1}$ is now \seqnum{A387017} and the sequence~$(F_{n-1}(5))_{n\geq 1}$ is \seqnum{A030221} in the OEIS~\cite{OEIS}. The latter consists of the values of Chebyshev even-indexed $U$-polynomials at~$\sqrt{7}/2$. \end{remark} \subsection{Odd divisors}\label{ssec-oddP} Now we give a formula for $\Pg_n(X)$ as a linear combination of the polynomials~$F_k(X)$, each appearance of~$F_k(X)$ being indexed by an odd divisor of~$n$. Recall that if we express $n$ ($\geq 1$) as the product $n = 2^a p_1^{a_1} p_2^{a_2} \cdots p_r^{a_r}$, where $p_1, p_2, \ldots, p_r$ are distinct odd primes and $a, a_1, a_2, \ldots, a_r$ are nonnegative integers, then the number of \emph{odd divisors} of~$n$ (including~$1$) is equal to \[ (a_1 + 1)(a_2 + 1) \cdots (a_r + 1) \geq 1. \] Observe that $(a_1 + 1)(a_2 + 1) \cdots (a_r + 1) = 1$ if and only if $n$ is a power of~$2$; otherwise, the number of odd divisors of~$n$ is at least~$2$. For $n\geq 1$ and every odd divisor~$d$ of~$n$, define $r_n(d)$ to be the integer \begin{equation}\label{eq-rnd} r_n(d) = \frac{n}{d} - \frac{d+1}{2} \, . \end{equation} When $d$ is the trivial divisor~$1$, we have $r_n(1) = n - 1$. Let us state our main result for $\Pg_n(X)$. \begin{theorem}\label{th-PFodd} For all $n\geq 1$, \[ \Pg_n(X) = \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) \geq 0} \, F_{r_n(d)}(X) - \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) <0} \, F_{-r_n(d) -1}(X) . \] \end{theorem} The number of terms~$F_r(X)$ in the previous formula is clearly equal to the number of odd divisors of~$n$. The proof of Theorem~\ref{th-PFodd} will be given in Section~\ref{pf-th-PFodd}. In Table~\ref{table-PPTTF} each polynomial~$\Pg_n(X)$ ($1 \leq n \leq 16$) is expressed as a sum of the polynomials~$\TT_k(X)$ as well as a linear combination of the polynomials~$F_k(X)$. In order to save space we left out the indeterminate~$X$ from the table. \begin{table}[H] \renewcommand\arraystretch{1.30} \noindent\[ \centering \begin{array}{|c||c|c|} \hline n & \text{As sums of} \, \TT_k & \text{In terms of}\, F_k \\ \hline\hline 1 & 1 & F_0 \\ \hline 2 & 1 + \TT_1 & F_1 \\ \hline 3 & \TT_1 + \TT_2 & F_2 - F_0 \\ \hline 4 & 1 + \TT_1 + \TT_2 + \TT_3 & F_3 \\ \hline 5 & \TT_2 + \TT_3 + \TT_4 & F_4 - F_1 \\ \hline 6 & \TT_0 + \TT_1 + \TT_2 + \TT_3 + \TT_4 + \TT_5 & F_5 + F_0 \\ \hline 7 & \TT_3 + \TT_4 + \TT_5 + \TT_6 & F_6 - F_2 \\ \hline 8 & 1 + \TT_1 + \TT_2 + \TT_3 + \TT_4 + \TT_5 + \TT_6 + \TT_7 & F_7 \\ \hline 9 & 1 + \TT_1 + \TT_4 + \TT_5 + \TT_6 + \TT_7 + \TT_8 & F_8 - F_3 + F_1 \\ \hline 10 & \TT_1 + \TT_2 + \TT_3 + \TT_4 + \TT_5 + \TT_6 + \TT_7 + \TT_8 + \TT_9 & F_9 - F_0 \\ \hline 11 & \TT_5 + \TT_6 + \TT_7 + \TT_8 + \TT_9 + \TT_{10} & F_{10} - F_4 \\ \hline & \TT_0 + 2 \TT_1 + 2\TT_2 + \TT_3 + \TT_4 + \TT_5 & \\ 12 & + \TT_6 + \TT_7 + \TT_8 + \TT_9 + \TT_{10} + \TT_{11} & F_{11} + F_2 \\ \hline 13 & \TT_6 + \TT_7 + \TT_8 + \TT_9 + \TT_{10} + \TT_{11} + \TT_{12} & F_{12} - F_5 \\ \hline & \TT_2 + \TT_3 + \TT_4 + \TT_5 + \TT_6 + \TT_7& \\ 14 & + \TT_8 + \TT_9 + \TT_{10} + \TT_{11} + \TT_{12} + \TT_{13} & F_{13} - F_1 \\ \hline & \TT_0 + \TT_1 + \TT_2 + \TT_3 + \TT_7 + \TT_8 & F_{14}(X) - F_6(X) \\ 15 & + \TT_9 + \TT_{10} + \TT_{11} + \TT_{12} + \TT_{13} + \TT_{14} & {}+ F_3(X) + F_0(X) \\ \hline & 1 + \TT_1 + \TT_2 + \TT_3 + \TT_4 + \TT_5 + \TT_6 + \TT_7 & \\ 16 & + \TT_8 + \TT_9 + \TT_{10} + \TT_{11} + \TT_{12} + \TT_{13} + \TT_{14} + \TT_{15} & F_{15} \\ \hline \end{array} \] \caption{Expressing $\Pg_n (X)$ in terms of $\TT_k(X)$ and of $F_k(X)$.}\label{table-PPTTF} \end{table} Since $d= 1$ is an odd divisor for every~$n\geq 1$ and $r_n(1) = n - 1 \geq 0$, the polynomial $F_{n-1}(X)$ always appears, with a positive sign, in the formula of Theorem~\ref{th-PFodd}. We remarked above that only powers of~$2$ have a unique odd divisor, namely $d=1$. This implies the following consequence of~Theorem~\ref{th-PFodd}, which explains the coincidences $\Pg_n(N) = F_{n-1}(N)$ (set in boldface) observed in Table~\ref{table-values} for $N = 3$, $4$, and $5$. \begin{corollary} We have $\Pg_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of~$2$. \end{corollary} As special cases of Theorem~\ref{th-PFodd}, we obtain the following formulas for $\Pg_n(X)$. \subsubsection{Two odd divisors} If $n = p$ is an odd prime, then it has two odd divisors, namely $1$ and $p$, so that the formula for~$\Pg_p(X)$ has two summands. We have \begin{equation*} \Pg_p(X) = F_{p-1}(X) - F_{(p-3)/2}(X) = \sum_{m=(p-1)/2}^{p-1} \, \TT_m(X) . \end{equation*} Now consider the case $n = 2^a p$, where $a \geq 1$ and $p$ is an odd prime. Such an integer~$n$ also has exactly two odd divisors. If $r_n(p) = 2^a - (p+1)/2 \geq 0$, which is equivalent to $p \leq 2^{a+1} - 1$, then \begin{equation*} \Pg_{2^a p}(X) = F_{2^a p-1}(X) + F_{2^a - (p+1)/2}(X) . \end{equation*} For instance, $\Pg_{6}(X) = F_{5}(X) + F_{0}(X)$ and $\Pg_{12}(X) = F_{11}(X) + F_{2}(X)$ If $r_n(p) = 2^a - (p+1)/2 < 0$, which is equivalent to $p > 2^{a+1} - 1$, then \begin{equation*} \Pg_{2^a p}(X) = F_{2^a p-1}(X) - F_{(p-1)/2 - 2^a}(X) . \end{equation*} As special cases, $\Pg_{10}(X) = F_{9}(X) - F_{0}(X)$ and $\Pg_{14}(X) = F_{13}(X) - F_{1}(X)$. \subsubsection{Three odd divisors} If $n = p^2$ for some odd prime~$p$, then \begin{equation*} \Pg_{p^2}(X) = F_{p^2-1}(X) - F_{(p^2 - 3)/2}(X) + F_{(p -1)/2}(X) . \end{equation*} For instance, $\Pg_9(X) = F_8(X) - F_3(X) + F_1(X)$. \subsubsection{Four odd divisors} If $n = p^3$ for some odd prime~$p$, then $n$ has four odd divisors. We have \begin{equation*} \Pg_{p^3}(X) = F_{p^3-1}(X) - F_{(p^3 - 3)/2}(X) + F_{(2p^2 - p - 1)/2}(X) - F_{(p^2 -2p -1)/2}(X) . \end{equation*} For $p=3$, we obtain $\Pg_{27}(X) = F_{26}(X) - F_{12}(X) + F_7(X) - F_1(X)$. Similarly, consider an integer of the form $n = 3p$, where $p$ is an odd prime~$\geq 7$. Then $n$ has four odd divisors, namely $1, 3, p, 3p$, and \begin{equation*} \Pg_{3p}(X) = F_{3p-1}(X) - F_{3(p-1)/2} + F_{p-2}(X) - F_{(p-7)/2} \, . \end{equation*} For $p= 5$ and $n = 15$, we have $\Pg_{15}(X) = F_{14}(X) - F_6(X) + F_3(X) + F_0(X)$. \begin{remark} In Table~\ref{table-values} we set the values $\Pg_n(N)$ and $F_{n-1}(N)$ differing by~$1$ in italics, which occurs for $n = 3$, $6$, and $10$. These quasi-coincidences between values can be explained as follows. We have $\Pg_n(X) = F_{n-1}(X) + 1 = F_{n-1}(X) + F_0(X) $ if and only if the integer~$n$ is an even \emph{perfect} number; equivalently, $n = 2^a p$, where $p = 2^{a+1} - 1$ is a \emph{Mersenne prime} such as $3, 7, 31, 127$ (see the OEIS~\cite[\seqnum{A000668}]{OEIS}). The corresponding~$n$ are $n = 6, 28, 496, 8128$. Similarly, $\Pg_n(X) = F_{n-1}(X) - 1 = F_{n-1}(X) - F_0(X) $ if and only if $n = 2^a p$, where $p = 2^{a+1} + 1$ is a \emph{Fermat prime} such as $3$, $5$, $17$, $257$ (see the OEIS~\cite[\seqnum{A000215}]{OEIS}). The corresponding~$n$ are $3$, $10$, $136$, $32896$. In general, the formula for $\Pg_n(X)$ in Theorem~\ref{th-PFodd} contains $\pm F_0(X)$ as a summand if and only if $n$ is \emph{triangular}, i.e., $n = r(r+1)/2$ for some $r\geq 1$. The smallest triangular integers are $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$. \end{remark} \begin{remark} For which natural numbers~$n$ do we have $\Pg_n(X) = F_{n-1}(X) \pm F_1(X)$? We have $\Pg_n(X) = F_{n-1}(X) + F_1(X)$ if and only if $n = 2^a p$, where $p$ is a prime of the form $p = 2^{a+1} - 3$. Primes $p= 5$, $13$, $29$, $61$ are of this form; the corresponding~$n$ are $n = 20$, $104$, $464$, $1952$ (see \seqnum{A050415} and \seqnum{A181703} in the OEIS~\cite{OEIS}). Similarly, $\Pg_n(X) = F_{n-1}(X) - F_1(X)$ if and only if $n = 2^a p$, where $p$ is a prime of the form $p = 2^{a+1} + 3$. Primes $p = 5$, $7$, $11$, $19$, $67$ are of this form (see the OEIS~\cite[\seqnum{A057733}]{OEIS}); they correspond to $n= 5$, $14$, $44$, $152$, $2144$. More generally, $F_1(X)$ appears in the formula for $\Pg_n(X)$ in Theorem~\ref{th-PFodd} if and only if $n = r(r+3)/2$ for some $r\geq 1$; see the OEIS~\cite[\seqnum{A000096}]{OEIS} for a list of such~$n$. \end{remark} \section{Formulas for $C_n(q)$}\label{sec-Cnq} In order to prove Theorem~\ref{th-PFodd} we need new expressions for the polynomial~$C_n(q)$. Recall that $C_n(q)$ and $\Pg_n(X)$ are related by the equalities \begin{equation*} \frac{C_n(q)}{q^n} = \frac{(q-1)^2}{q} \, \frac{P_n(q)}{q^{n-1}} = \frac{(q-1)^2}{q} \, \Pg_n(q + q^{-1}) . \end{equation*} \subsection{Increasing sequences}\label{ssec-IS} We need the following material borrowed from Mason~\cite{Ma}. We call \emph{increasing sequence} every finite nonempty ordered set of consecutive integers $\{ a+1, \ldots, a+h\}$ with smallest element~$a+1$ and largest element~$a+h$. Here $a$ is an integer (positive, negative or zero) and $h$ is a positive integer. We denote this set by~$\IS(a,h)$; it has $h$ elements. An increasing sequence~$\IS(a,h)$ is called \emph{positive} if $a\geq 0$ and \emph{negative} if $a < 0$. It is called \emph{odd} if $h$ is odd and \emph{even} if $h$ is even. We say that an integer~$n \geq 1$ is \emph{represented} by~$\IS(a,h)$, and we write \[ \IS(a,h) \models n, \] if $n$ is equal to the sum of all elements of~$\IS(a,h)$. There is an involution $\IS(a,h) \mapsto \IS(\check{a},\check{h})$ on increasing sequences determined by \begin{equation}\label{def-invIS} a + \check{a} +1 = 0 \quad \text{and} \quad \check{a} + \check{h} = a + h. \end{equation} If $\IS(a,h)$ represents~$n$, so does~$\IS(\check{a},\check{h})$. Since $\check{h} - h = a - \check{a} = 2a + 1$, we conclude that $h$ and $\check{h}$ are of opposite parity. In other words, $\IS(a,h)$ is even (resp., odd) if and only if $\IS(\check{a},\check{h})$ is odd (resp., even). Moreover, if $\IS(a,h)$ is positive (resp., negative), then $\IS(\check{a},\check{h})$ is negative (resp., positive). Let $n \geq 1$ be a positive integer. With every \emph{odd} divisor~$d$ of~$n$ we associate the odd increasing sequence $\IS(a,h)$, where \begin{equation}\label{eq-ah} a = \frac{n}{d} - \frac{d+1}{2} \quad\text{and}\quad h = d . \end{equation} It is centered around~$n/d$, namely we have \begin{equation}\label{def-ISd} \IS(a,h) = \left\{ \frac{n}{d} - \frac{d-1}{2}, \ldots, \frac{n}{d} -1, \frac{n}{d}, \frac{n}{d} +1, \ldots, \frac{n}{d} + \frac{d-1}{2} \right\}. \end{equation} This odd increasing sequence may be positive or negative, depending on~$d$. We have $\IS(a,h) \models n$. Starting from $a = n/d - (d+1)/2$ and $h = d$, we have \begin{equation}\label{eq-ah2} \check{a} = -\frac{n}{d} + \frac{d-1}{2} \quad\text{and}\quad \check{h} = \frac{2n}{d}. \end{equation} We thus obtain the even increasing sequence \begin{equation}\label{def-ISdv} \IS(\check{a},\check{h}) = \left\{ -\frac{n}{d} + \frac{d+1}{2}, \ldots, \frac{n}{d} + \frac{d-1}{2} \right\} \models n . \end{equation} Clearly, if $\IS(a,h)$ is positive, then $\IS(a,h) \subset \IS(\check{a},\check{h})$ and \begin{equation}\label{def-ISdv1} \IS(\check{a},\check{h}) \setminus \IS(a,h) = \left\{ -\frac{n}{d} + \frac{d+1}{2}, \ldots, \frac{n}{d} - \frac{d+1}{2} \right\} . \end{equation} If $\IS(a,h)$ is negative, then $\IS(\check{a},\check{h}) \subset \IS(a,h)$ and \begin{equation}\label{def-ISdv2} \IS(a,h) \setminus \IS(\check{a},\check{h}) = \left\{ \frac{n}{d} - \frac{d-1}{2}, \ldots, -\frac{n}{d} + \frac{d-1}{2} \right\} . \end{equation} For both pairs $(a,h)$ and $(\check{a}, \check{h})$ we have the same relation, namely \begin{equation}\label{eq-IS} \frac{h(h+2a+1)}{2} = n = \frac{\check{h}(\check{h}+2\check{a}+1)}{2} . \end{equation} Conversely, starting from an increasing sequence $\IS(a,h)$ we obtain an odd divisor $d$ of~$n$ as follows: $d$ is equal to the only \emph{odd} element of~$\{h, \check{h}\}$. In this way we have a bijection between the set of odd divisors of~$n$ and the set of positive increasing sequences representing~$n$, which is in bijection with the set of negative increasing sequences representing~$n$. Suppose that given $n\geq 1$ we have a pair $(a,h)$ of integers with $h\geq 1$ verifying Equation~\eqref{eq-IS}, which we rewrite as $h(h+2a+1) = 2n$. If $h$ is odd, hence coprime to~$2$, then $d = h$ is an odd divisor of~$n$ and $a = n/d - (d+1)/2$, as in~\eqref{eq-ah}. If $h$ is even, then $d = h+2a+1$ is odd; being a divisor of~$2n$, it is an odd divisor of~$n$; in this case $h = 2n/d$ and $a = - n/d + (d-1)/2$, as in~\eqref{eq-ah2}. \begin{example} If $n$ is a power of~$2$, then it has only one odd divisor, namely $d=1$; the corresponding positive increasing sequence is \[ \IS(n-1,1) = \{n\} \] and its negative counterpart is \[ \IS(-n,2n) = \{ -(n-1), \ldots, -1,0,1, \ldots, n-1,n\}. \] \end{example} \begin{example} Every other integer~$n$ has odd divisors $d\neq 1$, hence more than one positive (or negative) increasing sequences. For instance, consider $n= 18$ with odd divisors $d= 3$ and $d=9$. If $d = 3$, then $a = 4$, $h=3$, $\check{a} = -5$, $\check{h} = 12$, and \[ \IS(4,3) = \{ 5,6,7\} \quad \text{and}\quad \IS(-5, 12) = \{-4, -3, \ldots, 3, 4, 5, 6, 7\} . \] If $d = 9$, then $a = -3$, $h=9$, $\check{a} = 2$, $\check{h} = 4$, and \[ \IS(-3,9) = \{ -2, -1, 0, 1, 2, 3, 4, 5,6 \} \quad \text{and}\quad \IS(2,4) = \{ 3, 4, 5, 6\} . \] \end{example} \subsection{New formulas for~$C_n(q)$}\label{ssec-Cnq} The authors \cite[Th.\ 1.1]{KR3} gave the following explicit expression for~$C_n(q)$: \begin{equation*}\label{Cn-coeffi} \frac{C_n(q)}{q^n} = c_{n,0} + \sum_{i=1}^n \, c_{n,i} \, \left( q^{i} + q^{-i} \right) , \end{equation*} where \begin{equation}\label{formula-cn0} c_{n,0} = 2 (-1)^r \end{equation} if $n = r(r+1)/2$ for some positive integer~$r$ and $c_{n,0} = 0$ otherwise. For the remaining coefficients $c_{n,i}$ ($1 \leq i \leq n$) we have \begin{equation}\label{formula-cni} c_{n,i} = \begin{cases} (-1)^k , & \text{if $n = k(k+2i +1)/2$ for some integer $k \geq 1$;} \\ (-1)^{k-1} , & \text{if $n = k(k+2i -1)/2$ for some integer $k \geq 1$;} \\ 0, & \text{otherwise.} \end{cases} \end{equation} Observe that in (\ref{formula-cni}) the first two conditions are mutually exclusive, as shown by the authors~\cite[Lemma~3.5 (ii)]{KR3}. This may also be verified as follows: suppose \[ k(k+2i+1)=h(h+2i-1) \] with $i,h,k\geq 1$; then $k^2+2ik+k=h^2+2ih-h$, which can be rewritten as \[ (k-h+1)(k+h)+2i(k-h)=0 . \] It is readily seen that the left-hand side cannot vanish, neither in the case $k\geq h$, nor in the case~$k 0$, and $(-1)^{k-1}=(-1)^h$. This proves the lemma in view of the observation made just before its statement. \end{proof} We have the following new formulas for~$C_n(q)$. \begin{theorem}\label{th-Cn} For each integer $n\geq 1$ we have \[ \frac{C_n(q)}{q^n} = \sum \, (-1)^h \, \left( q^a + q^{-a} \right), \] where the summation is taken over all increasing sequences $\IS(a,h)$ such that $\IS(a,h) \models n$. Moreover, for given $|a|$, there is a unique increasing sequence $\IS(a,h)$ with $\IS(a,h) \models n$. We also have \[ \frac{C_n(q)}{q^n} = \sum_{d | n , \, d \, \mathrm{odd}} \, \left( q^{\frac{n}{d} - \frac{d-1}{2}} + q^{-\frac{n}{d} + \frac{d-1}{2}} - q^{\frac{n}{d} - \frac{d+1}{2}} - q^{-\frac{n}{d} + \frac{d+1}{2}} \right) . \] In the previous sum, all monomials $\neq q^0$ are distinct. \end{theorem} \begin{proof} The first formula follows from Lemma~\ref{formula1} since the sum of the elements in $\IS(a,h)$ is equal to \[ a+1+a+2+ \cdots+a+h = ha+h(h+1)/2 = h(h+2a+1)/2. \] As in \S~\ref{ssec-IS}, with each odd divisor~$d$ of~$n$ we associate the two increasing sequences $\IS(a,h)$ and $\IS(\check a,\check h)$, respectively odd and even, defined by \eqref{def-ISd} and \eqref{def-ISdv}. All increasing sequences representing~$n$ are obtained in this way. The second formula then follows from the first one: the first two terms correspond to~$\IS(\check{a}, \check{h})$ and the last two to~$\IS(a,h)$. \end{proof} The previous result immediately implies the following. \begin{corollary}\label{coro-vacni} The sum of the absolute values of the coefficients of $C_n(q)$ is equal to four times the number of odd divisors of~$n$. \end{corollary} \subsection{Zeta functions}\label{ssec-zeta} The following consequences of Theorem~\ref{th-Cn} may be of interest to algebraic geometers. Recall that the \emph{local zeta function}~$Z_{\HH^n/\FF_q}(t)$ of the Hilbert scheme~$\HH^n$ of $n$~points on the two-dimensional torus over the finite field~$\FF_q$ is the formal power series \begin{equation}\label{def-Z} Z_{X/\FF_q}(t) = \exp\left(\sum_{m\geq 1}\, C_n(q^m) \, \frac{t^m}{m} \right) . \end{equation} The authors \cite[Th.\ 2.1]{KR3} showed that $Z_{X/\FF_q}(t)$ is equal to the formal power series expansion of the following rational function: \[ Z_{\HH^n/\FF_q}(t) = \frac{1}{(1-q^nt)^{c_{n,0}}} \, \prod_{i=1}^n \, \frac{1}{[(1-q^{n+i}t)(1-q^{n-i}t)]^{c_{n,i}}} \, , \] where $c_{n,i}$ are the coefficients of~$C_n(q)$ defined in \S~\ref{ssec-Cnq}. Rewriting this formula in view of Theorem~\ref{th-Cn} yields the following statement, where $r_n(d) = n/d - (d+1)/2$ is the integer introduced in~\eqref{eq-rnd}. \begin{proposition}\label{prop-zeta} For $n\geq 1$ we have \begin{equation*} Z_{\HH^n/\FF_q}(t) = \prod_{d | n , \, d \, \mathrm{odd}} \, \frac{(1-q^{n + r_n(d)}t)(1-q^{n - r_n(d)}t)}{(1-q^{n + r_n(d)+1}t)(1-q^{n - r_n(d)-1}t)} . \end{equation*} \end{proposition} The right-hand side is a rational function whose numerator and denominator are factored into degree-one polynomials. Clearly, the number of such factors in the numerator (resp., in the denominator) is equal to twice the number of odd divisors of~$n$. Let us display the product formula for $Z_{\HH^n/\FF_q}(t)$ in two special cases. For $n=4$, which has only~$1$ as an odd divisor, we have $r_4(1) = 3$. Hence, \begin{equation*} Z_{\HH^4/\FF_q}(t) = \frac{(1-q^{7}t)(1-qt)}{(1-q^{8}t)(1-t)} . \end{equation*} For $n=3$, which has two odd divisors, we have $r_3(1) = 2$ and $r_3(3) = -1$. Hence, \begin{equation*} Z_{\HH^3/\FF_q}(t) = \frac{(1-q^{5}t)(1-qt)}{(1-q^{6}t)(1-t)} \cdot \frac{(1-q^{2}t)(1-q^4t)}{(1-q^{3}t)^2} . \end{equation*} Proposition~\ref{prop-zeta} allows us to give a similar formula for the \emph{Hasse--Weil zeta function} $\zeta_{\HH^n}(s)$ of the above Hilbert scheme; this zeta function is defined over the complex numbers by \begin{equation*}\label{def-HWZ} \zeta_{\HH^n}(s) = \prod_{p \; \text{prime}} \, Z_{{\HH^n}/\FF_p}(p^{-s}), \qquad (s \in \CC) \end{equation*} where the product is taken over all prime integers~$p$. As an immediate consequence of the formula for the local zeta functions~$Z_{\HH^n/\FF_q}(t)$ we have \begin{equation} \zeta_{\HH^n}(s) = \prod_{d | n , \, d \, \mathrm{odd}} \, \frac{ \zeta(s - n - r_n(d)) \, \zeta(s - n + r_n(d) ) }{ \zeta(s - n - r_n(d) -1) \, \zeta(s - n + r_n(d) +1) } \, , \end{equation} where $\zeta(s)$ is Riemann's zeta function. Recall from \cite[Sect.\ 2]{KR3} that the Hasse--Weil zeta function satisfies the simple functional equation \begin{equation} \zeta_{\HH^n}(s) = \zeta_{\HH^n}(2n-s). \end{equation} \section{Proof of Theorem~\ref{th-PFodd}}\label{pf-th-PFodd} The following lemma exhibits a crucial link between the polynomials $F_r(X)$ and the expression in parentheses in the second formula of Theorem~\ref{th-Cn}. \begin{lemma}\label{lem-CF} For $r \geq 0$ we have \[ q^{r+1} + q^{-r-1} - q^{r} - q^{-r} = q^{-1}(q-1)^2 F_r(q + q^{-1}) . \] \end{lemma} \begin{proof} The formula can be rephrased in terms of polynomials in the indeterminate~$X$ as $\TT_{r+1}(X) - \TT_r(X) = (X-2) F_r(X)$. By \eqref{def-F} the latter is equivalent to \[ \bigl( F_{r+1}(X) - F_r(X) \bigr) - \bigl( F_r(X) - F_{r-1}(X) \bigr) = X F_r(X) - 2 F_r(X), \] hence to the linear recurrence relation of Proposition~\ref{prop-recF}. \end{proof} \begin{proof}[Proof of Theorem~\ref{th-PFodd}] It follows from the second formula of Theorem~\ref{th-Cn} that \[ q^{-1}(q-1)^2 \frac{P_n(q)}{q^{n-1}} = \frac{C_n(q)}{q^n} = \sum_{d | n , \, d\, \mathrm{odd}} \, \left( q^{\frac{n}{d} - \frac{d-1}{2}} + q^{-\frac{n}{d} + \frac{d-1}{2}} - q^{\frac{n}{d} - \frac{d+1}{2}} - q^{-\frac{n}{d} + \frac{d+1}{2}} \right) . \] We partition the previous sum into two sums depending on the sign of $r_n(d) = n/d - (d+1)/2$. We obtain \begin{eqnarray*} q^{-1}(q-1)^2 \frac{P_n(q)}{q^{n-1}} & = & \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) \geq 0} \, \left( q^{\frac{n}{d} - \frac{d-1}{2}} + q^{-\frac{n}{d} + \frac{d-1}{2}} - q^{\frac{n}{d} - \frac{d+1}{2}} - q^{-\frac{n}{d} + \frac{d+1}{2}} \right) \\ & & \hskip -10pt + \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) < 0} \, \left( q^{\frac{n}{d} - \frac{d-1}{2}} + q^{-\frac{n}{d} + \frac{d-1}{2}} - q^{\frac{n}{d} - \frac{d+1}{2}} - q^{-\frac{n}{d} + \frac{d+1}{2}} \right) . \end{eqnarray*} Now replacing $r$ by $r_n(d)$ in Lemma~\ref{lem-CF}, we see that the sum of monomials corresponding to $r_n(d) \geq 0$ is equal to \[ q^{-1}(q-1)^2 \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) \geq 0} \, F_{r_n(d)}(q + q^{-1}). \] Replacing $r$ by $-r_n(d) - 1$ in Lemma~\ref{lem-CF}, we observe that the sum of monomials corresponding to $r_n(d) < 0$, equivalently to $-r_n(d) - 1 \geq 0$, is equal to \[ q^{-1}(q-1)^2 \sum_{d | n, \, d \, \mathrm{odd} \atop r_n(d) < 0} \, F_{-r_n(d) -1}(q + q^{-1}). \] To conclude, it suffices to divide by $q^{-1}(q-1)^2$ and to appeal to the equality $P_n(q)/q^{n-1} = \Pg_n(q + q^{-1})$. \end{proof} We close this section with another formula for $P_n(q)/q^{n-1}$. \begin{proposition}\label{formula2} We have \[ \frac{P_n(q)}{q^{n-1}} = \sum_{d|n, \,d\,\mathrm{odd} \atop r_n(d) \geq 0} \; \sum_{i \in \IS(\check{a},\check{h}) \setminus \IS(a,h)} \, q^i - \sum_{d|n, \,d\,\mathrm{odd} \atop r_n(d) < 0} \; \sum_{i \in \IS(a,h)\setminus \IS(\check{a},\check{h})}\, q^i , \] where $a = r_n(d)$ and $\check{a} = - r_n(d) - 1$, and $\IS(a,h)$ and $\IS(\check{a},\check{h})$ are defined respectively by \eqref{def-ISd} and \eqref{def-ISdv}. \end{proposition} \begin{proof} It follows from Theorem~\ref{th-PFodd} and the explicit descriptions~\eqref{def-ISdv1} and \eqref{def-ISdv2}. \end{proof} \begin{remark} Sums of powers of $q$ indexed by odd divisors have also appeared in work of Caballero~\cite{RC1} (especially in Chapter~II). \end{remark} \section{Acknowledgments} We are grateful to Jos\'e Bastidas Olaya for his help with SageMath and to V\'aclav Kot\v{e}\v{s}ovec for having corrected the value~$\Pg_7(4)$ in Table~\ref{table-values}. We thank Christophe Pouzat and Marcus Slupinski for suggesting the title of the present paper. \begin{thebibliography}{99} \bibitem{AS} M. Abramowitz and I. A. Stegun (eds.), \emph{Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, Dover, 1972. \bibitem{Ap} T. M. Apostol, \emph{Introduction to Analytic Number Theory}. Springer Verlag, 1986. \bibitem{BR} J. Berstel and C. Reutenauer, \emph{Noncommutative Rational Series with Applications}, Encyclopedia of Mathematics and its Applications, 137, Cambridge University Press, 2011. \bibitem{RC1} J. M. R. Caballero, Propri\'et\'es arithm\'etiques du $E$-polyn\^ome du sch\'ema de Hilbert de $n$~points dans le tore bidimensionnel, M\'emoire de ma\^\i trise en math\'ematiques, Universit\'e du Qu\'ebec \`a Montr\'eal, 2018. \bibitem{RC2} J. M. R. Caballero, On Kassel-Reutenauer $q$-analog of the sum of divisors and the ring $\FF_3[X]/X^2\FF_3[X]$, \emph{Finite Fields Appl.} \textbf{51} (2018), 183--190. \bibitem{Fi} N. J. Fine, \emph{Basic Hypergeometric Series and Applications}, Mathematical Surveys and Monographs,~27, American Mathematical Society, 1988. \bibitem{Ho} A. F. Horadam, Vieta polynomials, \emph{Fibonacci Quarterly} \textbf{40} (2002), 223--232. \bibitem{KR1} C. Kassel and C. Reutenauer, The Fourier expansion of $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$, \emph{Arch.\ Math.\ (Basel)} \textbf{108} (2017), 453--463. \bibitem{KR2} C.~Kassel and C.~Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, \emph{Michigan Math.~J}.\ \textbf{67} (2018), 715--741. \bibitem{KR3} C.~Kassel and C.~Reutenauer, Complete determination of the zeta function of the Hilbert scheme of $n$~points on a two-dimensional torus, \emph{Ramanujan~J}.\ \textbf{46} (2018), 633--655. \bibitem{Ma} T. E. Mason, On the representation of an integer as the sum of consecutive integers, \emph{Amer.\ Math.\ Monthly} \textbf{19} (1912), 46--50. \bibitem{OEIS} OEIS~Foundation Inc., The {O}n-{L}ine {E}ncyclopedia of {I}nteger {S}equences, 2026. Published electronically at \url{https://oeis.org/}. \bibitem{Ri} T. J. Rivlin, \emph{Chebyshev Polynomials}, Wiley, 1990. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2020 {\it Mathematics Subject Classification}: Primary 11T55; Secondary 05A30, 14N10. \noindent \emph{Keywords: } integer sequence, generating function, infinite product, polynomial, Chebyshev polynomial, approximation, multiplicative function. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000096}, \seqnum{A000203}, \seqnum{A000215}, \seqnum{A000668}, \seqnum{A001834}, \seqnum{A002325}, \seqnum{A002654}, \seqnum{A002878}, \seqnum{A030221}, \seqnum{A050415}, \seqnum{A057733}, \seqnum{A113063}, \seqnum{A181703}, \seqnum{A329156}, \seqnum{A386706}, and \seqnum{A387017}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received September 19 2025; revised version received March 2 2026. Published in {\it Journal of Integer Sequences}, July 4 2026. \bigskip \hrule \bigskip \noindent Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}. \vskip .1in \end{document}