A003080 Number of rooted triangular cacti with 2n+1 nodes (n triangles).

1, 1, 2, 5, 13, 37, 111, 345, 1105, 3624, 12099, 41000, 140647, 487440, 1704115, 6002600, 21282235, 75890812, 272000538, 979310627, 3540297130, 12845634348, 46764904745, 170767429511, 625314778963, 2295635155206, 8447553316546, 31153444946778, 115122389065883

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• eigen
• nice

Comments

a(n) is also the number of isomorphism classes of Fano Bott manifolds of complex dimension n (see [Cho-Lee-Masuda-Park]). - Eunjeong Lee, Jun 29 2021

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 305, (4.2.34).
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.20).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
• Yunhyung Cho, Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, On the enumeration of Fano Bott manifolds, arXiv:2106.12788 [math.AG], 2021. See Table 1 p. 8.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
• N. J. A. Sloane, Transforms
• Index entries for sequences related to cacti

Crossrefs

Column k=3 of A332648.

Cf. A003081, A034940, A034941.

Programs

Mathematica

terms = 30;
nmax = 2 terms;
A[] = 0; Do[A[x] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
DeleteCases[CoefficientList[A[x], x], 0] (* Jean-François Alcover, Sep 02 2018 *)

Formula

a(n)=b(2n+1). b shifts left under transform T where Tb = EULER(E_2(b)). E_2(b) has g.f. (B(x^2)+B(x)^2)/2.

a(n) ~ c * d^n / n^(3/2), where d = 3.90053254788870206167147120260433375638561926371844809... and c = 0.4861961460367182791173441493565088408563977498871021... - Vaclav Kotesovec, Jul 01 2021

Extensions

Sequence extended by Paul Zimmermann, Mar 15 1996

Additional comments from Christian G. Bower