given formula = easy but doesnt work

[Antti Karttunen]

Concerning

ID Number: A007595 (Formerly M2681)
Sequence:  1,1,3,7,22,66,217,715,2438,8398,29414,104006,371516,1337220,
           4847637,17678835,64823110,238819350,883634026,3282060210,
           12233141908,45741281820,171529836218,644952073662,
           2430973304732
Name:      (C_n + C_(n-1)/2 )/2, C = Catalan numbers (A000108).
References P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38
              (1987), 155-183.
Links:     P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Keywords:  nonn,easy
Offset:    1
Author(s): njas

Reiner Martin wrote:
> 
> The following seems to work: (for A007595)
> C_n / 2   if n is even
> ( C_n + C_((n-1)/2) ) / 2   if n is odd

Yes, this is just as I interpreted it, as I mention
it to be the 0th, 2th and 4th row (occurring also in infinitude
of other positions) of the table A073201,
and as such the formula also matches with the interpretation
given for it by Peter J. Cameron: "Binary trees up to reflection"
in his appendix: http://www.math.uwaterloo.ca/JIS/VOL3/groupdata.html
of his JIS article
"Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5."

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