The Weyl-Heisenberg algebra is the real algebra generated by boson creation and annihilation operators, or by the non-commuting operators x and D (the derivative with respect to x).  Any element can be rewritten in a normally ordered form, with the x's to the left of the D's.  One can also interpret any product of a specified number of x's and D's by introducing an operator ordering rule, such as the standard one due to Weyl.  Bender and Dunne (1988) asked whether there are rules for which the ordered version of any operator product would equal a power or factorial series in xD or xD+Dx, with coefficients that have a combinatorial interpretation, such as Stirling numbers. This is an instance of a general problem: constructing combinatorial identities in the Weyl-Heisenberg algebra.  We approach this problem by first studying triangles of numbers that satisfy triangular recurrences, such as generalized Stirling numbers and Eulerian ones of our own definition.  We find closed-form expressions for the entries of any such triangles, by first computing their exponential generating functions.  Identities involving such numbers turn out to yield combinatorial identities in the Weyl algebra, including ones of the Bender-Dunne type with generalized Eulerian numbers as
coefficients.