Numbers Generated by the Reciprocal of ex - x - 1

In this paper we examine the polynomials An(z) and the rational numbers An = An(0) defined by means of oo exzx2(ex x I)'1 =22 An(z)x"/n\. n=0 We prove that the numbers An 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 simUar relationship involving Bernoulli and Stirling numbers. Other similarities between An and the Bernoulli numbers are pointed out. We also reexamine and extend previous results concerning An and An(z). In particular, it has been conjectured that An has the same sign as —cos nd, where re is the zero of ex — x 1 with smallest absolute value. We verify this for 1 < n < 14329 and show that if the conjecture is not true for An, then Icos i7Ô| < 10—'n— '' . We also show that An(z) has no integer roots, and in the interval [0, 1 ), An(z) has either two or three real roots.