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\begin{center}
\vskip 1cm{\LARGE\bf Generalizing Tuenter's Binomial Sums
}
\vskip 1cm
\large
Richard P. Brent\\
Mathematical Sciences Institute\\
Australian National University\\
Canberra, ACT 2614\\
Australia\\
\href{mailto:sums@rpbrent.com}{\tt sums@rpbrent.com}
\end{center}

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\newcommand{\raisecomma}{\raisebox{2pt}{$,$}}
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\newcommand{\Pbar}{{\overline P}}
\newcommand{\Qbar}{{\overline Q}}
\newcommand{\half}{{\mbox{$\frac{1}{2}$}}}

\begin{abstract}
Tuenter
considered centered binomial sums of the form
\[S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}|n-k|^r,\]
where $r$ and $n$ are non-negative integers.
We consider sums of the form
\[U_r(n) = \sum_{k=0}^n \binom{n}{k}|n/2-k|^r,\]
which are a generalization of Tuenter's sums and may be interpreted as
moments of a symmetric Bernoulli random walk with~$n$ steps. 
The form of $U_r(n)$ depends on the parities of both $r$ and~$n$.  
In fact, $U_r(n)$ is the product of a polynomial (depending on the parities
of~$r$ and $n$) times a power of two or a binomial coefficient. In all cases
the polynomials can be expressed in terms of Dumont-Foata polynomials. We
give recurrence relations, generating functions and explicit formulas for
the functions $U_r(n)$ and/or the associated polynomials.
\end{abstract}

\section{Introduction}
\label{sec:intro}

We consider centered binomial sums of the form
\begin{equation}		\label{eq:U_def}
U_r(n) = \sum_{k=0}^n \binom{n}{k}\left|\frac{n}{2}-k\right|^r,
\end{equation}
where $r,n \ge 0$. 
These generalize the binomial sums
\begin{equation}		\label{eq:S_def}
S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}|n-k|^r,
\end{equation}
previously considered by Tuenter~\cite{Tuenter1}
and other authors, 
since $S_r(n) = U_r(2n)$, but $U_r(n)$ is well-defined for both even and odd
values of~$n$.
The generalization arises naturally in the study of certain two-fold
centered binomial sums of the form
$\sum_{j}\sum_{k}\binom{2n}{n+j}\binom{2n}{n+k}P(j,k)$,
see Brent {et al.\ }\cite[Lemma~6.4]{BOOP}.

In definitions such as~\eqref{eq:U_def} and~\eqref{eq:S_def} we always
interpret $0^0$ as $1$.
Thus $U_0(n) = 2^n$ for all $n\ge 0$.
We define $U_r(m) = 0$ if $m &lt; 0$. 

For $r &gt; 0$ we can avoid the absolute value function
in~\eqref{eq:U_def} by writing
\[U_r(n) = 2\sum_{k=0}^{\lfloor{n/2}\rfloor} 
  \binom{n}{k}\left(\frac{n}{2}-k\right)^r.\]

Tuenter~\cite{Tuenter1}
showed in a direct manner that, for $r \ge 0$ and $n &gt; 0$, $S_r(n)$
satisfies the recurrence
\begin{equation}	\label{eq:Tuenter_rec0}
S_{r+2}(n) = n^2 S_r(n) - 2n(2n-1)S_r(n-1).
\end{equation}
Observe that this recurrence splits into two separate recurrences, one
involving odd values of~$r$ and the other involving even values of~$r$.
Also, 
$S_0(n) = 2^{2n}$ and $S_1(n) = n\binom{2n}{n}$.
It follows that, for $r, n \ge 0$,
\begin{equation}		\label{eq:SPQ}
S_{2r}(n) = Q_r(n)2^{2n-r} \;\text{ and }\;
S_{2r+1}(n) = P_r(n)n\binom{2n}{n},
\end{equation}
where $P_r(n)$ and $Q_r(n)$ are polynomials of degree~$r$ with integer
coefficients, satisfying the recurrences
\begin{align}	\label{eq:Prec}
P_{r+1}(n) &amp;= n^2P_r(n) - n(n-1)P_r(n-1),\\
		\label{eq:Qrec}
Q_{r+1}(n) &amp;= 2n^2Q_r(n) - n(2n-1)Q_r(n-1)
\end{align}
for $r \ge 0$,
with initial conditions $P_0(n) = Q_0(n) = 1$.

The Dumont-Foata polynomials $F_r(x,y,z)$ are $3$-variable polynomials
satisfying the recurrence relation
\begin{equation}		\label{eq:DF-recurrence}
F_{r+1}(x,y,z) = (x+z)(y+z)F_r(x,y,z+1) - z^2F_r(x,y,z)
\end{equation}
for $r \ge 1$, with $F_1(x,y,z) = 1$.
Dumont and Foata~\cite{DF} gave a combinatorial interpretation for the
coefficients of $F_r(x,y,z)$ and showed that $F_r(x,y,z)$ is symmetric in the
three variables $x,y,z$.

Tuenter~\cite{Tuenter1} showed that $P_r(n)$ and $Q_r(n)$ may be expressed
in terms of Dumont-Foata polynomials.
In fact, for $r \ge 1$,
\begin{equation}			\label{eq:PQF}
P_r(n) = (-1)^{r-1}nF_r(1,1,-n)
\;\text{ and }\; 
Q_r(n) = (-2)^{r-1}nF_r(\half,1,-n).
\end{equation}
Thus, we can obtain explicit formulas and generating functions for
the polynomials $P_r(n)$ and $Q_r(n)$ as special cases of the
results of Carlitz~\cite{Carlitz} on Dumont-Foata polynomials.

We show that all the above results for $S_r(n)$ can be generalized
to cover $U_r(n)$. In particular, Theorem~\ref{thm:recurrence} shows that
$U_r(n)$ satisfies a recurrence~\eqref{eq:Urec}
similar to the recurrence~\eqref{eq:Tuenter_rec0} satisfied by $S_r(n)$.
Also, $U_r(n)$ is the product
of a polynomial (depending on the parity
of~$r$) times a power of two or a binomial coefficient, 
and Theorem~\ref{thm:polynomials} shows
that these polynomials satisfy three-term recurrence relations
analogous to~\eqref{eq:Prec}--\eqref{eq:Qrec}.
Using the recurrences, the
polynomials can be expressed in terms of Dumont-Foata polynomials,
so the results of Carlitz allow us to obtain 
explicit formulas (in~\S\ref{sec:explicit})
and exponential generating functions
(in~\S\ref{sec:egfs}).

\section{Recurrence relations}		\label{sec:main}

Theorems~\ref{thm:recurrence}--\ref{thm:polynomials} give
recurrence relations for
$U_r(n)$ and associated polynomials.
The recurrence~\eqref{eq:Urec} in
Theorem~\ref{thm:recurrence} implies
the recurrence~\eqref{eq:Tuenter_rec0} satisfied by
$S_r(n)$.

\begin{theorem}		\label{thm:recurrence}
For all $r, n \ge 0$, $U_r(n)$ satisfies the recurrence
\begin{equation}			\label{eq:Urec}
4U_{r+2}(n) = n^2U_r(n) - 4n(n-1)U_r(n-2),
\end{equation}
and  may be computed from the recurrence using the initial
conditions
\[
U_0(n) = 2^n,\;
U_1(2n) = n\binom{2n}{n},\;
U_1(2n+1) = (2n+1)\binom{2n}{n}
\text{ for all } n \ge 0.
\]
\end{theorem}

\begin{proof}
We have
\begin{align}
4U_{r+2}(n) &amp;= \sum_{k=0}^n 4\binom{n}{k}\left|\frac{n}{2}-k\right|^{r+2}
	= \sum_{k=0}^n \binom{n}{k}\left|\frac{n}{2}-k\right|^{r}(n-2k)^2,
	\label{eq:rec1}
\end{align}
\begin{align}
n^2U_r(n) &amp;= \sum_{k=0}^n\binom{n}{k}\left|\frac{n}{2}-k\right|^{r}n^2,
	\label{eq:rec2}
\end{align}
and
\begin{align}
4n(n-1)U_r(n-2) &amp;= 
 \sum_{k=0}^{n-2} 4n(n-1)\binom{n-2}{k}\left|\frac{n-2}{2}-k\right|^r
	\nonumber\\
 &amp;= \sum_{k=1}^{n-1} 4n(n-1)\binom{n-2}{k-1}\left|\frac{n}{2}-k\right|^{r}
	\nonumber\\
 &amp;= \sum_{k=1}^{n-1} 4k(n-k)\binom{n}{k}\left|\frac{n}{2}-k\right|^{r}.
	\label{eq:rec3}
\end{align}
Since $(n-2k)^2 - n^2 - 4k(n-k) = 0$, the recurrence~\eqref{eq:Urec}
follows from~\eqref{eq:rec1}--\eqref{eq:rec3}.

For the initial conditions, it is easily verified that $U_0(n) = 2^n$. The
solution by Hillman~\cite{Hillman} to the Putnam problem 35-A4 gives
$U_1(n) = n\binom{n-1}{\lfloor{n/2}\rfloor}$, and taking account of the
parity of $n$ gives the remaining conditions.
\end{proof}

It is now clear that the structure of $U_r(n)$ depends upon the parities of
both $r$ and $n$, and one can elucidate the recursion~\eqref{eq:Urec}
by the following substitutions:
\begin{align}
U_{2r+1}(2n-1) &amp;= 2^{-(2r+1)}n\Pbar_r(n)\binom{2n}{n},
	\label{eq:U_odd_odd_Pbar}\\
U_{2r}(2n+1) &amp;= 2^{2n+1-2r}\Qbar_r(n).
	\label{eq:U_even_odd_Qbar}
\end{align}
It is easily verified that $\Pbar_r(n)$ and $\Qbar_r(n)$ are polynomials
in~$n$ of degree~$r$. By substitution into~\eqref{eq:Urec}, we
obtain the following theorem.
\begin{theorem}		\label{thm:polynomials}
For $r \ge 0$, the polynomials $\Pbar_r(n)$ and $\Qbar_r(n)$ defined by
equations \eqref{eq:U_odd_odd_Pbar}--\eqref{eq:U_even_odd_Qbar}
satisfy the following recurrence relations:
\begin{align}
\Pbar_{r+1}(n) &amp;= (2n-1)^2\,\Pbar_r(n) - 4(n-1)^2\,\Pbar_r(n-1),
 \label{eq:Pbar_rec}\\
\Qbar_{r+1}(n) &amp;= (2n+1)^2\,\Qbar_r(n) - 2n(2n+1)\Qbar_r(n-1),
 \label{eq:Qbar_rec}
\end{align}
with initial conditions
$\Pbar_0(n) = \Qbar_0(n) = 1$.
\end{theorem}

\section{Explicit formulas}			\label{sec:explicit}

Using the recurrence~\eqref{eq:DF-recurrence} 
and matching the initial conditions,
we can verify that the polynomials
$\Pbar_r(n)$ and $\Qbar_r(n)$ can be expressed in
terms of Dumont-Foata polynomials, as follows:
\begin{align}
\Pbar_r(n) &amp;= (-4)^r F_{r+1}\left(\half,\half,\half-n\right),
					\label{eq:Pbar_F}\\
\Qbar_r(n) &amp;=
(-1)^{r-1}2^{2r-1} \left(n+\half\right)F_r\left(\half,1,-n-\half\right).
					\label{eq:Qbar_and_Q}
\end{align}
Also, replacing $n$ by $n+\half$ in the second half of~\eqref{eq:PQF}
shows that
\begin{equation}		\label{eq:QandQbar}
 \Qbar_r(n) = 2^r Q_r\!\left(n+\half\right).
\end{equation}

Carlitz~\cite[{eqns.\ }(1.13) and (1.16)]{Carlitz} gives
an explicit formula for the Dumont-Foata polynomials.
We follow Carlitz and let $(z)_k$ denote 
the Pochhammer symbol or ``rising factorial''.
Note that, although $F_{r+1}(x,y,z)$ is symmetric in $x,y$ and $z$,
the representation given in Proposition~\ref{prop:DF} is not symmetric.
Thus, it is sometimes necessary to permute variables before
applying Proposition~\ref{prop:DF}.
\begin{proposition}[Carlitz]	\label{prop:DF}
For $r \ge 0$
and $z&gt;0$, 
\begin{equation*}		
F_{r+1}(x,y,z) = \sum_{k=0}^r (-1)^{r-k}{(x+z)_k(y+z)_k}\,A_{r,k}(z),
\end{equation*}
\vspace*{-10pt}
where
\vspace*{-12pt}
\begin{equation*}
\;\;\;
A_{r,k}(z) = 
 \frac{2}{k!}\,
  \sum_{j=0}^k(-1)^{k-j}\binom{k}{j}\frac{(z+j)^{2r+1}}{(2z+j)_{k+1}}
	\,\raisedot
\end{equation*}
\end{proposition}
Application of Proposition~\ref{prop:DF}
gives, after some simplification, the following formulas
for $U_r(n)$, valid for $r\ge 1$ and $n \ge 0$:
\begin{align}					\label{eq:U_even_a2}
U_{2r}(n) &amp;= 2^{n+1} \sum_{1 \le j \le k \le r}
	(-1)^j \frac{\left(-\frac{n}{2}\right)_k 
	    \left(\half\right)_k}
		    {(k-j)!(k+j)!}\,j^{2r},\\
						\label{eq:U_odd_even2}
U_{2r+1}(2n) &amp;= 2n\binom{2n}{n}\sum_{1 \le j \le k \le r}
	(-1)^{j}\frac{(-n)_{k}}{(k-j)!\,(k+1)_j}\,j^{2r},\\
						\label{eq:U_odd_odd2}
U_{2r-1}(2n-1) &amp;= \binom{2n}{n}\sum_{1 \le j \le k \le r}
	(-1)^{j}\frac{(-n)_{k}}{(k-j)!\,(k)_j}\,
	  \left(j-\half\right)^{2r-1}.
\end{align}

\section{Generating functions}		\label{sec:egfs}

The exponential generating function (egf)
\begin{equation}			\label{eq:U_even_gf}
\sum_{r\ge 0} U_{2r}(n)\frac{z^{2r}}{(2r)!}
 = 2^n \cosh^n (z/2).
\end{equation}
generalizes the egf
\begin{equation}			\label{eq:S_even_gf}
\sum_{r\ge 0} S_{2r}(n)\frac{z^{2r}}{(2r)!}
 = 2^{2n} \cosh^{2n} (z/2)
\end{equation}
given by Tuenter~\cite[\S5]{Tuenter1},
since replacing $n$ by $2n$ in~\eqref{eq:U_even_gf}
gives~\eqref{eq:S_even_gf}. The proof of~\eqref{eq:U_even_gf}
is straightforward,
and does not require the results of Carlitz.

We can obtain other egfs from the results of
Carlitz.
First, we note that Carlitz~\cite[{eqn.\ }(4.2)]{Carlitz} gives the egf
\begin{equation}		\label{eq:Carlitz42}
\sum_{r \ge 1} (-1)^rF_r(x,y,1) \frac{z^{2r}}{(2r)!}
 = \frac{1}{xy}\sum_{k\ge 1}(-1)^k\frac{(x)_k(y)_k}{(2k)!}
	\left(2\sinh\frac{z}{2}\right)^{2k}.
\end{equation}
In view of~\eqref{eq:SPQ} and~\eqref{eq:PQF}, 
this allows us to obtain an egf for $U_{2r+1}(2n)$:
\begin{equation}	\label{eq:U_odd_even_egf}
\sum_{r\ge 0}U_{2r+1}(2n)\frac{z^{2r}}{(2r)!} =
 n\binom{2n}{n}\sum_{k=0}^n 2^{2k}\binom{n}{k}\binom{2k}{k}^{-1}
  \sinh^{2k}\left(\frac{z}{2}\right)\,\raisedot
\end{equation}
In order to calculate $U_{2r+1}(2n)$
from~\eqref{eq:U_odd_even_egf}, it is only necessary to sum
the terms on the right-hand side
for $k \le \min(r,n)$.

The remaining case $U_{2r+1}(2n-1)$ is more difficult because
\eqref{eq:Carlitz42} does not apply.
We can use the egf
\begin{align}
\sum_{r\ge 0}&amp;(-1)^r F_{r+1}(x,y,z)\frac{u^{2r+1}}{(2r+1)!} \nonumber\\
 &amp;=\, 2\,\sum_{k=0}^\infty \sum_{j=0}^k
     \frac{(-1)^j (x+z)_k (y+z)_k (2z)_j}{j! (k-j)! (2z)_{k+1} (2z+k+1)_j}
        \sinh((z+j)u),	\label{eq:Carlitz222}
\end{align}
which follows
from the discussion in Carlitz~\cite[{pp.\ }221--222]{Carlitz}.
Using~\eqref{eq:U_odd_odd_Pbar}, \eqref{eq:Pbar_F} 
and~\eqref{eq:Carlitz222}, after some simplification followed by
a change of variables ($u \mapsto z$),
we obtain the egf:
\begin{align}
\sum_{r\ge 0}&amp;\,U_{2r+1}(2n-1)\frac{z^{2r+1}}{(2r+1)!} \nonumber \\
&amp;=\,
n\binom{2n}{n}\sum_{0 \le j \le k &lt; n}
	\frac{(-1)^{k-j}\binom{n-1}{k}\binom{2k}{k-j}}{\binom{2k}{k}}\,
	\frac{\sinh\left(\left(j+\half\right)z\right)}
		{j+k+1}\,\raisecomma
	\label{eq:U_odd_odd_egf}
\end{align}
which is valid for $n\ge 1$.

\section{Acknowledgment}

Thanks are due to an anonymous referee for detailed and helpful comments
on a preliminary version of this paper.
This research was supported by Australian Research Council grant DP140101417.

\subsection*{Appendix 1: The polynomials $\Pbar_r, \Qbar_r$ for $r \le 5$}

\begin{align*}
\Pbar_0(n) &amp;= 1,\\
\Pbar_1(n) &amp;= 4n-3,\\
\Pbar_2(n) &amp;= 32n^2 - 56n + 25,\\
\Pbar_3(n) &amp;= 384n^3 - 1184n^2 + 1228n - 427,\\
\Pbar_4(n) &amp;= 6144n^4 - 29184n^3 + 52416n^2 - 41840n + 12465,\\
\Pbar_5(n) &amp;= 122880n^5 - 829440n^4 +\\
	   &amp; \phantom{= xx} 2258688n^3 - 3076288n^2
		+ 2079892n - 555731,\\
\Qbar_0(n) &amp;= 1,\\
\Qbar_1(n) &amp;= 2n+1,\\
\Qbar_2(n) &amp;= 12n^2 + 8n + 1,\\
\Qbar_3(n) &amp;= 120n^3 + 60n^2 + 2n + 1,\\
\Qbar_4(n) &amp;= 1680n^4 - 168n^2 + 128n + 1,\\ 
\Qbar_5(n) &amp;= 30240n^5 - 25200n^4 + 5040n^3 + 7320n^2 - 2638n + 1.
\end{align*}

A similar table for Tuenter's polynomials $P_r(n), Q_r(n)$ may
be found in Brent~\cite{rpb259-arxiv}.
The triangles of coefficients of $-P_r(-n)/n, -Q_r(-n)/n$, and
$\Qbar_r(-n)$ for $r \ge 1$ are OEIS~\cite{OEIS} 
sequences \seqnum{A036970}, \seqnum{A083061},
and \seqnum{A160485} respectively. 
We have contributed the coefficients of $\Pbar_r(n)$
as sequence \seqnum{A245244}.
The values $(-1)^r\Pbar_r(0)$ are
sequence \seqnum{A009843} (see Appendix~$2$ for details).

The bijection~\eqref{eq:QandQbar} between \seqnum{A083061} and
\seqnum{A160485} (by a shift of $\pm\half$ and scaling by a power of~$2$)
was not mentioned in the relevant OEIS entries as at July 14, 2014;
we have now contributed comments to this effect.

\subsection*{Appendix 2: Special values of $P_r, Q_r, \Pbar_r, \Qbar_r$}

In Table~\ref{tab:one}, we give values of the polynomials
$P_r, Q_r, \Pbar_r$, and $\Qbar_r$ at $0, 1, \infty$.
Here $P(\infty)$ denotes
the leading coefficient of the polynomial $P(z)$,
$\delta_{i,j}$ is the Kronecker delta, 
and ${\cal S}_r$ is the $r$-th Secant number.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline &amp;&amp;&amp;&amp;\\[-10pt]
$z$	&amp; $P_r(z)$	&amp; $Q_r(z)$	&amp; $\Pbar_r(z)$	&amp; $\Qbar_r(z)$\\[1pt]
\hline &amp;&amp;&amp;&amp;\\[-10pt]
$0$	&amp;
 $\delta_{0,r}$ &amp;
  $\delta_{0,r}$ &amp;
   $(-1)^r(2r+1){\cal S}_r$ &amp; $1$\\[1pt]
$1$	&amp;
 $1$	&amp;
  $\max(1,2^{r-1})$ &amp; $1$ &amp;
   $(3^{2r} + 3)/4$\\[1pt]
$\infty$	&amp; 
 $r!$	&amp;
  $(2r)!/(2^r r!)$ &amp;
   $2^{2r}r!$ &amp;
 $(2r)!/r!$\\[1pt]
\hline
\end{tabular}
\caption{Special values of the polynomials} \label{tab:one}
\end{center}
\end{table}
The values
$(-1)^r\Pbar_r(0)$ are OEIS sequence \seqnum{A009843}, 
and are given by the egf
\begin{equation}		\label{eq:Pbar_gf}
\sum_{r=0}^\infty \Pbar_r(0)\frac{x^{2r+1}}{(2r+1)!}
 = \frac{x}{\cosh x}\,\raisedot
\end{equation}
They may be expressed in terms of the
Secant numbers ${\cal S}_r$, which comprise OEIS sequence \seqnum{A000364}.
In view of~\eqref{eq:Pbar_F}, we obtain a special value of
the Dumont-Foata polynomials: 
\begin{equation}		\label{eq:F_half_half_half}
F_{r+1}\left(\half,\half,\half\right) = 2^{-2r}(2r+1){\cal S}_r.
\end{equation}

The values $\Qbar_r(1)$ comprise OEIS sequence \seqnum{A054879}.
The values in the last row of Table~\ref{tab:one} may also be found in OEIS:
they are sequences \seqnum{A000142}, \seqnum{A001147},
\seqnum{A047053}, and \seqnum{A001813}.

Tuenter~\cite{Tuenter1} observed that, for $r \ge 1$, the constant
terms of $-P_r(n)/n$ are the
Genocchi numbers (\seqnum{A001469}),
and the constant terms of $(-1)^{r-1}Q_r(n)/n$ are the
reduced tangent numbers (\seqnum{A002105}).

\begin{thebibliography}{9}

\bibitem{rpb259-arxiv}
R. P. Brent,
{Generalising Tuenter's binomial sums}, preprint, \newline
\url{http://arxiv.org/abs/1407.3533v4/},
January 19 2015.

\bibitem{BOOP}
R. P. Brent, H. Ohtsuka, J. H. Osborn, and H. Prodinger,
{Some binomial sums involving absolute values}, preprint, 
\url{http://arxiv.org/abs/1411.1477v1/}, \newline
November 6 2014.

\bibitem{Carlitz}
L. Carlitz,	
Explicit formulas for the Dumont-Foata polynomial,
\emph{Discrete Math.\ }\textbf{30} (1980), 211--225.

\bibitem{DF}
D. Dumont and D. Foata, 
Une propri\'et\'e de sym\'etrie des nombres de Genocchi,
\emph{Bull.\ Soc.\ Math.\ France}
\textbf{104} (1976), 433--451.

\bibitem{Hillman}
A. P. Hillman,
The William Lowell Putnam mathematical competition,
\emph{Amer.\ Math.\ Monthly} \textbf{82} (1975), 905--912.

\bibitem{OEIS}
OEIS Foundation Inc., 
\emph{The On-Line Encyclopedia of Integer Sequences},
published electronically at \url{http://oeis.org}.

\bibitem{Tuenter1}
H. J. H. Tuenter, 
Walking into an absolute sum,
\emph{Fibonacci Quart.\ }\textbf{40} (2002), 175--180.

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A10;
Secondary 11B65, 05A15, 05A19, 44A60, 60G50.


\noindent \emph{Keywords: }
Bernoulli random walk,
binomial sum identity, 
Catalan number,
Dumont-Foata polynomial,
explicit formula,
generating function,
Genocchi number,
moment,
polynomial interpolation,
secant number,
tangent number.

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\noindent (Concerned with sequences
\seqnum{A000142},
\seqnum{A000364},
\seqnum{A001147},
\seqnum{A001469},
\seqnum{A001813},
\seqnum{A002105},
\seqnum{A009843},
\seqnum{A036970},
\seqnum{A047053},
\seqnum{A054879},
\seqnum{A083061},
\seqnum{A160485}, and
\seqnum{A245244}.)

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\vspace*{+.1in}
\noindent
Received July 16 2014;
revised versions received  January 18 2015; January 25 2015.
Published in {\it Journal of Integer Sequences}, January 26 2015.

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\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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