login
Number of nonequivalent noncrossing cacti with n nodes up to rotation and reflection.
3

%I #8 Mar 11 2023 00:13:26

%S 1,1,1,2,5,17,79,421,2537,16214,108204,743953,5237414,37574426,

%T 273889801,2023645764,15128049989,114256903169,870786692493,

%U 6690155544157,51771411793812,403238508004050,3159259746188665,24884525271410389,196966954270163612

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

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

%H Andrew Howroyd, <a href="/A361243/b361243.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(4) = 5 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}, {1,3,4}},

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

%e Graphically these can be represented:

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

%e | \ | \ | | | \ | | |

%e 2 3 2 3 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/(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 Cf. A003168, A007297, A361239, A361242.

%K nonn

%O 0,4

%A _Andrew Howroyd_, Mar 07 2023