  Euler's difference table associated to the sequence $\{n!\}$ leads naturally
to the counting formula for the derangements. In this paper we study Euler's
difference table associated to the sequence $\{\ell^n n!\}$ and the generalized
derangement problem. For the coefficients appearing in the later table we will
give the combinatorial interpretations in terms of two kinds of $k$-successions
of the group $C_\ell\wr S_n$. In particular for $\ell=1$ we recover the known
results for the symmetric groups while for $\ell=2$ we obtain the corresponding
results for the hyperoctahedral groups.
