Fuss-Catalan Triangles

TL;DR

This paper introduces Fuss-Catalan triangles, a family of combinatorial arrays generalizing Catalan triangles, with explicit formulas and connections to lattice paths and known sequences for specific parameters.

Contribution

It defines a new class of triangles based on Fuss-Catalan numbers, providing explicit formulas and exploring their combinatorial properties and special cases.

Findings

Rows sum to Fuss-Catalan numbers

Explicit formulas for triangle entries

Connections to known sequences for small p

Abstract

For each $p>0$ we define by recurrence a triangle $T^p(n,k)$ whose rows sum to the Fuss-Catalan numbers $ \frac{1}{p n+1}\binom{pn+1}{n}$, generalizing the known Catalan triangle corresponding to the case $p=2$. (In fact, $T^p(n,k)$ has an explicit formula counting simple lattice paths). Moreover, for some small values of $p$, the signed sums turn out to be known sequences. \end{abstract}