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Number of nonequivalent noncrossing cacti with n nodes up to rotation.
7

%I #11 Mar 11 2023 00:13:31

%S 1,1,1,2,7,26,144,800,4995,32176,215914,1486270,10471534,75137664,

%T 547756650,4047212142,30255934851,228513227318,1741572167716,

%U 13380306774014,103542814440878,806476983310180,6318519422577854,49769050291536486,393933908000862866

%N Number of nonequivalent noncrossing cacti with n nodes up to rotation.

%C A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).

%C Since every cactus is an outerplanar graph, every cactus has at least one drawing as a noncrossing graph.

%H Andrew Howroyd, <a href="/A361242/b361242.txt">Table of n, a(n) for n = 0..500</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.

%e The a(3) = 2 nonequivalent cacti have the following blocks:

%e {{1,2}, {1,3}},

%e {{1,2,3}}.

%e Graphically these can be represented:

%e 1 1

%e / \ / \

%e 2 3 2----3

%e .

%e The a(4) = 7 nonequivalent cacti have the following blocks:

%e {{1,2}, {1,3}, {1,4}},

%e {{1,2}, {1,3}, {3,4}},

%e {{1,2}, {1,4}, {2,3}},

%e {{1,2}, {2,4}, {3,4}},

%e {{1,2}, {1,3,4}},

%e {{1,2}, {2,3,4}},

%e {{1,2,3,4}}.

%e Graphically these can be represented:

%e 1---4 1 4 1---4 1 4

%e | \ | \ | | | / |

%e 2 3 2 3 2---3 2 3

%e .

%e 1---4 1 4 1---4

%e | \ | | / | | |

%e 2 3 2---3 2---3

%o (PARI) \\ Here F(n) is the g.f. of A003168.

%o F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}

%o seq(n) = {my(f=F(n-1)); Vec(1 + intformal(f) - sum(d=2, n, eulerphi(d) * log(1-subst(x*f^2 + O(x^(n\d+1)),x,x^d)) / d), -n-1)}

%Y Cf. A003168, A007297, A361236, A361243.

%K nonn

%O 0,4

%A _Andrew Howroyd_, Mar 07 2023