A003081 Number of triangular cacti with 2n+1 nodes (n triangles).

1, 1, 1, 2, 4, 8, 19, 48, 126, 355, 1037, 3124, 9676, 30604, 98473, 321572, 1063146, 3552563, 11982142, 40746208, 139573646, 481232759, 1669024720, 5819537836, 20390462732, 71762924354, 253601229046, 899586777908, 3202234779826, 11435967528286, 40964243249727

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 306, (4.2.35).
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.21).
• 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.
• 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)
• Index entries for sequences related to cacti

Crossrefs

Column k=3 of A332649.

Cf. A003080, A034940, A034941.

Programs

Mathematica

terms = 31;
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}];
g[x_] = (A[x] /. x^k_ -> x^((k - 1)/2)) - x + 1;
g[x] + x((g[x^3] - g[x]^3)/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2020, after Andrew Howroyd *)

Formula

a(n)=b(2n+1). A003080(n)=c(2n+1).

G.f.: B(x)=C(x)+(C(x^3)-C(x)^3)/3.

G.f.: g(x) + x*(g(x^3) - g(x)^3)/3 where g(x) is the g.f. of A003080. - Andrew Howroyd, Feb 18 2020

Extensions

Extended with formula by Christian G. Bower, 10/98