Concerning A007123.

[Antti Karttunen]

I forgot to mention the _combinatorial interpretation_
for this Cat(n)+binomial(n,floor(n/2)))/2 suspected for A007123:
Dyck paths, general (rooted) plane trees and general
parenthesizations (and other denizens of the Catalania
in various disguises), upto the reflection.

(Note that A007595 is the plane _binary_ trees
(and their other incarnation: polygon triangulations)
upto the reflection.)

So, is there a combinatorial map between these
and with the objects given on current name-line below?
Working from the bracelets through the mapping explained
in A003239 sniffs promising. 

-- Same

Antti Karttunen wrote:

> Then there is another sequence, A007123:
> 
> ID Number: A007123 (Formerly M1218)
> Sequence:  1,1,2,4,10,26,76,232,750,2494,8524,29624,104468,372308,
>            1338936,4850640,17685270,64834550,238843660,883677784,
>            3282152588,12233309868,45741634536,171530482864,
>            644953425740,2430975800876
> Name:      Connected unit interval graphs with n nodes; also bracelets (turn over
>               necklaces) with n black beads and n-1 white beads.
> References R. W. Robinson, personal communication.
> Links:     F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
>            Index entries for sequences related to bracelets
> Keywords:  nonn,nice
> Offset:    1
> Author(s): njas
> Extension: Extended by Christian G. Bower (bowerc at usa.net)
> 
> which I know to occur in A073201, but only if its formula is:
> 
>    A000108(n)+A001405(n)
>    ---------------------
>              2
> 
> i.e. (Cat(n)+binomial(n,floor(n/2)))/2 with maybe a shorter
> formula through some intricate binomial identity...
> 
> (So, who knows, does A007123 have this one?)
> 
> Yours,
> 
> Antti Karttunen