On Solutions to a General Combinatorial Recurrence

∣ + [n = k = 0]. Many interesting combinatorial numbers, such as binomial coefficients, both kinds of Stirling and associated Stirling numbers, Lah numbers, Eulerian numbers, and second-order Eulerian numbers, satisfy special cases of this recurrence. Our techniques yield explicit expressions in the instances α = −β, β = β = 0, and α β = α ′ β + 1, adding to the result of Neuwirth on the case α = 0. Our approach employs finite differences, continuing work of the author on using finite differences to study combinatorial numbers satisfying simple recurrences. We also find expressions for the power sum ∑n j=0 ∣