%I A007123 M1218
%S A007123 1,1,2,4,10,26,76,232,750,2494,8524,29624,104468,372308,1338936,
%T A007123 4850640,17685270,64834550,238843660,883677784,3282152588,
%U A007123 12233309868,45741634536,171530482864,644953425740,2430975800876
%N A007123 Number of connected unit interval graphs with n nodes; also bracelets 
               (turn over necklaces) with n black beads and n-1 white beads.
%C A007123 Also number of rooted planar general trees (of n vertices or n-1 edges) 
               up to reflection, - AK, Aug 09, 2002 (for the correspondence with 
               bracelets, start by considering Raney's lemma as explained by Graham, 
               Knuth & Patashnik).
%C A007123 Number of connected lattice path matroids on n elements up to isomorphism.
%C A007123 a(n) = number of noncrossing set partitions of [n] up to reflection (i<->
               n+1-i). Example: a(4) counts 123, 1-23, 13-2, 1-2-3 but not 12-3 
               because it is the reflection of 1-23. - David Callan (callan(AT)stat.wisc.edu), 
               Oct 08 2005
%D A007123 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
%D A007123 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 345 & 346.
%D A007123 R. W. Robinson, personal communication.
%D A007123 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
%D A007123 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007123 R. W. Robinson, <a href="b007123.txt">Table of n, a(n) for n = 1..190</
               a>
%H A007123 J. E. Bonin, A. de Mier and M. Noy, <a href="http://arXiv.org/abs/math.CO/
               0211188">Lattice path matroids: enumerative aspects and Tutte polynomials</
               a>.
%H A007123 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A007123 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">
               Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A007123 <a href="Sindx_Br.html#bracelets">Index entries for sequences related 
               to bracelets</a>
%F A007123 a(n) = (Cat(n)+binomial(n, floor(n/2)))/2 = (A000108(n)+A001405(n))/2. 
               - Antti Karttunen, Aug 09, 2002
%F A007123 G.f.: (1+2*x-sqrt(1-4*x)*sqrt(1-4*x^2))/(4*sqrt(1-4*x^2)).
%t A007123 f[k_Integer, n_] := (Plus @@ (EulerPhi[ # ]Binomial[n/#, k/# ] & /@ Divisors[GCD[n, 
               k]])/n + Binomial[(n - If[OddQ@n, 1, If[OddQ@k, 2, 0]])/2, (k - If[OddQ@k, 
               1, 0])/2])/2 - Robert A. Russell (russell(AT)post.harvard.edu), Sep 
               27 2004
%t A007123 Table[ f[n, 2n - 1], {n, 10}]
%Y A007123 Cf. A007595, A073201.
%Y A007123 Occurs as row 164 in A073201. Next-to-center columns of triangle A052307.
%Y A007123 Sequence in context: A049401 A148099 A007579 this_sequence A007578 A007580 
               A000085
%Y A007123 Adjacent sequences: A007120 A007121 A007122 this_sequence A007124 A007125 
               A007126
%K A007123 nonn,nice
%O A007123 1,3
%A A007123 N. J. A. Sloane (njas(AT)research.att.com).
%E A007123 Extended by Christian G. Bower (bowerc(AT)usa.net)