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\begin{center}
\vskip 1cm{\LARGE\bf 
Log-Concavity of Recursively Defined \\
\vskip .1in
Polynomials}
\vskip 1cm
\large
Bernhard Heim and Markus Neuhauser \\
Department of Mathematics and Science\\
Faculty of Science\\
German University of Technology in Oman (GUtech)\\
PO Box 1816\\
Athaibah PC 130\\
Sultanate of Oman\\
\href{mailto:bernhard.heim@gutech.edu.om}{\tt bernhard.heim@gutech.edu.om} \\
\href{mailto:markus.neuhauser@gutech.edu.om}{\tt markus.neuhauser@gutech.edu.om}
\end{center}

\vskip .2 in
\begin{abstract}
Fourier coefficients of powers of the Dedekind eta function can be studied
by polynomials
introduced by M. Newman. We generalize
the defining recurrence relations in this paper.
From this we derive
new families of polynomials, which approximate
these polynomials from below and above.
Although these families are recursively defined, we are able to
determine explicit closed formulas for
both approximating polynomials.
(For the original
polynomials closed formulas are not yet known.)
Furthermore, we obtain
that both approximating
families and the coefficients involved are log-concave and unimodal.
\end{abstract}


\section{Introduction}
Let $\eta(\tau)$ be the Dedekind eta function \cite{On03}. Put $q := e^{2 \pi \, i \, \tau}$, where $\tau$ is in the upper half-space $\mathbb{H}$.
Let further $z \in \mathbb{C}$. We consider the Fourier expansion 
\begin{equation} \label{start}
\left( q^{-\frac{1}{24}} \eta(\tau)\right)^{-z} = \prod_{n=1}^{\infty} \left( 1 - q^n \right)^{-z} = \sum_{n=0}^{\infty} P_n(z) \, q^n.
\end{equation}
We recover several famous sequences \cite[Introduction]{HLN18}.
For example, the partition numbers $p(n)$ and the Ramanujan numbers $\tau(n)$ are directly linked to $z=1$ and $z=-24$,
\begin{eqnarray*}
\left( P_n(1)\right) _{n=0}^{\infty}  =  \left( p(n)\right) _{n=0}^{\infty}&:&   1,1,2,3,5,\ldots \quad (\seqnum{A000041});\\
\left( P_{n-1}(-24)\right) _{n=1}^{\infty}  =  \left( \tau(n)\right)     _{n=1}^{\infty} &:&  1,-24,252, \ldots \quad (\seqnum{A000594}). 
\end{eqnarray*}
The integer-valued polynomials $P_n(X)$
have degree $n$ and positive integral coefficients $A_k^n$ for $0 \leq k \leq n-1$:
\begin{equation}
P_n(X) = \frac{X}{n!} \,\, \sum_{k=0}^{n-1}  A_k^{n} \, \, X^k.
\end{equation}
Newman \cite{Ne55} calculated the coefficients recursively for $ 0 \leq k \leq n-1$. 
For example 
$A_0^n = (n-1)! \sigma(n)$,
$A_{n-1}^n = 1$, and $A_{n-2}^n = 3 n (n-1)/2$. Here $\sigma(n)$ is the sum of the divisors of $n$.
Further Newman determined the first ten polynomials and their integral 
zeros and found a recursion formula for the involved polynomials.
Apparently there is yet no closed explicit
formula for these polynomials 
defined by
\begin{equation} \label{recursion}
P_n(X) = \frac{X}{n} \,\,  \sum_{k=1}^{n} \sigma(k) P_{n-k}(X)
\end{equation}
and the involved coefficients.

Numerical calculations indicate that the sequence of the coefficients $A_k^n$ seems to be close to 
unimodal and stable (Hurwitz polynomial).
Here we normalize $P_n(X)$ by $n!/X$. A direct approach seems to be out of reach at the moment,
since there is still a lack of understanding
the properties of the arithmetic function $\sigma(n)$ 
completely. Nevertheless the value distribution of the polynomials $P_n(X)$, especially the 
zeros, have significant applications.
For $r$ even and positive, the sequences
$\left( P_n(-r)\right) _{n}$ are lacunary if and only if $r= 2,4,6,8,10,14,26$. This result is due to
Serre \cite{Se85} and is in relation
to modular forms with complex multiplication (CM-forms). The non-existence of $-24$ as a 
zero is equivalent to the Lehmer conjecture \cite{Le47}.




In this paper we generalize
the recurrence relation (\ref{recursion}), studying closed formulas for
the polynomials associated with the functions $g(n)=n$ and $g(n)=n^2$. They are integer valued and approximate $\sigma(n)$ from below and above.


\begin{definition}
Let $g(n)$ be an arithmetic function. We define a family of polynomials $P_n^g(X)$ associated with $g$. Let $P_0^g(
X):=1$ and
\begin{eqnarray}\label{generalrecursion}
P_n^g(X) &:=&   \frac{X}{n} \sum_{k=1}^{n}  g(k)  \,\, P_{n-k}^g(X),\\
&=& \frac{X}{n!} \sum_{k=0}^{n-1}  A_k^{n} (g) \, X^k. 
\end{eqnarray}
\end{definition}


Let $\varphi_1(n) = n $ and $\varphi_2(n)=n^2$.
In what follows,
we study the properties of the associated 
polynomials $P_n^{\varphi_1}(X)$ and $P_n^{\varphi_2}(X)$. 
Their properties
are related to $P_n(X) = P_n^{\sigma}(X)$, since
$\varphi_1(n) < \sigma(n) < \varphi_2(n)$ 
for 
$n >1$.
We obtain $A_{n-1}^n(\varphi_1) = A_{n-1}^n = A_{n-1}^n\left( \varphi _{2}\right) =1$. Further, let $ 0 \leq k \leq n-2$. Then
\begin{equation}
A_k^n (\varphi_1) < A_k^n < A_k^n (\varphi_2).   
\end{equation}

\begin{theorem}\label{varphi1}
Let $\varphi_1(n) =n$.
Then the coefficients of $P_n^{\varphi_1}(X)$ are given by
\begin{equation}
A_k^n(\varphi_1) =  \frac{n!}{(k+1)!}  \,   \binom{n-1}{k}.
\label{eq:linear}
\end{equation}
\end{theorem}
Although the binomial coefficients are twisted by $1/(k+1)!$, they
remain log-concave.
\begin{corollary}\label{uni1}
The sequence of the coefficients of $P_n^{\varphi_1}(X)$
$$\left( A_k^n(\varphi_1)\right) _{k=0}^{n-1}$$ is strongly log-concave and hence unimodal.
\end{corollary}
Further we can determine the index,
and therefore the size of the largest coefficient.
\begin{corollary}  \label{max1}
The index $K_1$ of the maximal coefficient $A_k^n(\varphi_1)$ is given by
\begin{equation}
\sqrt{n+1} -2 \leq K_1 < \sqrt{n+1} -1.
\end{equation}
In general, $K_1$ is not always unique.
\end{corollary}
Numerical calculations indicate that the 
zeros of $P_n^{\varphi_1}(X)$ are simple and that the polynomials are stable in the sense of Hurwitz (neglecting $X=0$).

It is also possible to get similar results 
for $P_n^{\varphi_{2}}(X)$, although the involved 
coefficients $A_k^n(\varphi_2)$ are more complicated.
We also studied the polynomials $P^{\varphi}_n(X)$ associated
with  $\varphi(n) := n \, \mbox{ln}(n)$ and $\varphi(n) := n \, \sqrt{n}$. But 
a closed formula for the coefficients is very difficult to obtain. 

\begin{theorem}\label{varphi2}
Let $\varphi_2(n) =n^2$. Then the coefficients of $P_n^{\varphi_2}(X)$ are given by
\begin{equation}
A_k^n(\varphi_2) =  \frac{n!}{(k+1)!}  \,   \binom{n+k}{2k+1}\   .
\label{eq:quadratisch}
\end{equation}
\end{theorem}

\begin{corollary}\label{uni2}
The sequence of the coefficients of $P_n^{\varphi_2}(X)$
$$\left( A_k^n(\varphi_{2})\right) _{k=0}^{n-1}$$ is strongly log-concave and
hence unimodal.
\end{corollary}

\begin{corollary}  \label{max2}
The coefficients $A_k^n(\varphi_2)$ assume
their maximum at the index $K_2$,
such that $K_{2}\leq k <K_{2}+1$ is the real solution 
of the cubic equation $4k^{3}+19k^{2}+28k+13=n^{2}$. Let 
$$D:= \frac{1}{8}n^{2}-\frac{91}{1728}+\frac{1}{144}
\sqrt{324n^{4}-273n^{2}-51}, $$
then $\sqrt[3]{D} + \frac {25}{ 144 \, \sqrt[3]{D}}\leq K_2 < \sqrt[3]{D} + \frac {25}{ 144 \, \sqrt[3]{D}}+1$. 
For large $n$ we obtain $K_2 \approx (n/2)^{2/3}$.
\end{corollary}
Numerical calculations indicate that the 
zeros of $P_n^{\varphi_{1}}(X)$ are simple and that the polynomials are stable.

Let $x_0$ be a positive real number. Let $g_1,g_2$ be two arithmetic functions, satisfying
$1 \leq g_1(n) \leq g_2(n)$. Then $P_n^{g_1}(x_0) \leq P_n^{g_2}(x_0)$.
We let $p(n)$ denote the partition numbers \cite{An98}.
As an application we obtain an approximation of the 
partition numbers from below.
Let $N= 200 $. Then
\begin{equation}
\frac{p(n)}{P_n^{\varphi_1}(1)} < 647.71   \quad \quad \mbox{ for } n \leq N.
\end{equation}

Finally we obtain for $x=-1$ the following result. 
\begin{figure}[H]
\begin{center}
\includegraphics[width=10cm,angle=270]{x-1_50.eps}
\caption{`$+$' marks the value of 
$P_{n}^{\varphi _{1}}\left( -1\right) $,
`$\times $' marks the value of 
$P_{n}^{\sigma }\left(- 1\right) $
depending on $n$.}
\end{center}
\end{figure}

Euler showed that
$P_n(-1)$ takes only the values $-1,0,1$ and has high vanishing rate,
called superlacunary \cite{OR95} in the
language of modular forms. This is reflected by the asymptotic behavior of the values of the sequence $P_n^{\varphi_1}(-1)$, although $x_0<0$. 

\section{Log-Concavity and maximal coefficients}
A sequence $a_0,a_1, a_2, \ldots, a_n$ of real numbers 
is called unimodal if 
the sequence increases steadily at first and then 
decreases steadily \cite{Wi06}.
The sequence of binomial coefficients and Stirling numbers are unimodal.
In the case where
\begin{equation}
a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_K \geq a_{K+1} \geq a_{K+2} \geq \cdots  \geq a_m,
\end{equation}
the $K$ is called the \textit{index\/} of the sequence.
The index does not have to be unique.
In general, it is not clear how to determine the index.

Another important property of some sequences is 
(strong) log-concavity. 
A sequence is called \textit{log-concave\/} if for all $k=1,2, \ldots, m-1$:
\begin{equation}
a_k^2 \geq a_{k+1} \, a_{k-1}.
\end{equation}
Note that log-concavity implies unimodality.

Due to Newton's inequality, the sequence of coefficients of a polynomial
with (real coefficients and) only real zeros is strongly log-concave. It
is very likely that $P_n^{\varphi_1}(X)$ and $P_n^{\varphi_2}(X)$ have
only real zeros.  But $P_n(X)$ definitely does not have only real zeros.
Heim et al.\ \cite{HNR17} showed that, for example, in the case $n=10$
non-real zeros appear.  Nevertheless, directly
applying the log-concave
criterion works for $\varphi_1, \varphi_2$, due to the explicit formulas
proved in this paper.

\subsection{Proof of Corollary \ref{uni1}}
Let $n \geq 3$ be given.
We consider the sequence
\begin{equation*}
 a_k :=A_k^n(\varphi_1) = \frac{n!}{(k+1)!}  \, \binom{ n-1}{k}
\end{equation*}
for $k=0, \ldots, n-1$.
We show that for each pair $(n,k)$ with $1 \leq k \leq n-2$ an $\alpha  >1$ exists, such that
\begin{equation}
\frac{ A_k^n (\varphi_1)^2 }{ A_{k+1}^n(\varphi_1) \,  A_{k-1}^n(\varphi_1) } \geq \alpha.
\end{equation}
This quotient results in 
\begin{eqnarray*}
\frac{\left( \frac{1}{\left( k+1\right) !}\binom{n-1}{k}\right) ^{2}}{\frac{1}{k!}\binom{n-1}{k-1}\frac{1}{\left( k+2\right) !}\binom{n-1}{\left( k+1\right) }} 
&=&\frac{\frac{1}{k+1 }\left( \frac{\left( n-1\right) !}{k!\left( n-k-1\right) !}\right) ^{2}}{\frac{\left( n-1\right) !}{\left( k-1\right) !\left( n-k\right) !}\frac{1}{k+2}\frac{\left( n-1\right) !}{\left( k+1\right) !\left( n-k-2\right) !}} 
\\
& =  & \frac{\frac{1}{k+1}\frac{1}{k\left( n-k-1\right) }}{\frac{1}{n-k}\frac{1}{k+2}\frac{1}{k+1}} \\ 
&=&\frac{\left( n-k\right) \left( k+2\right) }{k\left( n-k-1\right) } \geq \alpha >1.
\end{eqnarray*}
Hence the sequence $A_k^n(\varphi_1)$ is strongly log-concave. This implies unimodality.

We obtain $\alpha =\alpha _{n}\geq 1+\left( 3n+1+2\sqrt{2n\left(
n+1\right) }\right) \left( n-1\right) ^{-2}$.  Depending on~$n$ $\alpha $
decreases to $1$.

To show that $\alpha $ decreases to $1$, consider
the denominator of the derivative of
$\frac{\left( n-k\right) \left( k+2\right) }{k\left( n-1-k\right) }$
with respect to~$k$. The denominator yields
$2n\left( n+1\right) -\left( k-2n\right) ^{2}$.
Hence the minimum of the denominator 
for $1\leq k\leq n-2$ is obtained at
$k=2n-\sqrt{2n\left( n+1\right) }$.
This yields the claimed value.

\subsection{Proof of Corollary \ref{uni2}}
Let $n \geq 2$ be given.
We consider the sequence
\begin{equation*}
 a_k :=A_k^n(\varphi_2) = \frac{n!}{(k+1)!}  \, \binom{ n + k }{2k  +1}
\end{equation*}
for $k=0, \ldots, n-1$.
Hence $a_0,a_1,\ldots, a_m$ with $m=n-1$.
We show that for each pair $(n,k)$ with $0 \leq k \leq n-2$
a $\beta  >1$ exists such that
\begin{equation}
\frac{ A_k^n (\varphi_{2})^2 }{ A_{k+1}^n(\varphi_{2}) \,  A_{k-1}^n(\varphi_{2}) } > \beta.
\end{equation}

This quotient results in
\begin{eqnarray*}
\frac{\left( \frac{1}{\left( k+1\right) !}\binom{n+k}{2k+1}\right) ^{2}}{\frac{1}{k!}\binom{n+k-1}{2k-1}\frac{1}{\left( k+2\right) !}\binom{n+k+1}{2k+3}}
 & =  &
\frac{\frac{1}{k+1}\left( \frac{\left( n+k\right) !}{\left( 2k+1\right) !\left( n-k-1\right) !}\right) ^{2}}{\frac{\left( n+k-1\right) !}{\left( 2k-1\right) !\left( n-k\right) !}\frac{1}{k+2}\frac{\left( n+k+1\right) !}{\left( 2k+3\right) !\left( n-k-2\right) !}} \\
&=&  \frac{\frac{1}{k+1}\frac{n+k}{\left( 2k+1\right) 2k\left( n-k-1\right) }}{\frac{1}{n-k}\frac{1}{k+2}\frac{n+k+1}{\left( 2k+3\right) \left( 2k+2\right) }} \\
& =  &
\frac{\left( n+k\right) \left( n-k\right) \left( k+2\right) \left( 2k+3\right) \left( 2k+2\right) }{\left( k+1\right) \left( n+k+1\right) \left( 2k+1\right) 2k\left( n-k-1\right) } \\
&>&      \frac{n^{2}-k^{2}}{n^{2}-\left( k+1\right) ^{2}}\geq   \beta >1.
\end{eqnarray*}
Hence the sequence $A_k^n(\varphi_2)$ is strongly log-concave. This implies unimodality.

We have $\frac{n^{2}-k^{2}}{n^{2}-\left( k+1\right) ^{2}}=1+\frac{2k+1}{n^{2}\left( k+1\right) ^{2}}$.
This is increasing as a function of~$k$
for $1\leq k \leq n-2$.
So the minimum is attained at 
$k=1$ and results in 
$1+\frac{3}{n^{2}-4}$.
For $n\rightarrow \infty $ $1+\frac{3}{n^{2}-4}$ again 
decreases to $1$.
Even if we could determine the minimum 
of the expression before the `strictly less'
sign, we would obtain the same
limiting behavior. 

\subsection{Proof of Corollary \ref{max1}}
The idea is simple. Let $n \geq 2$. Let $a_k = A_k^n(\varphi_1)$, $0 \leq k \leq n-1$.
We analyze the possible sign changes of
\begin{equation}
\Delta_k  (\varphi_1) := a_{k+1} - a_k = 
A_{k+1}^n (\varphi_1) - A_{k}^n (\varphi_1),  \quad  0 \leq k \leq n-2.
\end{equation}
We obtain 
\begin{equation}
\Delta_k  (\varphi_1) = \gamma(k,n)  \left( \frac{1}{\left( k+2\right) \left( k+1\right) }-\frac{1}{n-k-1}\right) ,
\end{equation}
where $\gamma(k,n)$ is a positive rational number. Hence $\Delta_k(\varphi_1)$ has the same sign as
$$ n+1 - (k+2)^2.$$
Hence the coefficients increase for $k<\sqrt{n+1}-2$, and decrease for
$k>\sqrt{n+1}-2$.
In the case of equality we have two maxima.


\subsection{Proof of Corollary \ref{max2}}
Let $n \geq 2$. Let $a_k = A_k^n(\varphi_2)$, $0 \leq k \leq n-1$.
We analyze the possible sign changes of
\begin{equation}
\Delta_k  (\varphi_2) := a_{k+1} - a_k =
A_{k+1}^n (\varphi_2) - A_{k}^n (\varphi_2), \quad   0 \leq k \leq n-2.
\end{equation} 
We obtain 
\begin{equation}
\Delta_k  (\varphi_2) = \gamma'(k,n)  \,
\left( \frac{n+k+1}{\left( k+2\right)
\left( 2k+3\right) \left( 2k+2\right) }-\frac{1}{n-k-1}\right) .
\end{equation}
where $\gamma'(k,n)$ is a positive rational number. Hence $\Delta_k(\varphi_2)$ has the same sign as
\begin{equation*}
-4k^{3}-19k^{2}-28k-13+n^{2}.
\end{equation*}
As a function of $k \geq 0$, the expression $\Delta_k(\varphi_2)$ is decreasing. Hence there exists exactly one $0 \leq K \leq n-1$ such that
$$ A_{K-1}^n (\varphi_2)  <   A_K^n (\varphi_2) \geq  A_{K+1}^n (\varphi_2).$$
We used the computer algebra system 
\texttt{Maple} to calculate the algebraic expression defining  $K$.
Nevertheless, it is obvious that $K \approx  2^{-2/3} \,\, n^{\frac{2}{3}}$ for large $n$.

\section{Explicit formulas for $P_n^{\varphi_1}(X)$}
Let us start with a list of the polynomials
for $n=1,2,\ldots ,7$. Note that they are
no longer integer-valued polynomials, although the modified coefficients are integral.
\begin{eqnarray*}
P_{1}^{\varphi_1}  (X) &  =  &   X  \\
P_{2}^{\varphi_1}  (X)             & = &    \frac{1}{2}X\left( X+2\right) \\
P_{3}^{\varphi_1}  (X)             & = & \frac{1}{6} X\left( X^{2}+6X+6\right) \\
P_{4}^{\varphi_1}  (X)             & = &  \frac{1}{24}X\left( X^{3}+12X^{2}+36X+24\right) \\
P_{5}^{\varphi_1}  (X)             & = &  \frac{1}{120}X\left( X^{4}+20X^{3}+120X^{2}+240X+120\right) \\
\end{eqnarray*}
\begin{eqnarray*}
P_{6}^{\varphi_1}  (X)             & = & \frac{1}{720}X\left(X^{5}+30X^{4}+300X^{3}+1200X^{2}+1800X+720\right) \\
P_{7}^{\varphi_1}  (X)             & = & \allowbreak \frac{1}{5040}X\left( X^{6}+42X^{5}+630X^{4}+4200X^{3}+12600X^{2}
+15120X+5040\right) . 
\end{eqnarray*}

\begin{remark}
The first polynomials $\frac{n!}{X} P_{n}^{\varphi_1}(X)$ are irreducible.
\end{remark}

\begin{proof}[Proof of Theorem \ref{varphi1}] 
We start with the identity
\begin{equation} \label{qn}
\sum_{n=0}^{\infty }P_{n}^{\varphi_1} \left( X  \right) \,\, q^{n}=\exp \left(
X         \,\, \sum_{n=1}^{\infty }q^{n}\right). 
\end{equation}
It is useful to substitute $\left( \sum_{n=1}^{\infty}q^{n}\right) ^{k}$ by the multi-index sum
$$\sum_{m_{1},\ldots ,m_{k}=1}^{\infty }q^{m_{1}+\cdots + m_{k}}.$$
Comparing the coefficients of $q^n$ in (\ref{qn}) leads to
\begin{eqnarray*}
P_{n}\left( X\right) &=&\sum_{k=1}^{n}\frac{1}{k!}\left( \sum_{m_{1}+\cdots
+m_{k}=n}1\right) X^{k} \\
&=&\frac{X}{n!}\sum_{k=0}^{n-1}\frac{n!}{\left(
k+1\right) !}\left( \sum_{m_{1}+\cdots +m_{k+1}=n}1\right) X^{k}.
\end{eqnarray*}
Note that
\begin{equation}
\sum_{m_{1}+\cdots +m_{k}=n}1=\binom{n-1}{k-1}.
\label{eq:linearcoeff}
\end{equation}
We prove (\ref{eq:linearcoeff})
by induction. For $k=1$ we have $1=\binom{n-1}{0}=\binom{n-1}{k-1}$.
Suppose the formula holds for~$k$ then 
\begin{eqnarray*}
\sum_{m_{1}+\cdots +m_{k+1}=n}1 &=&\sum_{m_{k+1}=1}^{n-k}\sum_{m_{1}+\cdots
+m_{k}=n-m_{k+1}}1=\sum_{m_{k+1}=1}^{n-k}\binom{n-m_{k+1}-1}{k-1} \\
&=&\sum_{m_{k+1}=k}^{n-1}\binom{m_{k+1}-1}{k-1}=\binom{n-1}{k}. 
\end{eqnarray*}
The proof (again by induction) of the identity
$\sum_{m=k}^{n-1}\binom{m-1}{k-1}=\allowbreak 
\binom{n-1}{k}$ we leave to the reader.
\end{proof}


\section{Explicit formulas for $P_n^{\varphi_2}(X)$}
Let us start with a list of the polynomials for
$n=1,2,\ldots 7$. Note that they are no longer integer-valued polynomials,
although the modified coefficients are integral.
\begin{eqnarray*}
P_{1}^{\varphi_2}  (X) &  =  &   X  \\
P_{2}^{\varphi_2}  (X) &  =  &  1/2\,X \left( 4+X \right) \\
P_{3}^{\varphi_2}  (X) &  =  &  1/6\,X \left( {X}^{2}+12\,X+18 \right) \\
P_{4}^{\varphi_2}  (X) &  =  & 1/24\,X \left( {X}^{3}+24\,{X}^{2}+120\,X+96 \right) \\ 
P_{5}^{\varphi_2}  (X) &  =  & \frac{1}{120}X \left( {X}^{4}+40\,{X}^{3}+420\,{X}^{2}+1200\,X+600 \right) \\ 
P_{6}^{\varphi_2}  (X) &  =  & \frac{1}{720}X \left( {X}^{5}+60\,{X}^{4}+1080\,{X}^{3}+6720\,{X}^{2}+12600
\,X+4320 \right)  \\ 
P_{7}^{\varphi_2}  (X) &  =  & \frac{1}{5040}X \left( {X}^{6}+84\,{X}^{5}+2310\,{X}^{4}+25200\,{X}^{3}+
105840\,{X}^{2}
+141120\,X+35280 \right)
\end{eqnarray*}
Before we prove Theorem \ref{varphi2} we show the following useful property.
\begin{lemma}\label{lemma}
For $n\geq 1$ and $1\leq k\leq n$ we obtain
\begin{equation}
\sum_{m_{1}+\cdots +m_{k}=n}m_{1} m_2 \cdots m_{k}=\binom{n+k-1}{2k-1}.
\label{eq:quadraticcoeff}
\end{equation}
\end{lemma}
\begin{proof}
The proof is by induction on~$n$ and $k$.
For $k=1$ we have $n=\binom{n}{1}=\binom{n+k-1}{2k-1}$.
Let now $N\geq K\geq 2$ be fixed.
We assume that
formula (\ref{eq:quadraticcoeff}) holds for $k<K$. For $k=K$ we also
assume that the
formula (\ref{eq:quadraticcoeff}) holds
for $n<N$. Then we will prove
formula (\ref{eq:quadraticcoeff}) for
$n=N$ and $k=K$. We
obtain for
\begin{equation*}
\sum_{m_{1}+\cdots +m_{K}=N}m_{1} m_2 \cdots m_{K} 
\end{equation*}
the expressions
\begin{eqnarray*}
&&\sum_{m_{K}=1}^{N+1-K}m_{K}\sum_{m_{1}+\cdots
+m_{K-1}=N-m_{K}}m_{1} m_2 \cdots m_{K-1} \\
&=&\sum_{m_{K}=1}^{N+1-K}
\sum_{\substack{m_{1}+\cdots +m_{K-1} \\ =N-m_{K}}}
m_{1}\cdots
m_{K-1}+\sum_{m_{K}=1}^{N-K}m_{K}
\sum_{
\substack{
m_{1}+\cdots
+m_{K-1} \\ =N-1-m_{K}}}       m_{1} m_2 \cdots m_{K-1} \\
&=&\sum_{m_{K}=1}^{N+1-K}\binom{N-m_{K}+K-2}{2K-3}+
\sum_{m_{1}+\cdots
+m_{K}=N-1}m_{1} m_2 \cdots m_{K} \\
&=&\sum_{m_{K}=2K-2}^{N+K-2}\binom{m_{K}-1}{2K-3}+\binom{N+K-2}{2K-1} \\
&=&\binom{N+K-2}{2K-2}+\binom{N+K-2}{2K-1}=\binom{N+K-1}{2K-1}.
\end{eqnarray*}
Note that here we again used the identity 
$$\sum_{m=k}^{n-1}\binom{m-1}{k-1}=\allowbreak 
\binom{n-1}{k}.$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{varphi2}] 
We obtain 
\begin{eqnarray*}
\exp \left( X\sum_{n=1}^{\infty }\frac{n^{2}}{n}q^{n}\right)  &=&\allowbreak
1+\sum_{k=1}^{\infty }\frac{1}{k!}X^{k}\left( \sum_{n=1}^{\infty
}nq^{n}\right) ^{k} \\
&=&\allowbreak 1+\sum_{k=1}^{\infty }\frac{1}{k!}X^{k}\left(
\sum_{m_{1}=1}^{\infty }\cdots \sum_{m_{k}=1}^{\infty }m_{1}\cdots
m_{k}q^{m_{1}+\cdots +m_{k}}\right)  
\\
&=&\allowbreak 1+\sum_{n=1}^{\infty }\sum_{k=1}^{n}\frac{1}{k!}X^{k}\left(
\sum_{m_{1}+\cdots +m_{k}=n}m_{1} m_2\cdots m_{k}\right) q^{n}.
\end{eqnarray*}
In the next step we apply Lemma \ref{lemma} and obtain
\begin{eqnarray*}
&&\allowbreak 1+\sum_{n=1}^{\infty }\sum_{k=1}^{n}\frac{1}{k!}\binom{n+k-1}{2k-1}X^{k}q^{n} \\
&=&\allowbreak 1+\sum_{n=1}^{\infty }\frac{X}{n!}\sum_{k=0}^{n-1}\frac{n!}{\left( k+1\right) !}\binom{n+k}{2k+1}X^{k}q^{n},
\end{eqnarray*}
which yields the desired result.
\end{proof}

\section{Data}

\begin{figure}[H]
\begin{center}
\includegraphics[width=6.1cm,angle=270]{comparison.eps}
\caption{Quotient $P_{n}^{\sigma }\left( 1\right) /P_{n}^{\varphi _{1}}\left( 1\right) $ depending on $n$. }
\end{center}
\end{figure}

\begin{table}[H]
\[
\begin{array}{|r|r|r|r|}
\hline 
n & P_{n}^{1}\left( 1\right)  & P_{n}^{\sigma }\left( 1\right)  & P_{n}^{2}\left( 1\right)  \\ \hline \hline 
0 & 1 & 1 & 1 \\ \hline 
1 & 1 & 1 & 1 \\ \hline 
2 & 1.5 & 2 & 2.5 \\ \hline 
3 & 2.16667 & 3 & 5.16667 \\ \hline 
4 & 3.04167 & 5 & 10.0417 \\ \hline 
5 & 4.175 & 7 & 18.8417 \\ \hline 
6 & 5.62639 & 11 & 34.4181 \\ \hline 
7 & 7.46687 & 15 & 61.4752 \\ \hline 
8 & 9.78058 & 22 & 107.694 \\ \hline 
9 & 12.6669 & 30 & 185.485 \\ \hline 
10 & 16.2426 & 42 & 314.694 \\ \hline 
11 & 20.6448 & 56 & 526.768 \\ \hline 
12 & 26.0337 & 77 & 871.113 \\ \hline 
13 & 32.5961 & 101 & 1424.73 \\ \hline 
14 & 40.5493 & 135 & 2306.78 \\ \hline 
15 & 50.1454 & 176 & 3700.32 \\ \hline 
16 & 61.676 & 231 & 5884.91 \\ \hline 
17 & 75.4781 & 297 & 9284.78 \\ \hline 
18 & 91.9399 & 385 & 14540.1 \\ \hline 
19 & 111.508 & 490 & 22612 \\ \hline 
20 & 134.694 & 627 & 34935.4 \\ \hline 
21 & 162.087 & 792 & 53643.4 \\ \hline 
22 & 194.356 & 1002 & 81891.7 \\ \hline 
23 & 232.27 & 1255 & 124329 \\ \hline 
24 & 276.702 & 1575 & 187773 \\ \hline 
25 & 328.648 & 1958 & 282189 \\ \hline 
\end{array}
\]
\caption{Values
of $P_{n}^{\varphi _{1}}\left( 1\right) $,
the partition numbers
$P_{n}^{\sigma }\left( 1\right) $, and
$P_{n}^{\varphi _{2}}\left( 1\right) $.}
\end{table}


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\bibitem{HNR17} B. Heim, M. Neuhauser, and F. Rupp, Fourier
coefficients of powers of the Dedekind eta function,
\textit{Ramanujan J.\/} (2017).  Available at
\url{http://dx.doi.org/10.1007/s11139-017-9923-4}.

\bibitem{Le47} D. Lehmer, 
The vanishing of Ramanujan's $\tau(n)$, \textit{Duke Math.\ J.\/}
 \textbf{14} (1947), 429--433.
 
\bibitem{Ne55} M. Newman, 
An identity for the coefficients of certain modular forms,
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\bibitem{OR95} K. Ono and S. Robins, Superlacunary cusp forms,
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\bibitem{On03}K. Ono, \textit{The Web of Modularity: Arithmetic of the
Coefficients of Modular Forms and $q$-series\/},
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\bibitem{Se85} 
J. Serre,
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\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 05A10, 11B73, 11F20.

\noindent \emph{Keywords: } 
binomial coefficient, Dedekind eta function, Lehmer conjecture, 
linear recurrence, log-concavity,  special sequence.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequence
\seqnum{A000041},
\seqnum{A000594},
\seqnum{A089231}, and
\seqnum{A322970}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received  March 21 2018;
revised version received January 3 2019.  
Published in {\it Journal of Integer Sequences}, January 6 2019.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in

\end{document}

