Journal of Integer Sequences, Vol. 21 (2018), Article 18.6.5

Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions

Christian Ballot
Département de Mathématiques et Informatique
Université de Caen-Normandie


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


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 In = n, then the CI,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 0 if $k\ge1$, 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.

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(Concerned with sequences A001764 A003150 A014847 A107920.)

Received March 6 2018; revised version received June 6 2018. Published in Journal of Integer Sequences, August 22 2018.

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