cp's OEIS Frontend

A381469 Number of unlabeled 2,3 cacti (triangular cacti with bridges) rooted at a triangle with n triangles and every node contained in exactly one triangle.

%I A381469 #7 Feb 25 2025 01:57:28
%S A381469 0,1,1,4,15,66,304,1503,7622,39856,212447,1151614,6324924,35127396,
%T A381469 196917025,1112776860,6332114208,36252066562,208665030299,
%U A381469 1206819559836,7009605269315,40871341270810,239144296550695,1403719120877546,8263431521645830,48774908707685849
%N A381469 Number of unlabeled 2,3 cacti (triangular cacti with bridges) rooted at a triangle with n triangles and every node contained in exactly one triangle.
%C A381469 The number of vertices is 3*n and for n > 0, the number of bridges is n-1.
%H A381469 Andrew Howroyd, <a href="/A381469/b381469.txt">Table of n, a(n) for n = 0..500</a>
%H A381469 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>.
%H A381469 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>.
%F A381469 G.f.: x*(B(x)^3 + 3*B(x)*B(x^2) + 2*B(x^3))/6 where B(x) is the g.f. of A287891.
%o A381469 (PARI) \\ here R(n) gives A287891 as g.f.
%o A381469 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A381469 raise(p,d) = {my(n=serprec(p,x)-1); subst(p + O(x^(n\d+1)), x, x^d)}
%o A381469 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 A381469 seq(n)={ my(p=R(n-1)); Vec(x*(p^3 + 3*p*raise(p,2) + 2*raise(p,3))/6 + O(x*x^n), -n-1) }
%Y A381469 Cf. A287891, A380634 (unrooted).
%K A381469 nonn
%O A381469 0,4
%A A381469 _Andrew Howroyd_, Feb 25 2025