Let
A
(
2
r
+
1
,
n
)
A(2r + 1,n)
denote the
n
×
n
n \times n
band matrix, of bandwidth
2
r
+
1
2r + 1
, with the binomial coefficients in the expansion of
(
x
−
1
)
2
r
{(x - 1)^{2r}}
as the elements in each row and column. Using the fact that the rows of
A
(
2
r
+
1
,
n
)
A(2r + 1,n)
provide the coefficients for the
2
r
2r
th central difference, a number of properties of
A
(
2
r
+
1
,
n
)
A(2r + 1,n)
are obtained for all positive integers
r
r
and
n
n
. These include obtaining explicit formulas for
det
A
(
2
r
+
1
,
n
)
,
A
−
1
(
2
r
+
1
,
n
)
,
|
|
A
−
1
(
2
r
+
1
,
n
)
|
|
∞
\det A(2r + 1,n),{A^{ - 1}}(2r + 1,n),||{A^{ - 1}}(2r + 1,n)|{|_\infty }
and an upper triangular matrix
U
U
such that
A
(
2
r
+
1
,
n
)
U
A(2r + 1,n)U
is lower triangular.