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\begin{center}
\vskip 1cm{\Large\bf 
On The Pfaffians and Determinants of Some   \\
\vskip .1in
Skew-Centrosymmetric Matrices
}
\vskip 1cm
Fatih Y\i lmaz\\
Polatl\i\ Arts and Sciences Faculty \\
Gazi University\\
06900 Ankara \\
Turkey\\
\href{mailto:fatihyilmaz@gazi.edu.tr}{\tt fatihyilmaz@gazi.edu.tr} \\
\ \\
Tomohiro Sogabe\\
Graduate School of Engineering\\
Nagoya University \\
Nagoya, 464-8603 \\
Japan\\
\href{mailto:sogabe@na.cse.nagoya-u.ac.jp}{\tt sogabe@na.cse.nagoya-u.ac.jp} \\
\ \\
Emrullah K\i rklar\\
Polatl\i\ Arts and Sciences Faculty \\
Gazi University \\
06900 Ankara \\
Turkey\\
\href{mailto:e.kirklar@gazi.edu.tr}{\tt e.kirklar@gazi.edu.tr} \\
\end{center}

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\begin{abstract}  
This paper shows that the Pfaffians and determinants of some skew centrosymmetric matrices can be computed by a paired two-term recurrence relation, or a general number sequence of second order.
As a result, the complexities of the formulas are of order $n$.
Furthermore, the formulas have no divisions at all, i.e., they fall into the class of breakdown-free algorithms.
\end{abstract}
	
	\section{Introduction}
The \textit{determinant\ }is one of the basic parameters in matrix theory.
The \textit{determinant} of a square matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}$ is defined as
\begin{equation*}
\det (A)=\underset{\sigma \in S_{n}}{\sum }\mathrm{sgn}(\sigma )\underset{i=1}{\overset{n}{\prod }}a_{i,\sigma (i)},
\end{equation*}
where the symbol $S_{n}$ denotes the group of permutations of sets with $n$ elements and  the symbol sgn$(\sigma)$ denotes  the signature of $\sigma \in S_n$.

The  \textit{Pfaffian} of a skew-symmetric matrix $A=(a_{i,j}) \in \mathbb{C}^{2k\times 2k}$ is defined by 
\begin{equation}
\label{Def:Pfaffian}
\mathrm{Pf}(A)=\frac{1}{2^k k!}\underset{\sigma \in S_{2k}}{\sum }\mathrm{sgn}(\sigma )\underset{i=1}{\overset{k}{\prod }}a_{\sigma  (2i-1),\sigma  (2i)},
\end{equation}
and is closely related to the determinant.
In fact, Cayley's theorem states that the square of the Pfaffian of a matrix is equal to the determinant of the matrix, i.e.,
\begin{equation*}
\det (A)=\mathrm{Pf}(A)^{2}.
\end{equation*}

Matrix $A$ is called a {\it centrosymmetric} matrix if $A=JAJ^{-1}$, where $J$ is the anti-diagonal matrix whose anti-diagonal elements are one with all others being zero.
If $A=-JAJ^{-1}$, the matrix is said to be {\it skew-centrosymmetric}.
Skew-centrosymmetric matrices arise in many fields of science including numerical solutions of certain differential equations, digital signal processing, information theory, statistics, linear systems theory, and some Markov processes \cite{1,2,3,4,5,6}.

In general, the complexities of the Pfaffian and the determinant are of the order $\mathcal{O}(n^3)$.
This paper describes efficient computational formulas for the Pfaffians and determinants of special matrices for which the complexities  of the formulas are of the order $\mathcal{O}(n)$.
The formulas have no divisions at all, i.e., the formulas fall into the class of breakdown-free algorithms.
	\section{Pfaffians of skew-centrosymmetric matrices} 

\begin{definition}
	$A_{n}=(a_{i,j})$ and $B_{n}=(b_{i,j})$ denote $n$-by-$n$ matrices with the following elements:
	\begin{align*}
	&a_{i,j}=
	\begin{cases} 
	a, & \text{if  $j=i+1$;} \\
	-a, & \text{if $i=j+1$;} \\
	0, & \text{otherwise,}
	\end{cases} \\[1em]
	&b_{i,j}=
	\begin{cases} 
	(-1)^{i+1}b, & \text{if $i+j=n+1$;} \\ 
	0, & \text{otherwise,}
	\end{cases}
	\end{align*}
	where $1\leq i,j\leq n.$
\end{definition}

\begin{definition}\label{def:F}
	$\mathcal{F}_{n}$ and $\mathcal{G}_{n}$ denote  $2$-by-$2$ block matrices of the following form:
	\begin{equation*}
	\mathcal{F}_{n}=
	\left( 
	\begin{array}{cc}
	A_{k} & B_{k} \\ 
	(-1)^{k}B_{k} & A_{k}
	\end{array}
	\right), \quad 
	\mathcal{G}_{n}=
	\left( 
	\begin{array}{cc}
	A_{k} & -B_{k} \\ 
	(-1)^{k+1}B_{k} & A_{k}
	\end{array}
	\right),
	\end{equation*}
	where $n=2k$.
\end{definition}

For example, if $n=10$, it follows from Definition \ref{def:F} that
\begin{align*}
\mathcal{F}_{10}
&=\left( 
\begin{array}{cc}
A_{5} & B_{5} \\ 
(-1)^{5}B_{5} & A_{5}
\end{array}
\right)\\[1em]
&=\left( 
\begin{array}{ccccc|ccccc}
0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b \\ 
-a & 0 & a & 0 & 0 & 0 & 0 & 0 & -b & 0 \\ 
0 & -a & 0 & a & 0 & 0 & 0 & {b} & 0 & 0 \\ 
0 & 0 & -a & 0 & a & 0 & -b & 0 & 0 & 0 \\ 
0 & 0 & 0 & -a & 0 & b & 0 & 0 & 0 & 0 \\ \hline
0 & 0 & 0 & 0 & -b & 0 & a & 0 & 0 & 0 \\ 
0 & 0 & 0 & b & 0 & -a & 0 & a & 0 & 0 \\ 
0 & 0 & {-b} & 0 & 0 & 0 & -a & 0 & a & 0 \\ 
0 & b & 0 & 0 & 0 & 0 & 0 & -a & 0 & a \\ 
-b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -a & 0%
\end{array}
\right) .
\end{align*}

We now describe algorithms for computing the Pfaffians of $\mathcal{F}_{n}$ and $\mathcal{G}_{n}$.
\begin{theorem}
	Let $\{f_n\}$ and $\{g_{n}\}$ be the recursively defined sequences below:
	\begin{align*}
	&f_{n}=bg_{n-1}+a^{2}f_{n-2}\text{ \ \ for \ }f_{1}=b, \\
	&g_{n}=-bf_{n-1}+a^{2}g_{n-2}\text{ \ \ for \ }g_{1}=-b.
	\end{align*}
	Then, for $n=2k$, we obtain
	\begin{equation*}
	f_{k}=\mathrm{Pf}(\mathcal{F}_{n})\text{ \ \ and \ \ }g_{k}=\mathrm{Pf}(%
	\mathcal{G}_{n}),
	\end{equation*}
	where $f_{-1}=0,$ $f_{0}=1$ and $g_{-1}=0,$ $g_{0}=1.$
\end{theorem}

\begin{proof}
	The proof is done by induction on $k$.
	For $k=1$, 
	\begin{equation*}
	\mathcal{F}_{2}=\left( 
	\begin{array}{cc}
	A_{1} & B_{1} \\ 
	-B_{1} & A_{1}
	\end{array}
	\right) =\left( 
	\begin{array}{cc}
	0 & b \\ 
	-b & 0
	\end{array}
	\right) \text{  and  } \mathcal{G}_{2}=\left( 
	\begin{array}{cc}
	A_{1} & -B_{1} \\ 
	B_{1} & A_{1}
	\end{array}
	\right) =\left( 
	\begin{array}{cc}
	0 & -b \\ 
	b & 0
	\end{array}
	\right).
	\end{equation*}
	The definition of the Pfaffian in \eqref{Def:Pfaffian} clearly indicates that $\mathrm{Pf}(\mathcal{F}_{2})=b$ and $\mathrm{Pf}(\mathcal{G}_{2})=-b$.
	Thus, $f_{1}=b=\mathrm{Pf}(\mathcal{F}_{2}), g_{1}=-b=\mathrm{Pf}(\mathcal{G}_{2}).$
	
Let us assume that the recurrence relations hold for all $t\leq k$. Then we show that they hold for $k=t+1$.
	\begin{align}
	\mathcal{F}_{2t+2}
	& =\left( 
	\begin{array}{c|c}
	A_{t+1} & B_{t+1} \\ \hline
	(-1)^{t+1}B_{t+1} & A_{t+1}
	\end{array}
	\right)  \nonumber  \\[1em]
	&=\left( 
	\begin{tabular}{c|cccc|c}
	$0$ & $a$ & $0$ & $\cdots $ & $0$ & $b$ \\ \hline
	$-a$ &  &  &  &  & $0$ \\ 
	$0$ &  & $A_{t}$ & $-B_{t}$ &  & $\vdots $ \\ 
	$\vdots $ &  & $(-1)^{t+1}B_{t}$ & $A_{t}$ &  & $0$ \\ 
	$0$ &  &  &  &  & $a$ \\ \hline
	$-b$ & $0$ & $\cdots $ & $0$ & $-a$ & $0$
	\end{tabular}
	\right).  \label{10}
	\end{align}
	From the expansion formula along with $2t+2$ column of (\ref{10}), it
	follows that 
	\begin{equation}
	\mathrm{Pf}(\mathcal{F}_{2t+2})=b\mathrm{Pf}(\mathcal{G}_{2t})+a\mathrm{Pf}(%
	\mathcal{M}_{2t})=bg_{t}+a\mathrm{Pf}(\mathcal{M}_{2t}),  \label{20}
	\end{equation}
	where
	\begin{equation}
	\mathcal{M}_{2t}=\left( 
	\begin{tabular}{cc|cccc}
	$0$ & $a$ & $0$ & $\cdots $ & $\cdots $ & $0$ \\ 
	$-a$ & $0$ & $a$ & $0$ & $\cdots $ & $0$ \\ \hline
	$0$ & $a$ &  &  &  &  \\ 
	$\vdots $ & $0$ &  & $A_{t-1}$ & $B_{t-1}$ &  \\ 
	$\vdots $ & $\vdots $ &  & $(-1)^{t-1}B_{t-1}$ & $A_{t-1}$ &  \\ 
	$0$ & $0$ &  &  &  & 
	\end{tabular}
	\right).  \label{30}
	\end{equation}
	From the expansion formula along with the first row  of (\ref{30}), it follows that 
	\begin{equation}
	\mathrm{Pf}(\mathcal{M}_{2t})=a\mathrm{Pf}(\mathcal{F}_{2t-2})=af_{t-1}.
	\label{40}
	\end{equation}
	From (\ref{20}) and (\ref{40}), we have
	\begin{equation*}
	f_{t+1}=bg_{t}+a^{2}f_{t-1}.
	\end{equation*}
	The recurrence relation for $g_{t+1}$ can be obtained similarly.
\end{proof}

\begin{corollary} \label{Cor:1}
	$f_{n}=(-1)^{n-1}bf_{n-1}+a^{2}f_{n-2}$ with $f_{-1}=0$\ and $f_{1}=1$.
\end{corollary}

Corollary \ref{Cor:1} shows that the computational costs of $\mathrm{Pf}(\mathcal{F}_{n})$ and $\det (\mathcal{F}_{n}) (=\mathrm{Pf}(\mathcal{F}_{n})^2)$ are of the order $\mathcal{O}(n)$.
Furthermore, the recurrences in Corollary \ref{Cor:1} have no divisions. Thus, no breakdown occurs during the computation. 

	
	\section{Determinant of the skew-centrosymmetric matrix}
In this section, we consider the determinant of the matrix $\mathcal{F}_{n}$ with $n=2k$.
It is  well known from \cite{3} that the determinant of the 2-by-2 block matrix holds
\begin{equation*}
\left\vert 
\begin{array}{cc}
A & B \\ 
C & D
\end{array}
\right\vert =\det (AD-CB)
\end{equation*}
if $AC=CA$. 
Applying the above formula to $\mathcal{F}_{n}$ in Definition \ref{def:F}, the determinant of
matrix $\mathcal{F}_{n}$ equals that of $\mathcal{T}_{k}:=A_k^2-(-1)^{k}B_k^2$.
Thus, we have
\begin{equation*}
\left\vert \mathcal{F}_{n}\right\vert =\left\vert \mathcal{T}_{k}\right\vert
=\det \left( 
\begin{array}{ccccc}
-a^{2}+b^{2} & 0 & a^{2} &  &  \\ 
0 & -2a^{2}+b^{2} & 0 & \ddots &  \\ 
a^{2} & 0 & \ddots & \ddots & a^{2} \\ 
& \ddots & \ddots & -2a^{2}+b^{2} & 0 \\ 
&  & a^{2} & 0 & -a^{2}+b^{2}
\end{array}
\right) _{k\times k}.
\end{equation*}
The matrix $\mathcal{T}_{k}$ belongs to  the set of $k$-tridiagonal matrices. 
Sogabe and El-Mikkawy \cite{10} considered a fast block diagonalization of $k$-tridiagonal matrices using permutation matrices.
Exploiting the block diagonalization method, we can rearrange the matrix $\mathcal{T}_{k}$ as below.
\begin{enumerate}
	\item[(i)] 
	We consider the case where $k$ is odd.
	Let us define the following matrices: 
	\begin{equation*}
	H_{\frac{k-1}{2}}=(h_{i,j})=
	\begin{cases}
	-2a^{2}+b^{2},& \text{if $i=j$;} \\
	a^{2},        & \text{if $i=j+1$ or $j=i+1$}; \\
	0,            & \text{otherwise}
	\end{cases}
	\end{equation*}
	and
	\begin{equation*}
	K_{\frac{k+1}{2}}=(k_{i,j})= 
	\begin{cases}
	-a^{2}+b^{2}, & \text{if $i=j=1$ or $i=j=\frac{k+1}{2}$;} \\ 
	-2a^{2}+b^{2},& \text{if $i=j=2 \dots \frac{k-1}{2}$}; \\ 
	a^{2},        & \text{if $i=j+1$ or $j=i+1$;} \\ 
	0,            & \text{otherwise.}
	\end{cases}
	\end{equation*}
	Then,
	
	\begin{equation*}
	P^{T}\mathcal{T}_{k}P=\left( 
	\begin{array}{c|c}
	H_{\frac{k-1}{2}} & 0 \\ \hline
	0 & K_{\frac{k+1}{2}}
	\end{array}
	\right), \text{ }
	\end{equation*}
	where the permutation matrix $P$ is determined by using the method in \cite{10}.
	Obviously, 
	\begin{equation*}
	\det (P^{T}\mathcal{T}_{k}P)=\det \mathcal{T}_{k}=\det \mathcal{F}_{n}=\det
	(H_{\frac{k-1}{2}})\det (K_{\frac{k+1}{2}}).
	\end{equation*}
	
	\item[(ii)]
	We consider  the case where $k$ is even.
	Let us define
	\begin{equation*}
	N_{\frac{k}{2}}=(n_{i,j})=
	\begin{cases}
	-a^{2}+b^{2},  &\text{if $i=j=\frac{k}{2}$;} \\ 
	-2a^{2}+b^{2}, &\text{if $i=j=1 \dots \frac{k}{2}-1$;} \\ 
	a^{2},         &\text{if $i=j+1$ or $j=i+1$;} \\ 
	0,             &\text{otherwise}
	\end{cases}
	\end{equation*}
	and
	\begin{equation*}
	Q_{\frac{k}{2}}=(q_{i,j})=
	\begin{cases}
	-a^{2}+b^{2},    & \text{if $i=j=1$;} \\ 
	-2a^{2}+b^{2},   & \text{if $i=j=2\dots\frac{k}{2}$;} \\ 
	a^{2},           & \text{if $i=j+1$ or $j=i+1$;} \\ 
	0,               & \text{otherwise.}
	\end{cases}
	\end{equation*}
	Then,
	\begin{equation*}
	P^{T}\mathcal{T}_{k}P=\left( 
	\begin{array}{c|c}
	N_{\frac{k}{2}} & 0 \\ \hline
	0 & Q_{\frac{k}{2}}
	\end{array}
	\right).
	\end{equation*}
	Obviously, 
	\begin{equation*}
	\det (P^{T}\mathcal{T}_{k}P)=\det \mathcal{T}_{k}=\det \mathcal{F}_{n}=\det
	(N_{\frac{k}{2}})\det (Q_{\frac{k}{2}}).
	\end{equation*}
	It can be seen that $\det (N_{\frac{k}{2}})=\det (Q_{\frac{k}{2}})$.
\end{enumerate}

El-Mikkawy \cite{11} obtained two-term recurrence relation for the determinants of tridiagonal matrices, i.e., 
\begin{equation*}
v_{i}=\left\vert 
\begin{array}{ccccc}
d_{1} & a_{1} & 0 & \ldots & 0 \\ 
b_{2} & d_{2} & a_{2} & \ddots & \vdots \\ 
0 & b_{3} & d_{3} & \ddots & 0 \\ 
\vdots & \ddots & \ddots & \ddots & a_{i-1} \\ 
0 & \ldots & 0 & b_{i} & d_{i}
\end{array}
\right\vert,
\end{equation*}
where $v_{i}=d_{i}v_{i-1}-b_{i}a_{i-1}v_{i-2}$ for $v_{0}=1$ and $v_{-1}=0$.
Using this relation and the Laplace expansion, we obtain the result.
If $k$ is even, then
\begin{equation*}
\det (N_{\frac{k}{2}})=\det (Q_{\frac{k}{2}})=(-a^{2}+b^{2})w_{\frac{k}{2}
	-1}-a^{4}w_{\frac{k}{2}-2}.
\end{equation*}
If $k$ is odd, then
\begin{align*}
&\det (K_{\frac{k+1}{2}})=\left( -a^{2}+b^{2}\right) ^{2}w_{\frac{k-3}{2}
}-2a^{4}(-a^{2}+b^{2})w_{\frac{k-5}{2}}+a^{8}w_{\frac{k-7}{2}}, \\
&\det (H_{\frac{k-1}{2}})=w_{\frac{k-1}{2}},
\end{align*}
where 
\begin{equation*}
w_{i}=\left\vert 
\begin{array}{cccc}
-2a^{2}+b^{2} & a^{2} & \ldots & 0 \\ 
a^{2} & -2a^{2}+b^{2} & \ddots & \vdots \\ 
\vdots & \ddots & \ddots & a^{2} \\ 
0 & \ldots & a^{2} & -2a^{2}+b^{2}
\end{array}
\right\vert.
\end{equation*}
Here $w_{i}=(-2a^{2}+b^{2})w_{i-1}-a^{4}w_{i-2}$ for $w_{0}=1$ and $w_{-1}=0$
.

Consequently, for $n=2k$, we obtain

\begin{enumerate}
	\item[(i)] If $k$ is odd, 
	\begin{align*}
	\det \mathcal{F}_{n}
	&= \det \mathcal{T}_{k} \\
	&=w_{\frac{k-1}{2}}\left( \left(
	-a^{2}+b^{2}\right) ^{2}w_{\frac{k-3}{2}}-2a^{4}(-a^{2}+b^{2})w_{\frac{k-5}{2
		}}+a^{8}w_{\frac{k-7}{2}}\right).
		\end{align*}
		
		\item[(ii)] If $k$ is even, $\det \mathcal{F}_{n}=\det \mathcal{T}
		_{k}=\left( (-a^{2}+b^{2})w_{\frac{k}{2}-1}-a^{4}w_{\frac{k}{2}-2}\right)
		^{2}. $
	\end{enumerate}
	\section{Examples}
	Some examples of the Pfaffian and the determinant of the matrix $\mathcal{F}_{n}~$($n=2k$) are shown in the following tables.
	Here $F_{n}$, $P_{n}$, and $J_{n}$ are the $n$th Fibonacci, Pell, and Jacobsthal numbers, respectively.

\begin{table}[H]
		\label{tPfaffian}\centering
		\begin{equation*}
		\begin{tabular}{c|l|l|l}
			& $a=i,b=1$ & \ $a=i,b=2$ & $a=i\sqrt{2},b=1$ \\ \cline{2-4}
			$k$ & \hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ & 
			\hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ & 
			\hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ \\ 
			\hline\hline
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 1$} & $\hphantom{-} F_{2}=1$ & $
			\hphantom{-} P_{2}=2$ & $\hphantom{-} J_{2}=1 $ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 2$} & $-F_{3}=-2$ & $-P_{3}=-5$ & $
			-J_{3}=-3$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 3$} & $-F_{4}=-3$ & $-P_{4}=-12$ & $
			-J_{4}=-5$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 4$} & $\hphantom{-} F_{5}=5$ & $\hphantom{-} P_{5}=29$ & $\hphantom{-}
			J_{5}=11$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 5$} & $\hphantom{-} F_{6}=8$ & $\hphantom{-} P_{6}=70$ & $\hphantom{-}
			J_{6}=21$ \\ 
	      	\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 6$} & $-F_{7}=-13$ & $-P_{7}=-169$ & $-J_{7}=-43$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 7$} & $-F_{8}=-21$ & $-P_{8}=-408$ & $-J_{8}=-85$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 8$} & $\hphantom{-} F_{9}=34$ & $\hphantom{-} P_{9}=985$ & $\hphantom{-}
			J_{9}=171$ \\ 
			\multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ \vdots$} & \hphantom{-}\hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ & 
			\hphantom{-}\hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ & \hphantom{-}
			\hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ \\ \hline
			$\equiv 0,1$ (mod  $4$) & $\hphantom{-} F_{k+1}$ & $\hphantom{-} P_{k+1}$
			& $\hphantom{-} J_{k+1}$ \\ 
			$\equiv 2,3$ (mod  $4$) & $-F_{k+1}$ & $-P_{k+1}$ & $-J_{k+1}$
		\end{tabular}
		\end{equation*}
\centerline{Examples of the Pfaffians}	
		\label{tbl:determinant}
\end{table}

\vfill\eject

\begin{table}[H]
		\centering
			\begin{equation*}
		\begin{tabular}{c|c|c|c}
			& $a=i,b=1$ & $a=i,b=2$ & $a=i\sqrt{2},b=1$ \\ \cline{2-4}
			$k$ & $\det { (}\mathcal{F}_{2k}{ )}$ & $\det { (}\mathcal{
				F}_{2k}{ )}$ & $\det { (}\mathcal{F}_{2k}{ )}$ \\ 
			\hline\hline
			\multicolumn{1}{c|}{${ 1}$} & ${ F}_{2}^{2}$ & ${ P}_{2}^{2}$ & ${ J}_{2}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 2}$} & ${ F}_{3}^{2}$ & ${ P}_{3}^{2}$ & ${ J}_{3}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 3}$} & ${ F}_{4}^{2}$ & ${ P}_{4}^{2}$ & ${ J}_{4}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 4}$} & ${ F}_{5}^{2}$ & ${ P}_{5}^{2}$ & ${ J}_{5}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 5}$} & ${ F}_{6}^{2}$ & ${ P}_{6}^{2}$ & ${ J}_{6}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 6}$} & ${ F}_{7}^{2}$ & ${ P}_{7}^{2}$ & ${ J}_{7}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 7}$} & ${ F}_{8}^{2}$ & ${ P}_{8}^{2}$ & ${ J}_{8}^{2}$ \\ 
			\multicolumn{1}{c|}{${ 8}$} & ${ F}_{9}^{2}$ & ${ P}_{9}^{2}$ & ${ J}_{9}^{2}$ \\ 
	     	\multicolumn{1}{c|}{${ \vdots}$} & ${ \vdots }$ & ${ \vdots }$ & ${	\vdots }$ \\ 
			\multicolumn{1}{c|}{${ t}$} & ${ F}_{t+1}^{2}$ & ${ P}_{t+1}^{2}$ & ${ J}_{t+1}^{2}$
		\end{tabular}
			\end{equation*}
	{Examples of the determinants}
\end{table}
	
\section{Acknowledgment} 
This work has been supported in part by JSPS KAKENHI (Grant No.~26286088).

The authors sincerely appreciate the referee's comments that enhanced the quality of the manuscript.
	

	
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	\noindent 2010 {\it Mathematics Subject Classification}:
	Primary 15A15; Secondary 15A23.
	
	\noindent \emph{Keywords: }
	Pfaffian, determinant, skew-centrosymmetric matrix.
	
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	\noindent (Concerned with sequences
	\seqnum {A000045}, \seqnum {A000129}, \seqnum{A001045}.)
	
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	\vspace*{+.1in}
	\noindent
Received June 20 2016;
revised versions received July 8 2016; January 18 2017;
February 2 2017.
Published in {\it Journal of Integer Sequences}, 
February 11 2017.
	
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	\noindent
	Return to
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