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Lucasnomial Fuss-Catalan Numbers and
Related Divisibility Questions Christian Ballot
Département de Mathématiques et Informatique
Université de Caen-Normandie
France
mailto:christian.ballot@unicaen.frchristian.ballot@unicaen.fr

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Abstract:

For all integers $n\ge1$ we define the generalized Lucasnomial Fuss-Catalan numbers

$$C_{U,a,r}(n):=\frac{U_r}{U_{(a-1)n+r}}\binom{an+r-1}{n}_U,$$

and prove their integrality. Here U is a fundamental Lucas sequence, $a\ge2$ and $r\ge1$ are integers, and $\binom{*}{*}_U$ denotes a Lucasnomial coefficient. If U=I, where I_n=n, then the C_I,a,r(n) are the usual generalized Fuss-Catalan numbers. With the assumption that U is regular, we show that U_(a-1)n+k divides $\binom{an}{n}_U$ for all $n\ge1$ but a set of asymptotic density 0if $k\ge1$, but only for a small set if $k\le0$. This small set is finite when $U\not=I$ and at most of upper asymptotic density $1-\log 2$ when U=I. We also determine all triples (U,a,k), where $k\ge2$, for which the exceptional set of density 0 is actually finite, and in fact empty.