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

Département de Mathématiques et Informatique

Université de Caen-Normandie

France

**Abstract:**

For all integers
we define the generalized Lucasnomial
Fuss-Catalan numbers

and prove their integrality. Here*U* is a fundamental Lucas sequence,
and
are integers, and
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
for all
but a set of asymptotic density 0 if ,
but only for a small set if .
This small set is
finite when
and at most of upper asymptotic density
when *U* = *I*. We also determine
all triples (*U*,*a*,*k*), where ,
for which the
exceptional set of density 0 is actually finite, and in fact empty.

and prove their integrality. Here

(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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