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A380634 Number of unlabeled 2,3 cacti (triangular cacti with bridges) with n triangles and every node contained in exactly one triangle.

%I A380634 #10 Feb 25 2025 01:56:24
%S A380634 1,1,1,2,6,18,66,265,1140,5186,24588,120062,600884,3066490,15907266,
%T A380634 83665520,445317808,2394928214,12997988041,71116953074,391931826699,
%U A380634 2174062325068,12130745830640,68049392678632,383601371168527,2172093593344465,12349917974708867
%N A380634 Number of unlabeled 2,3 cacti (triangular cacti with bridges) with n triangles and every node contained in exactly one triangle.
%C A380634 The number of vertices is 3*n and for n > 0, the number of bridges is n-1.
%H A380634 Andrew Howroyd, <a href="/A380634/b380634.txt">Table of n, a(n) for n = 0..500</a>
%H A380634 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A380634 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A380634 a(n) = A380631(3*n,n) = A381467(3*n,n).
%e A380634 The a(3) = 2 cactus graphs are:
%e A380634     o       o       o        o   o---o   o
%e A380634    / \     / \     / \      / \   \ /   / \
%e A380634   o---o---o---o---o---o    o---o---o---o---o
%o A380634 (PARI) \\ here R(n) gives A287891 as g.f.
%o A380634 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A380634 raise(p,d) = {my(n=serprec(p,x)-1); subst(p + O(x^(n\d+1)), x, x^d)}
%o A380634 R(n)={my(p=1+O(x)); for(n=1, n, p = 1 + x*Ser(EulerT(Vec(p*(p^2 + raise(p,2))/2)))); p}
%o A380634 seq(n)={ my(p=R(n-1), g=p*(p^2 + raise(p,2))/2); Vec(1 + x*(x*(raise(g,2) - g^2) + p*raise(p,2) + (p^3 + 2*raise(p,3))/3)/2) }
%Y A380634 Cf. A091487, A287891, A380631, A381467.
%K A380634 nonn
%O A380634 0,4
%A A380634 _Andrew Howroyd_, Feb 24 2025