Single-boson creation and annihilation operators satisfy a canonical commutation relation and generate the Weyl algebra. It is also generated by the noncommuting operators x and D (the derivative with respect to x), acting on any suitable space of functions. An ordering identity in this algebra can be viewed as an element of a two-sided ideal in the free associative algebra generated by x and D. An example of an identity is the one that rewrites a word (a specified product of x's and D's) as a linear combination of normally ordered words: ones in which D's appear to the right of x's. In the relatively easy `single annihilator' case, the coefficients are Stirling numbers, with a combinatorial interpretation. But ordering identities can be far more complicated. We introduce triangles of recursively defined generalized Stirling and Eulerian numbers, and ordering identities in which they appear as coefficients. Some of these number triangles are related by a `binomial transform' operation, but most have no known combinatorial interpretation.