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

0,4

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.

Formula

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

Example

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

Prog

(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) }

Crossrefs

Cf. A091487, A287891, A380631, A381467.

Sequence in context: A150076 A150077 A173385 * A057693 A053496 A079577

Adjacent sequences: A380631 A380632 A380633 * A380635 A380636 A380637

Keyword

nonn,new

Author

Andrew Howroyd, Feb 24 2025

Status

approved