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\title{\bf  A criterion for the log-convexity
 of  combinatorial sequences}

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% the address should include the country, and does not have to include
% the street address

\author{Ernest X.~W. Xia \thanks{Supported   by
 the National  Natural  Science Foundation of China (11201188).}\\
\small  Department of Mathematics\\[-0.8ex]
\small Jiangsu University\\[-0.8ex]
\small Zhenjiang, Jiangsu,   P. R. China\\
\small\tt ernestxwxia@163.com\\
\and
Olivia X.~M. Yao\\
\small Department of Mathematics\\[-0.8ex]
\small Jiangsu University\\[-0.8ex]
\small Zhenjiang, Jiangsu,   P. R. China\\
\small\tt yaoxiangmei@163.com }

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{May 28, 2013 }{Sep 30, 2013}{Oct 14, 2013}\\
\small Mathematics Subject Classifications: 05A20, 11B83}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
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\begin{abstract}
  Recently, Do\v{s}li\'{c}, and Liu and Wang developed
techniques for dealing with the log-convexity of sequences. In this
paper, we present a  criterion    for the log-convexity   of some
combinatorial sequences.
 In order to prove  the  log-convexity
  of a sequence satisfying a three-term recurrence, by our method,
it suffices to compute a constant number of terms at the beginning
 of the sequence. For example, in order to prove the log-convexity
  of  the Ap\'{e}ry
 numbers $A_n$, by our method, we just need  to evaluate  the values
 of $A_n$ for $0\leq n \leq 6$. As applications, we   prove the
 log-convexity of some famous sequences including the
  Catalan-Larcombe-French   numbers.
  This confirms a conjecture given by Sun.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:}    log-convexity;
   three-term recurrence;
  combinatorial sequences

\end{abstract}

\section{Introduction}

A positive sequence $\{S_n\}_{n=0}^\infty $ is said to be log-convex
(respectively log-concave)
 if for $n\geq 1$,
 \begin{align}
 \frac{S_n}{S_{n-1}}\leq \frac{S_{n+1}}{S_n} \qquad ({\rm respectively }\ \ \frac{S_n}{S_{n-1}}\geq
 \frac{S_{n+1}}{S_n}). \label{1-1}
 \end{align}
 Meanwhile, the sequence  $\{S_n\}_{n=0}^\infty $
  is called strictly log-convex (log-concave) if the inequality
in (1.1)  is strict for all $n\geq 1$. In 1994, Engel \cite{Engel}
proved the log-convexity of the Bell numbers. Recently,
Do\v{s}li\'{c} \cite{D,D3,D4},   Do\v{s}li\'{c}
 and  Veljan \cite{D2},  and Liu and Wang  \cite{Liu}
developed techniques for
 proving  the log-convexity of sequences.
  Do\v{s}li\'{c} \cite{D,D3,D4} presented several methods for
   dealing with  the  log-convexity of combinatorial sequences.
   He proved that the Motzkin numbers, the Fine numbers, the
    Franel numbers of order 3 and 4,  the Ap\'{e}ry numbers,
     the large Schr\"{o}der numbers, the  derangements numbers  and  the
     central Delannoy numbers are log-convex. In their wonderful paper
      \cite{Liu}, Liu and Wang
    proved that the
log-convexity is preserved under componentwise sum, under binomial
convolution, and by the linear transformations given by the matrices
of binomial coefficients and Stirling numbers of two kinds. Many
combinatorial sequences satisfy a three-term recurrence.   Liu and
Wang \cite{Liu}  presented  some criterions for the log-convexity of
the sequences $\{z_n\}_{n=0}^\infty$ satisfying the following
recurrence
\begin{align}
a(n)z_{n+1}=b(n)z_n+c(n)z_{n-1}, \label{1-2}
\end{align}
where  $a(n)$, $b(n)$ and $c(n)$ are positive for $n\geq 1$, Liu and
Wang \cite{Liu} proved the following theorem.

\begin{theorem} \label{Th-1} Let $\{z_n\}_{n=0}^\infty$ be defined by
\eqref{1-2} and
\begin{align}
\lambda_n=\frac{b(n)+\sqrt{b^2(n)+4a(n)c(n)}}{2a(n)}.\label{1-3}
\end{align}
 Suppose that  $z_0,\ z_1,\ z_2,\ z_3$  is
log-convex and that the inequality
\begin{align}
a(n)\lambda_{n-1}\lambda_{n+1}-b(n)\lambda(n-1)-c(n)\geq
0\label{1-4}
\end{align}
is true for $n\geq 2$. Then the sequence $\{z_n\}_{n=0}^\infty$ is
log-convex.

\end{theorem}

 Liu and Wang  \cite{Liu} also considered the
log-convexity of the sequence $\{z_n\}_{n=0}^\infty $ defined by
 \begin{align}
(\alpha_n+\alpha_0)z_{n+1}=(\beta_1n+\beta_0)z_n-(\gamma_1n+\gamma_0)z_{n-1}.
\label{1-5}
 \end{align}
for $n\geq 1$. They gave   criterions for the log-convexity of the
sequences $\{z_n\}_{n=0}^\infty$.
 Employing their criterions, they proved the
 log-convexity of some combinatorial sequences. Liu \cite{Liu-0}
  gave sufficient conditions for
  the positivity of  the  sequences defined by \eqref{1-5}.

Motivated by these results established  by Liu and Wang \cite{Liu},
in this paper,  we investigate the log-convexity problem of the
sequence $\{S_n\}_{n=0}^\infty$ having the following  three-term
recurrence
\begin{align}\label{1-6}
S_n=\frac{\sum_{i=0}^k a_i
n^i}{\sum_{i=0}^kb_in^i}S_{n-1}-\frac{\sum_{i=0}^l c_i
n^i}{\sum_{i=0}^l d_i n^i}S_{n-2}  \qquad (n\geq 2),
\end{align}
where $\gcd(\sum_{i=0}^k a_i n^i,
\sum_{i=0}^kb_in^i)=\gcd(\sum_{i=0}^l c_i n^i,\sum_{i=0}^l d_i
n^i)=1$ and  $k$, $l$, $a_k $, $b_k$, $c_l$ and $d_l$ are positive
numbers.
 The authors \cite{Xia} gave a criterion for the positivity of the sequence
  $\{S_n\}_{n=0}^\infty$ defined by \eqref{1-6}.
 The aim of this paper is to present   a    criterion
  for the log-convexity
    of some famous combinatorial  sequences.    By our method, in order to
   determine the  log-convexity
  of the  sequence  $\{S_n\}_{n=0}^\infty$ defined by \eqref{1-6},
it suffices to compute a constant number of terms at the beginning
 of the sequence $\{S_n\}_{n=0}^\infty$. As applications, we prove some
   famous combinatorial sequences are strictly log-convex. Specially,  we show that
   the Catalan-Larcombe-French numbers $\{P_n\}_{n=0}^\infty$
    is strictly log-convex which confirms a conjecture given by Sun \cite{Sun-2}.


In order to state our main result, we first introduce some
notations. Given a polynomial $f(n)$ defined by
\begin{align}
f(n)=\sum_{i=0}^k f_i n^i,\label{1-7}
\end{align}
where $f_i$ $(0\leq i \leq k)$ are real numbers and $f_k>0$.  Define
an operator $L$ on $f(n)$ by
\begin{align}
L(f(n))=\frac{1}{f_k}\sum_{0 \leq i\leq k-1,\ f_i<0} |f_i|.
\label{1-8}
\end{align}
For example,
\begin{align}
L(5n^4-2n^3+4n^2-6n-3)= \frac{11}{5}. \label{1-9}
\end{align}
It is easy to see that  $f(n)>0$ for $n\geq \left[L(f(n))\right]+1$.

Throughout this paper, we always let
 \begin{align}
\frac{\sum_{i=0}^k a_i
(n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}&-\frac{\sum_{i=0}^k a_i
(n+1)^i}{\sum_{i=0}^kb_i(n+1)^i}=\frac{\sum_{j=0}^r e_j
n^j}{\sum_{t=0}^s h_t n^t },\label{1-10}\\[6pt]
\frac{\sum_{i=0}^l c_i (n+2)^i}{\sum_{i=0}^l d_i (n+2)^i}&-
\frac{\sum_{i=0}^l c_i (n+1)^i}{\sum_{i=0}^l d_i (n+1)^i}
=\frac{\sum_{j=0}^u p_j n^j}{\sum_{t=0}^v q_t n^t }, \label{1-11}
 \end{align}
 and
\begin{align}
 &\frac{\sum_{i=0}^k a_i
(n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}-\frac{\sum_{i=0}^l c_i
(n+2)^i}{\sum_{i=0}^l d_i (n+2)^i}\frac{\left(\sum_{t=0}^v q_t
n^t\right)\left(\sum_{j=0}^r e_j n^j\right) }{\left(\sum_{j=0}^u p_j
n^j\right)\left(\sum_{t=0}^s h_t n^t \right)}\nonumber\\[6pt]
&\qquad \qquad\qquad  -\frac{\left(\sum_{j=0}^u p_j
(n+1)^j\right)\left(\sum_{t=0}^s h_t (n+1)^t
\right)}{\left(\sum_{t=0}^v q_t (n+1)^t\right)\left(\sum_{j=0}^r e_j
(n+1)^j\right) }=\frac{\sum_{i=0}^\alpha x_i n^i }{\sum_{i=0}^\beta
y_i n^i}, \label{1-12}
\end{align}
where $h_s>0$, $q_v>0$,  $y_\beta>0$ and
\begin{align*}\gcd\left(
\sum_{j=0}^r e_j n^j,\sum_{t=0}^s h_t n^t \right)=\gcd\left(
\sum_{j=0}^u p_j n^j, \sum_{t=0}^v q_t n^t
\right)=\gcd\left(\sum_{i=0}^\alpha x_i n^i,
 \sum_{i=0}^\beta y_i n^i \right)=1.
 \end{align*}



Our main result can be stated as follows.

\begin{theorem} \label{Th-2} Let $\{S_n\}_{n=0}^\infty$ be a positive sequence and
 satisfy
 \eqref{1-6}.  If   $p_u>0$, $e_r>0$, $x_\alpha >0$ and
 there exists an integer $N_0$ such that
 \begin{align}
 N_0\geq r_1&= \max \bigg\{\left[L(\sum_{i=0}^l c_i n^i)\right],
\left[L(\sum_{i=0}^l d_i n^i)\right], \left[L(\sum_{j=0}^r e_j
n^j)\right],
 \left[L(\sum_{t=0}^s h_t n^t)\right],\nonumber\\[6pt]
&\qquad \quad \left[L(\sum_{j=0}^u p_j n^j)\right],
\left[L(\sum_{t=0}^v q_t n^t)\right], \left[L(\sum_{i=0}^\alpha  x_i
n^i)\right], \left[L(\sum_{i=0}^\beta y_i n^i)\right]
\bigg\}+1\label{1-13}
  \end{align}
   and
\begin{align}
 \frac{S_{N_0}}{S_{N_0-1}} &<
  \frac{S_{N_0+1}}{S_{N_0}},\label{1-16}\\[6pt]
\frac{S_{N_0+1}}{S_{N_0}}&>  \frac{\left(\sum_{j=0}^u p_j
N_0^j\right)\left(\sum_{t=0}^s h_t N_0^t \right)}{\left(\sum_{t=0}^v
q_t N_0^t\right)\left(\sum_{j=0}^r e_j N_0^j\right) },\label{1-17}
\end{align}
 then
 the sequence  $\{S_n\}_{n=N_0}^\infty $ is strictly log-convex, namely,
 \begin{align}
\frac{S_n}{S_{n-1}} < \frac{S_{n+1}}{S_n}, \qquad (n\geq
N_0).\label{1-18}
 \end{align}
 \end{theorem}

 This paper is organized as follows.
 We give the proof  of Theorem  \ref{Th-2} in Sections 2.
   As
applications of Theorem \ref{Th-2},   in Section 3,  we    prove the
 log-convexity of some famous sequences including the
  Catalan-Larcombe-French   numbers.
  This confirms a conjecture given by Sun \cite{Sun-2}.










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{Th-2}}


 In this section, we present the proof of Theorem \ref{Th-2}.

\begin{proof}  By the definition of $r_1$, we see that for all $n\geq
N_0\geq r_1$,
\begin{align}
\frac{\sum_{i=0}^l c_i n^i}{\sum_{i=0}^l d_i n^i}&>0,  \label{2-1}\\[6pt]
\frac{\sum_{i=0}^k a_i
(n+2)^i}{\sum_{i=0}^kb_i(n+2)^i}-&\frac{\sum_{i=0}^k a_i
(n+1)^i}{\sum_{i=0}^kb_i(n+1)^i}=\frac{\sum_{j=0}^r e_j
n^j}{\sum_{t=0}^s h_t n^t }>0, \label{2-2}\\[6pt]
\frac{\sum_{i=0}^k c_i (n+2)^i}{\sum_{i=0}^k d_i (n+2)^i}-&
\frac{\sum_{i=0}^k c_i (n+1)^i}{\sum_{i=0}^k d_i
(n+1)^i}=\frac{\sum_{j=0}^u p_j n^j}{\sum_{t=0}^v q_t n^t }>0,
\label{2-3}
\end{align}
and
\begin{align}
\frac{\sum_{i=0}^\alpha x_i n^i }{\sum_{i=0}^\beta y_i n^i}>0.
\label{2-4}
\end{align}
We first give a lower bound for $\frac{S_{n+1}}{S_{n}}$. Moreover,
we prove
  that for $n\geq N_0$,
\begin{align}
\frac{S_{n+1}}{S_{n}}> \frac{\left(\sum_{j=0}^u p_j
n^j\right)\left(\sum_{t=0}^s h_t n^t \right)}{\left(\sum_{t=0}^v q_t
n^t\right)\left(\sum_{j=0}^r e_j n^j\right) }. \label{2-5}
\end{align}
  We prove   \eqref{2-5}
by induction on $n$. By  \eqref{1-17}, we  see that  \eqref{2-5}
holds for $n=N_0$. Suppose that  \eqref{2-5} holds for $n=m\geq
N_0$, that is,
\begin{align}
\frac{S_{m+1}}{S_{m}}>\frac{\left(\sum_{j=0}^u p_j
m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}{\left(\sum_{t=0}^v q_t
m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }.\label{2-6}
\end{align}
It follows from \eqref{2-1}, \eqref{2-2} and \eqref{2-6} that for
$m\geq N_0$,
\begin{align}
-\frac{\sum_{i=0}^l c_i (m+2)^i}{\sum_{i=0}^l d_i
(m+2)^i}\frac{S_{m}}{S_{m+1}}>- \frac{\sum_{i=0}^l c_i
(m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t
m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j
m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}.\label{2-7}
\end{align}
Now, we are ready to show that  \eqref{2-5} also holds for $n=m+1$.
Employing \eqref{1-6} and \eqref{2-7}, we deduce that
\begin{align}
\frac{S_{m+2}}{S_{m+1}}&=\frac{\sum_{i=0}^k a_i
(m+2)^i}{\sum_{i=0}^kb_i(m+2)^i} -\frac{\sum_{i=0}^l c_i
(m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{S_{m}}{S_{m+1}}\nonumber\\[6pt]
&> \frac{\sum_{i=0}^k a_i
(m+2)^i}{\sum_{i=0}^kb_i(m+2)^i}-\frac{\sum_{i=0}^l c_i
(m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t
m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j
m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}.\label{2-8}
\end{align}
In view of \eqref{1-12},  \eqref{2-4} and  \eqref{2-8}, we find that
for $m\geq N_0$
\begin{align}
&\frac{S_{m+2}}{S_{m+1}}-\frac{\left(\sum_{j=0}^u p_j
(m+1)^j\right)\left(\sum_{t=0}^s h_t (m+1)^t
\right)}{\left(\sum_{t=0}^v q_t (m+1)^t\right)\left(\sum_{j=0}^r e_j
(m+1)^j\right) }\nonumber\\[6pt]
> & \frac{\sum_{i=0}^k a_i
(m+2)^i}{\sum_{i=0}^kb_i(m+2)^i}-\frac{\sum_{i=0}^l c_i
(m+2)^i}{\sum_{i=0}^l d_i (m+2)^i}\frac{\left(\sum_{t=0}^v q_t
m^t\right)\left(\sum_{j=0}^r e_j m^j\right) }{\left(\sum_{j=0}^u p_j
m^j\right)\left(\sum_{t=0}^s h_t m^t \right)}\nonumber\\[6pt]
&\qquad -\frac{\left(\sum_{j=0}^u p_j
(m+1)^j\right)\left(\sum_{t=0}^s h_t (m+1)^t
\right)}{\left(\sum_{t=0}^v q_t (m+1)^t\right)\left(\sum_{j=0}^r e_j
(m+1)^j\right) }\nonumber\\[6pt]
=&  \frac{\sum_{i=0}^\alpha x_i m^i }{\sum_{i=0}^\beta y_i m^i}>0,
\label{2-9}
\end{align}
which implies that  \eqref{2-5} is true for $n=m+1$. By induction,
we have proved \eqref{2-5} holds for $n\geq N_0$.


Now, we turn to prove \eqref{1-18}.  We also prove  \eqref{1-18} by
induction on $n$. It follows from \eqref{1-16} that \eqref{1-18}
holds for $n=N_0$. Assume that \eqref{1-18} is true for $n=m\geq
N_0$, namely,
 \begin{align}
\frac{S_{m}}{S_{m-1}}< \frac{S_{m+1}}{S_m}.\label{2-10}
 \end{align}
 By \eqref{2-1} and \eqref{2-10}, we  find that for $m\geq N_0$
\begin{align}
 \frac{\sum_{i=0}^l c_i
(m+1)^i}{\sum_{i=0}^l d_i (m+1)^i} \frac{S_{m-1}}{S_{m}} >
\frac{\sum_{i=0}^l c_i (m+1)^i}{\sum_{i=0}^l d_i (m+1)^i}
\frac{S_{m}}{S_{m+1}}.\label{2-11}
\end{align}
Employing  \eqref{1-6}, \eqref{1-10}, \eqref{1-11}, \eqref{2-3},
\eqref{2-5} and \eqref{2-11}, we deduce that for $m\geq N_0$,
 \begin{align}
\frac{S_{m+2}}{S_{m+1}}-\frac{S_{m+1}}{S_{m}}&=\frac{\sum_{i=0}^k
a_i(m+2)^i }{\sum_{i=0}^k b_i(m+2)^i} - \frac{\sum_{i=0}^l
c_i(m+2)^i }{\sum_{i=0}^l d_i(m+2)^i}  \frac{S_{m}}{S_{m+1}}
\nonumber\\[6pt]
&\qquad \qquad   -\frac{\sum_{i=0}^k a_i(m+1)^i }{\sum_{i=0}^k
b_i(m+1)^i} + \frac{\sum_{i=0}^l c_i(m+1)^i }{\sum_{i=0}^l
d_i(m+1)^i} \frac{S_{m-1}}{S_{m}} \nonumber
\\[6pt]
&>  \frac{\sum_{i=0}^k a_i(m+2)^i }{\sum_{i=0}^k b_i(m+2)^i}
-\frac{\sum_{i=0}^k a_i(m+1)^i }{\sum_{i=0}^k b_i(m+1)^i}
\nonumber\\[6pt]
&\qquad \qquad + \left( \frac{\sum_{i=0}^l c_i(m+1)^i }{\sum_{i=0}^l
d_i(m+1)^i} -\frac{\sum_{i=0}^l c_i(m+2)^i }{\sum_{i=0}^l
d_i(m+2)^i} \right)\frac{S_m}{S_{m+1}} \nonumber\\[6pt]
&= \frac{\sum_{j=0}^r e_j m^j}{\sum_{t=0}^s h_t m^t }-
\frac{\sum_{j=0}^u p_j m^j}{\sum_{t=0}^v q_t m^t }
\frac{S_m}{S_{m+1}}\nonumber\\[6pt]
&> \frac{\sum_{j=0}^r e_j m^j}{\sum_{t=0}^s h_t m^t }-
\frac{\sum_{j=0}^u p_j m^j}{\sum_{t=0}^v q_t m^t } \left(
\frac{\left(\sum_{t=0}^v q_t m^t\right)\left(\sum_{j=0}^r e_j
m^j\right) }{\left(\sum_{j=0}^u p_j m^j\right)\left(\sum_{t=0}^s h_t
m^t \right)} \right)=0,
\end{align}
which implies that \eqref{1-18} holds for $n=m+1$. Theorem
\ref{Th-2} is proved by induction.  This completes the proof.
\end{proof}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications of Theorem \ref{Th-2}}

In this section, employing the criterion given in this paper, we
prove some results on the log-convexity    of
 some combinatorial sequences.



The Catalan-Larcombe-French numbers $P_n$ for $n\geq 0$ were first
defined by Catalan in \cite{Catalan},
  in terms of the ``Segner numbers". Catalan stated that the
$P_n$ could be defined by the following  recurrence relation:
\begin{align}\label{3-2}
P_n=\frac{8(3n^2-3n+1)}{n^2}P_{n-1}-\frac{128(n-1)^2}{n^2}P_{n-2},
\end{align}
for $n\geq 2$, with the initial values given by $P_0=1$ and $P_1=8$.
Larcombe and French \cite{Larcombe} gave a detailed account of
properties of   $P_n$, and obtained  the following formulas for
these numbers:
\begin{align}\label{v-1}
P_n=\sum_{k=0}^n\frac{{2k\choose k}^2{2(n-k)\choose
n-k}^2}{{n\choose k}}=2^n\sum_{k=0}^{\lfloor n/2\rfloor}{n\choose
2k} {2k\choose k}^2 4^{n-2k}
\end{align}
 and
\begin{align}\label{v-2}
P_n=\frac{1}{n!}\sum_{r+s=n}{2r\choose r}{2s\choose s}
\frac{(2r)!(2s)!}{r!s!}=\sum_{r+s=n}\frac{{2r\choose r}^2{2s\choose
s}^2 }{{n\choose r}}
\end{align}
for $n\geq 0$.
 The first few
$P_n$ are $1,
 \ 8,\  80,\
896,\  10816,\  137728$. This is the sequence A053175 in Sloane's
database \cite{Sloane}. The sequence $\{P_n\}_{n=0}^\infty$ is also
related to the theory of modular forms; see \cite{Zagier}.

Recently, Sun \cite{Sun-2} conjectured that

\begin{conjecture}\label{c-2} The sequences $\{P_{n+1}/P_n\}_{n=0}^\infty $
 and $\{\sqrt[n]{P_n}\}_{n=1}^\infty$  are strictly increasing.
\end{conjecture}

Employing Theorem \ref{Th-2}, we  prove that

\begin{corollary}\label{c-3}
Conjecture \ref{c-2} is true.
\end{corollary}

\noindent{\it Proof.} By \eqref{1-13}, we find $r_1=3$. Set $N_0=3$.
It is easy to check that \eqref{1-16} and \eqref{1-17} hold for
$N_0=3$. By Theorem \ref{Th-2}, we see that the sequence
 $\{P_n\}_{n=3}^\infty$ is strictly log-convex. It is a routine to verify
 that  $\frac{P_{i+1}}{P_i}>
\frac{P_i}{P_{i-1}}$ for $1\leq i \leq 3$. Thus, the  sequence
 $\{P_n\}_{n=0}^\infty$ is strictly log-convex and the
  sequence $\{P_{n+1}/P_n\}_{n=0}^\infty $ is strictly increasing,
  namely,
  \begin{align}
\frac{P_{n+1}}{P_n}> \frac{P_n}{P_{n-1}}, \qquad n\geq 1.\label{3-3}
  \end{align}
By \eqref{3-3} and the fact $P_0=1$, we deduce  that
\begin{align}\label{3-4}
P_n=P_0\prod_{i=1}^n\frac{P_i}{P_{i-1}}<\left(\frac{P_{n+1}}{P_n}\right)^n,
\end{align}
which implies that
\begin{align}
P_n^{n+1}<P_{n+1}^n.\label{3-5}
\end{align}
It follows from \eqref{3-5} that the sequences
$\{\sqrt[n]{P_n}\}_{n=1}^\infty$
 is strictly increasing. This completes the proof. \qed


  The  Ap\'{e}ry number $A_n$ is defined by
 \begin{align}
A_n=\frac{34n^3-51n^2+27n-5}{n^3}
A_{n-1}-\frac{(n-1)^3}{n^3}A_{n-2}, \qquad n\geq 2, \label{3-1}
 \end{align}
with $A_0=1$ and $A_1=5$.  The Ap\'{e}ry numbers play a key role in
Ap\'{e}ry's proof of the irrationality of $\zeta
(3)=\sum_{n=1}^\infty \frac{1}{n^3}$;  see \cite{ap}. The
log-convexity of $\{A_n\}_{n=0}^\infty $ was proved by
Do\v{s}li\'{c}  \cite{D}. Chen and Xia  \cite{Chen} proved that the
sequence $\{A_n\}_{n=0}^\infty $ is 2-log-convex. Now, we present
another proof of the log-convexity of $\{A_n\}_{n=0}^\infty $. Set
$k=l=3$, $a_3 = 34$ and  $b_3 =c_3= d_3 = 1$ in Theorem \ref{Th-2}.
By the definition of $r_1$, we obtain $r_1=5$. Set $N_0=5$. We can
check that \eqref{1-16} and \eqref{1-17} hold for $N_0=5$. Thus, by
Theorem \ref{Th-2},  the sequence $\{A_n\}_{n=5}^\infty $ is
log-convex. We can also verify that $\frac{A_{i+1}}{A_i}>
\frac{A_i}{A_{i-1}}$ for $1\leq i \leq 5$. Thus,  the following
corollary is true.

\begin{corollary}\label{c-1}
 The sequence  $\{A_{n}\}_{n=0}^\infty $ is strictly log-convex.
\end{corollary}

 The central Delannoy number $D_n$ is defined by
 \begin{align}
D_n=\frac{3(2n-1)}{n}D_{n-1}-\frac{n-1}{n}D_{n-2}, \qquad n\geq
2,\label{3-6}
 \end{align}
with $D_0=1$ and $D_1=3$; see \cite{P}.  Do\v{s}li\'{c} \cite{D} ,
and Liu and Wang \cite{Liu} proved the log-convexity of the sequence
$\{D_n\}_{n=0}^\infty$. By \eqref{1-13}, we find $r_1=2$. Let
$N_0=2$. It is easy to check that \eqref{1-16} and \eqref{1-17} hold
for $N_0=2$.  The following corollary follows from Theorem
\ref{Th-2} and the fact $\frac{D_2}{D_1}>\frac{D_1}{D_0}$.

\begin{corollary}\label{c-4}
 The sequence  $\{D_{n}\}_{n=0}^\infty $ is strictly log-convex.
\end{corollary}

The little Schr\"{o}er number $s_n$ is defined by
\begin{align}
s_n=\frac{3(2n-1)}{n+1}s_{n-1}-\frac{n-2}{n+1}s_{n-2}, \qquad n\geq
2, \label{3-7}
\end{align}
with $s_0=1$ and $s_1=1$; see \cite{Foata,Sul}. Do\v{s}li\'{c}
\cite{D} , and Liu and Wang \cite{Liu} proved the log-convexity of
the sequence $\{s_n\}_{n=0}^\infty$.  It is easy to see that
$r_1=3$.  Let $N_0=3$. We can check  that \eqref{1-16} and
\eqref{1-17} hold for $N_0=3$.  The following corollary follows from
Theorem \ref{Th-2} and the fact
$\frac{s_3}{s_2}>\frac{s_2}{s_1}>\frac{s_1}{s_0}$.

\begin{corollary}\label{c-5}
 The sequence  $\{s_{n}\}_{n=0}^\infty $ is strictly log-convex.
\end{corollary}

Let $R_n$
 be the number of the set of all tree-like polyhexes with $n+1$ hexagons.
  The sequence $\{R_n\}_{n=0}^\infty $ satisfies the recurrence
  \begin{align}
    R_n=\frac{3(2n-1)}{n+1}R_{n-1}-\frac{5(n-2)}{n+1}R_{n-2},\qquad
    n\geq 2, \label{3-8}
  \end{align}
  with $R_0=1$ and $R_1=1$; see \cite{Harary}. The sequence $\{R_n\}_{n=0}^\infty$
  is   the sequence  A002212 in Sloane's
database \cite{Sloane}. Liu and Wang \cite{Liu} proved the
log-convexity of the sequence $\{R_n\}_{n=0}^\infty$. Let  $N_0=3$.
Employing Theorem
  \ref{Th-2} and evaluating the values of $R_2,\ R_3$ and $R_4$, we
  can prove the following corollary.

  \begin{corollary}\label{c-6}
 The sequence  $\{R_{n}\}_{n=0}^\infty $ is strictly log-convex.
\end{corollary}

Let $w_n$ be the number of walks on cubic lattice with $n$ steps,
starting and finishing on the $x-y$ plane and never going below it.
 The sequence $\{w_n\}_{n=0}^\infty $ has three-term recurrence relation
 \begin{align}
w_n=\frac{4(2n+1)}{n+2}w_{n-1}-\frac{12(n-1)}{n+2}w_{n-2}, \qquad
n\geq 2, \label{3-9}
 \end{align}
with $w_0=1$ and $w_1=4$; see \cite{Guy}. The sequence
$\{w_n\}_{n=0}^\infty$
  is  the sequence A005572 in Sloane's
database \cite{Sloane}. Liu and Wang \cite{Liu} proved the
log-convexity
   of the sequence $\{w_n\}_{n=0}^\infty$.  Set $N_0=2$.
 The following corollary follows from Theorem \ref{Th-2} and the
 fact $\frac{w_{i+1}}{w_i}>\frac{w_i}{w_{i-1}}$ for $i=1,\ 2 $.

   \begin{corollary}\label{c-7}
 The sequence  $\{w_{n}\}_{n=0}^\infty $ is strictly log-convex.
\end{corollary}

Let $F_n$ be defined by
\begin{align}
F_n=\frac{4n^4-n^3-n^2+3n+2 }{n^4+2n^2-1 } F_{n-1} -\frac{
2n^3-5n^2-n+1 }{ 2n^3-3n^2+2n}F_{n-2}, \qquad n\geq 2, \label{3-10}
\end{align}
with $F_0=1$ and $F_1=1$. By \eqref{1-13}, we find $r_1=42$.
  Set $N_0=42$. It is easy to check that \eqref{1-16} and \eqref{1-17}
  hold for $N_0=42$. We can also verify that $\frac{F_{i+1}}{F_i}>\frac{F_i}{F_{i-1}}$
   for $3\leq i \leq 42$. Hence, we can prove the following
   corollary.

\begin{corollary}\label{c-8}
 The sequence  $\{F_{n}\}_{n=2}^\infty $ is strictly log-convex.
\end{corollary}


To conclude this paper, we  remark that  the method presented in
this paper can be used  to prove the log-convexity of some
 combinatorial sequences satisfied
longer recurrence relations. The principle is the same.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}

 The authors would like to thank
 the anonymous referees for valuable suggestions
  and comments which resulted in a great improvement of
the original manuscript.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
