Search: id:A054726
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%I A054726
%S A054726 1,1,2,8,48,352,2880,25216,231168,2190848,21292032,211044352,
%T A054726 2125246464,21681954816,223623069696,2327818174464,24424842461184,
%U A054726 258054752698368,2742964283768832,29312424612462592,314739971287154688
%N A054726 Number of graphs with n nodes on a circle without crossing edges.
%C A054726 Related to Schroeder's second problem.
%C A054726 A001006 gives number of ways of drawing any number of nonintersecting 
               chords between n points on a circle, while this sequence gives number 
               of ways of drawing noncrossing chords between n points on a circle. 
               The difference is that nonintersection chords have no point in common, 
               while noncrossing chords may share an endpoint. - David W. Wilson, 
               Jan 30, 2003
%C A054726 a(n)=number of lattice paths from (0,0) to (n-2,n-2) that consist of 
               steps (i,j), i,j nonnegative integers not both 0 and that stay strictly 
               below the line y=x except at their endpoints. For example, a(4)=8 
               counts the paths with following step sequences: {(2, 2)}, {(2, 1), 
               (0, 1)}, {(2, 0), (0, 2)}, {(2, 0), (0, 1), (0, 1)}, {(1, 0), (1, 
               2)}, {(1, 0), (1, 1), (0, 1)}, {(1, 0), (1, 0), (0, 2)}, {(1, 0), 
               (1, 0), (0, 1), (0, 1)}. If the word "strictly" is replaced by "weakly", 
               the counting sequence becomes A059435. - David Callan (callan(AT)stat.wisc.edu), 
               Jun 07 2006
%H A054726 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/
               NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies 
               in Automatic Combinatorics, Volume II (1997).
%H A054726 P. Flajolet and M. Noy, <a href="http://algo.inria.fr/flajolet/Publications">
               Analytic Combinatorics of Noncrossing Configurations</a>, Discr. 
               Math. 204 (1999), 203-229
%H A054726 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               486
%F A054726 G.f. if offset 0: 1+3/2*z-z^2-z/2*sqrt(1-12*z+4*z^2);
%p A054726 br := {EA = Union(Sequence(EA, card >= 2), Prod(V, Sequence(EA), Sequence(EA))), 
               V=Union(Prod(Z, G)), G=Union(Epsilon, Prod(Z, G), Prod(V,V,Sequence(EA), 
               Sequence(EA), Sequence(Union(Sequence(EA,card>=1), Prod(V,Sequence(EA),
               Sequence(EA)))))) }; ggSeq := [seq(count([G, br], size=i), i=0..20)];
%Y A054726 Cf. A006013, A001003.
%Y A054726 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, 
               A006600, A007569, A007678. See also entries for chord diagrams in 
               Index file.
%Y A054726 a(n) = 2^(n-1)*A001003(n-2).
%Y A054726 a(n) = 2^(n-1)*A001003(n-3).
%Y A054726 Sequence in context: A007170 A136722 A085615 this_sequence A003576 A095989 
               A124453
%Y A054726 Adjacent sequences: A054723 A054724 A054725 this_sequence A054727 A054728 
               A054729
%K A054726 nonn
%O A054726 1,3
%A A054726 Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Apr 20 2000

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