We examine the integersV(n,k)V(n,k)defined by means of\[k!∑n=0∞V(n,k)xn/n!=[x(ex+1)−2(ex−1)]k,k!\sum \limits _{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},}\]and, 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 ofex−x−1{e^x} - x - 1. We prove that theV(n,k)V(n,k)are also related to the numbersW(n,k)W(n,k)defined by\[k!∑n=0∞W(n,k)xn/n!=[(x−2)(ex−1)]kk!\sum \limits _{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}}\]in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomialsBn,k(a1,a2,3−α,4−α,5−α,…){B_{n,k}}({a_1},{a_2},3 - \alpha ,4 - \alpha ,5 - \alpha , \ldots )and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.