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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  | &gt; 10 −  ( n −  1 ) / 5 |\cos n\theta | &gt; {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.

Numbers generated by the reciprocal of 𝑒^{𝑥}-𝑥-1