Suppose
(
p
n
)
n
≥
0
(p_n)_{n \geq 0}
is a non-increasing sequence of non-negative numbers with
p
0
=
1
p_0 = 1
,
P
n
=
∑
j
=
0
n
p
j
P_n = \sum _{j=0}^n p_j
,
n
=
0
,
1
…
n = 0, 1 \dots
, and
A
=
A
(
p
n
)
=
(
a
n
k
)
A = A(p_n) = (a_{nk})
is the lower triangular matrix defined by
a
n
k
=
p
n
−
k
/
P
n
a_{nk} = p_{n-k} / P_n
,
0
≤
k
≤
n
0 \leq k \leq n
, and
a
n
k
=
0
a_{nk} = 0
,
n
>
k
n > k
. We show that the operator norm of
A
A
as a linear operator on
ℓ
p
\ell _p
is no greater than
p
/
(
p
−
1
)
p / (p-1)
, for
1
>
p
>
∞
1 > p > \infty
; this generalizes, yet again, Hardy’s inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the
p
n
p_n
tend to a positive limit, the operator norm of
A
A
on
ℓ
p
\ell _p
is exactly
p
/
(
p
−
1
)
p/(p-1)
. We also give some cases when the operator norm of
A
A
on
ℓ
p
\ell _p
is less than
p
/
(
p
−
1
)
p/(p-1)
.