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A361242 Number of nonequivalent noncrossing cacti with n nodes up to rotation.

%I A361242 #11 Mar 11 2023 00:13:31
%S A361242 1,1,1,2,7,26,144,800,4995,32176,215914,1486270,10471534,75137664,
%T A361242 547756650,4047212142,30255934851,228513227318,1741572167716,
%U A361242 13380306774014,103542814440878,806476983310180,6318519422577854,49769050291536486,393933908000862866
%N A361242 Number of nonequivalent noncrossing cacti with n nodes up to rotation.
%C A361242 A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).
%C A361242 Since every cactus is an outerplanar graph, every cactus has at least one drawing as a noncrossing graph.
%H A361242 Andrew Howroyd, <a href="/A361242/b361242.txt">Table of n, a(n) for n = 0..500</a>
%H A361242 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A361242 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%e A361242 The a(3) = 2 nonequivalent cacti have the following blocks:
%e A361242    {{1,2}, {1,3}},
%e A361242    {{1,2,3}}.
%e A361242 Graphically these can be represented:
%e A361242         1           1
%e A361242       /  \        /  \
%e A361242      2    3      2----3
%e A361242 .
%e A361242 The a(4) = 7 nonequivalent cacti have the following blocks:
%e A361242   {{1,2}, {1,3}, {1,4}},
%e A361242   {{1,2}, {1,3}, {3,4}},
%e A361242   {{1,2}, {1,4}, {2,3}},
%e A361242   {{1,2}, {2,4}, {3,4}},
%e A361242   {{1,2}, {1,3,4}},
%e A361242   {{1,2}, {2,3,4}},
%e A361242   {{1,2,3,4}}.
%e A361242 Graphically these can be represented:
%e A361242    1---4    1   4    1---4    1   4
%e A361242    | \      | \ |    |        | / |
%e A361242    2   3    2   3    2---3    2   3
%e A361242 .
%e A361242    1---4    1   4    1---4
%e A361242    | \ |    | / |    |   |
%e A361242    2   3    2---3    2---3
%o A361242 (PARI) \\ Here F(n) is the g.f. of A003168.
%o A361242 F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}
%o A361242 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 A361242 Cf. A003168, A007297, A361236, A361243.
%K A361242 nonn
%O A361242 0,4
%A A361242 _Andrew Howroyd_, Mar 07 2023