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A361243 Number of nonequivalent noncrossing cacti with n nodes up to rotation and reflection.

%I A361243 #8 Mar 11 2023 00:13:26
%S A361243 1,1,1,2,5,17,79,421,2537,16214,108204,743953,5237414,37574426,
%T A361243 273889801,2023645764,15128049989,114256903169,870786692493,
%U A361243 6690155544157,51771411793812,403238508004050,3159259746188665,24884525271410389,196966954270163612
%N A361243 Number of nonequivalent noncrossing cacti with n nodes up to rotation and reflection.
%C A361243 A noncrossing cactus is a connected noncrossing graph (A007297) that is a cactus graph (a tree of edges and polygons).
%H A361243 Andrew Howroyd, <a href="/A361243/b361243.txt">Table of n, a(n) for n = 0..500</a>
%H A361243 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A361243 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%e A361243 The a(4) = 5 nonequivalent cacti have the following blocks:
%e A361243   {{1,2}, {1,3}, {1,4}},
%e A361243   {{1,2}, {1,3}, {3,4}},
%e A361243   {{1,2}, {1,4}, {2,3}},
%e A361243   {{1,2}, {1,3,4}},
%e A361243   {{1,2,3,4}}.
%e A361243 Graphically these can be represented:
%e A361243    1---4    1   4    1---4    1---4    1---4
%e A361243    | \      | \ |    |        | \ |    |   |
%e A361243    2   3    2   3    2---3    2   3    2---3
%o A361243 (PARI) \\ Here F(n) is the g.f. of A003168.
%o A361243 F(n) = {1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n))}
%o A361243 seq(n) = {my(f=F(n-1)); Vec(1/(1 - x*subst(f + O(x^(n\2+1)), x, x^2)) + 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)/2}
%Y A361243 Cf. A003168, A007297, A361239, A361242.
%K A361243 nonn
%O A361243 0,4
%A A361243 _Andrew Howroyd_, Mar 07 2023