Many classical numbers like binomial coefficients, Stirling numbers, Eulerian numbers etc can be defined by a recurrence with positive coefficients. A dual recursion defines a class of transient Markov chains, which may be described in terms of the Martin boundary construction. In the prototypical example of Pascal's triangle the boundary is [0,1] (a fact equivalent to Hausdorff's moments problem and to de Finetti's theorem), but for some other triangles it is discrete. The talk will survey a number of examples, including recent joint papers with Jim Pitman and Grisha Olshanski on generalised Stirling triangles and the Eulerian triangle.