We consider a cuspidal class number, which is the order of a subgroup of the full cuspidal divisor class group of X-1(Np-n) with p inverted iota N and n >= 1. By studying the second generalized Bernoulli numbers, we obtain results similar to ones (Ferrero and Washington in Ann Math (2) 109(2):377-395, 1979; Washington in Invent Math 49:87-97, 1978) about the relative class numbers of cyclotomic Z(p)-extension of an abelian number field