A091485 Number of labeled 2,3 cacti (triangular cacti with bridges).

1, 1, 4, 28, 290, 3996, 68992, 1434112, 34895772, 973450000, 30636233936, 1074020373504, 41510792057176, 1753764940408768, 80412829785000000, 3977094146761424896, 211058327532167398928, 11963018212810373415168, 721321146876339731628352

Offset

1,3

Comments

As Alois P. Heinz has pointed out, the e.g.f in the Example section does not match the offset. However, the identity a(n) = A091481(n)/n holds with the present offset of 1. - N. J. A. Sloane, Jun 23 2017

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

Links

Table of n, a(n) for n=1..19.

Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.

Index entries for sequences related to cacti

Formula

a(n) = A091481(n)/n.

From Paul D. Hanna, Jun 01 2012: (Start)

E.g.f.: (1/x)*Series_Reversion( x/exp(x+x^2/2) ).

E.g.f. satisfies: A(x) = exp( x*A(x) + x^2*A(x)^2/2 ).

E.g.f. satisfies: A( x/exp(x+x^2/2) ) = exp(x+x^2/2).

(End)

a(n+1) = n! * Sum_{k=0..n} (1/2)^(n-k) * (n+1)^(k-1) * binomial(k,n-k)/k!. - Seiichi Manyama, Aug 19 2023

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 290*x^4/4! + 3996*x^5/5! +...

Mathematica

CoefficientList[1/x InverseSeries[x/Exp[x+x^2/2]+O[x]^20], x] Range[0, 18]! (* Jean-François Alcover, Aug 06 2018 *)

Crossrefs

Cf. A091481, A091487, A000085.

Sequence in context: A201595 A076729 A078634 * A201354 A112938 A307083

Adjacent sequences: A091482 A091483 A091484 * A091486 A091487 A091488

Keyword

nonn

Author

Christian G. Bower, Jan 14 2004

Status

approved