  We solve the boson normal ordering problem for F[(a*)^r a^s], with r,s
positive integers, where a* and a are boson creation and annihilation operators
satisfying [a,a*]=1. That is, we provide exact and explicit expressions for the
normal form wherein all a's are to the right. The solution involves integer
sequences of numbers which are generalizations of the conventional Bell and
Stirling numbers whose values they assume for r=s=1. A comprehensive theory of
such generalized combinatorial numbers is given including closed-form
expressions (extended Dobinski-type formulas)and generating functions. These
last are special expectation values in boson coherent states.
