On Solutions to a General Combinatorial Recurrence

We develop techniques that can be applied to find solutions to the recurrence ∣∣n k ∣∣ = ( n+ k + )∣∣n−1 k ∣∣+ ( ′n+ ′k + ′)∣∣n−1 k−1∣∣+ [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 = − , = ′ = 0, and = ′ + 1, adding to the result of Neuwirth on the case ′ = 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 ∣∣n j ∣∣jm for some special cases of the recurrence, and we prove some apparently new identities involving Stirling numbers of the second kind, Bell numbers, Rao-Uppuluri-Carpenter numbers, second-order Eulerian numbers, and both kinds of associated Stirling numbers.