A091486 Number of unlabeled rooted 2,3 cacti (triangular cacti with bridges).

1, 1, 3, 7, 21, 60, 190, 600, 1977, 6589, 22408, 77050, 268178, 941599, 3333585, 11882427, 42615480, 153653039, 556664752, 2025330509, 7397242875, 27111563026, 99681629658, 367563272278, 1358945378906, 5036549490009, 18708739990129, 69640873691941

Offset: 1

Author

Christian G. Bower, Jan 14 2004

Keywords

• nonn
• eigen

Comments

Also number of unlabeled involution rooted trees.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
• N. J. A. Sloane, Transforms
• Index entries for sequences related to cacti
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A091487, A091488.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*(Ser(EulerT(Vec(p + (p^2 + subst(p,x,x^2))/2))))); Vec(p)} \\ Andrew Howroyd, Aug 30 2018

Formula

Shifts left under transform T where Ta = EULER(E_1, 2(a)). E_1, 2(a) has g.f. A(x)+(A(x^2)+A(x)^2)/2.