Abstract. We examine the integers V(n, k) defined by means ofand, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal of e -x -1.We prove that the V(n, k) are also related to the numbers W(n, k) defined by fc! 2 W(n, k)x"/n< = [(* -2)(e* -l)]fc n=0 in much the same way the associated Stirling numbers are related to the Stirling numbers.Finally, we examine, more generally, the Bell polynomials Bn k(ax, a2, 3 -a, 4 -a, 5 -a, ... ) and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.