In this paper we examine the polynomials
A
n
(
z
)
{A_n}(z)
and the rational numbers
A
n
=
A
n
(
0
)
{A_n} = {A_n}(0)
defined by means of
\[
e
x
z
x
2
(
e
x
−
x
−
1
)
−
1
=
2
∑
n
=
0
∞
A
n
(
z
)
x
n
/
n
!
.
{e^{xz}}{x^2}{({e^x} - x - 1)^{ - 1}} = 2\sum \limits _{n = 0}^\infty {{A_n}(z){x^n}/n!} .
\]
We prove that the numbers
A
n
{A_n}
are related to the Stirling numbers and associated Stirling numbers of the second kind, and we show that this relationship appears to be a logical extension of a similar relationship involving Bernoulli and Stirling numbers. Other similarities between
A
n
{A_n}
and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerning
A
n
{A_n}
and
A
n
(
z
)
{A_n}(z)
. In particular, it has been conjectured that
A
n
{A_n}
has the same sign as
−
cos
n
θ
- \cos n\theta
, where
r
e
i
θ
r{e^{i\theta }}
is the zero of
e
x
−
x
−
1
{e^x} - x - 1
with smallest absolute value. We verify this for
1
⩽
n
⩽
14329
1 \leqslant n \leqslant 14329
and show that if the conjecture is not true for
A
n
{A_n}
, then
|
cos
n
θ
|
>
10
−
(
n
−
1
)
/
5
|\cos n\theta | > {10^{ - (n - 1)/5}}
. We also show that
A
n
(
z
)
{A_n}(z)
has no integer roots, and in the interval
[
0
,
1
]
[0,1]
,
A
n
(
z
)
{A_n}(z)
has either two or three real roots.