Helmut Prodinger

1. The sum in question 1 is (with Fibonacci numbers Fk) ∑k=0nkmFk,which is popular since Brother A. Brousseau [2]. We treat the integer m≥0 as a parameter and n as the variable. A sum running over ...

The sequence F_{dn+h} and its convolutions have (for h=0) been studied in a recent paper [3]. The instance with general h is more involved and uses Chebyshev polynomials.

Peca suggested in a recent paper on the arxiv to consider binary butterfly trees and their Horton-Strahler numbers. The trees are obtained by glueing two binary trees together in a special way; the results are again binary trees but with a different probability distribution. A thorough combinatorial analysis is provided and leads asymptotically to the same results as for classical binary trees.

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the x -axis and return to the x -axis. Versions where the return to the x -axis isn’t required are also considered. A path is peakless (valleyless) if UD (if DU ) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of UD resp. DU . The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.

Descents of odd length in Dyck paths are discussed, taking care of some variations. The approach is based on generating functions and the kernel method and augments relations about them from the Encyclopedia of Integer Sequences, that were pointed out by David Callan.

Following an orginal idea by Kn¨odel, an online bin-packing problem is considered where the large items arrive in double-packs. The dual problem where the small items arrive in double-packs is also considered. The enumerations have a ternary random walk flavour, and for the enumeration, the kernel method is employed.

K-Dyck paths differ from ordinary Dyck paths by using an up-step of length k. We analyze at which level the path is after the s-th up-step and before the (s+1)-st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago, the terms max-terms and min-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.