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

1, 1, 1, 2, 6, 18, 66, 265, 1140, 5186, 24588, 120062, 600884, 3066490, 15907266, 83665520, 445317808, 2394928214, 12997988041, 71116953074, 391931826699, 2174062325068, 12130745830640, 68049392678632, 383601371168527, 2172093593344465, 12349917974708867

Offset: 0

Author

Andrew Howroyd, Feb 24 2025

Keywords

• nonn

Comments

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

Examples

			The a(3) = 2 cactus graphs are:
    o       o       o        o   o---o   o
   / \     / \     / \      / \   \ /   / \
  o---o---o---o---o---o    o---o---o---o---o
		

Links

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

Crossrefs

Cf. A091487, A287891, A380631, A381467.

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), 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) }

Formula

a(n) = A380631(3*n,n) = A381467(3*n,n).