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.

0, 1, 1, 4, 15, 66, 304, 1503, 7622, 39856, 212447, 1151614, 6324924, 35127396, 196917025, 1112776860, 6332114208, 36252066562, 208665030299, 1206819559836, 7009605269315, 40871341270810, 239144296550695, 1403719120877546, 8263431521645830, 48774908707685849

Offset: 0

Author

Andrew Howroyd, Feb 25 2025

Keywords

• nonn

Comments

The number of vertices is 3*n and for n > 0, the number of bridges is n-1.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Wikipedia, Cactus graph.
• Index entries for sequences related to cacti.

Crossrefs

Cf. A287891, A380634 (unrooted).

Programs

PARI

\\ here R(n) gives A287891 as g.f.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
raise(p,d) = {my(n=serprec(p,x)-1); subst(p + O(x^(n\d+1)), x, x^d)}
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}
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) }

Formula

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.