# central binomial coefficients modulo (was catalans modulo)

Antti Karttunen karttu at megabaud.fi
Tue Feb 5 12:34:08 CET 2002

```
Hélène Lenormand wrote:

> Let be the Catalan's numbers defined by
>
> c(1)=1, (n+1).c(n+1)=2.(2n-1).c(n).

Sorry, the formula you gave, converted to explicit form
as c := n -> `if`((1 = n),1,(2*((2*n)-1)*c(n-1))/n);
is not for Catalan's numbers (A000108)
but for one variant of the "central binomial coefficients":

ID Number: A001700 (Formerly M2848 and N1144)
Sequence:  1,3,10,35,126,462,1716,6435,24310,92378,352716,1352078,
5200300,20058300,77558760,300540195,1166803110,4537567650,
17672631900,68923264410,269128937220,1052049481860,
4116715363800
Name:      C(2n+1, n+1): number of ways to put n indistinguishable balls
into n
distinguishable boxes = number of n-th degree monomials in n
variables
= number of monotone maps from 1..n to 1..n.

>
>
> For n>1, if (n+1) modulo 6 = 1 or 5, then (c(n) modulo (n+1))=0.
>
> Beginning of c(n) modulo (n+1), n>0:
>
> 1  1  2  0  2  0  4  6  0  0  8  0  0  10  8  0  0  0  0  0  0  0 .....
>
> (sequence not in EOIS)
>
> Are there others n with  c(n) modulo (n+1) not zero ?

Concerning this problem, see
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A046097

and its assorted links, and old postings on this mailing list
from october & november of 1999.
(Answer is: most probably not)

>
>
> Claude lenormand (e-mail: hlne.lenormand at voonoo.net)

Salut,

Antti Karttunen

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