A034941 Number of labeled triangular cacti with 2n+1 nodes (n triangles).

1, 1, 15, 735, 76545, 13835745, 3859590735, 1539272109375, 831766748637825, 585243816844111425, 520038240188935042575, 569585968715180280038175, 753960950911045074462890625, 1186626209895384011075327630625, 2190213762744801162239116550679375

Offset

0,3

Comments

Also the number of 3-uniform hypertrees spanning 2n + 1 labeled vertices. - Gus Wiseman, Jan 12 2019

Number of rank n+1 simple series-parallel matroids on [2n+1]. - Matt Larson, Mar 06 2023

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.

Katie Gedeon, N. Proudfoot, and B. Young, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, arXiv preprint arXiv:1611.07474 [math.CO], 2016-2017.

Nicholas Proudfoot and Ben Young, Configuration spaces, FS^op-modules, and Kazhdan-Lusztig polynomials of braid matroids, arXiv:1704.04510 [math.RT], 2017.

Eric Weisstein's World of Mathematics, Cactus Graph

Index entries for sequences related to cacti

Formula

a(n) = A034940(n)/(2n+1).

The closed form a(n) = (2n-1)!! (2n+1)^(n-1) can be obtained from the generating function in A034940. - Noam D. Elkies, Dec 16 2002

Example

a(3) = 5!! * 7^2 = (1*3*5) * 49 = 735.
From Gus Wiseman, Jan 12 2019: (Start)
The a(2) = 15 3-uniform hypertrees:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypertrees spanning 7 vertices, and their multiplicities in the labeled case, which add up to a(3) = 735:
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  630 X {{1,2,6},{3,4,7},{5,6,7}}
(End)

Mathematica

Table[(2n+1)^(n-1)(2n)!/(2^n n!), {n, 0, 14}] (* Jean-François Alcover, Nov 06 2018 *)

Prog

(Magma) [(2*n+1)^(n-1)*Factorial(2*n)/(2^n*Factorial(n)): n in [0..15]]; // Vincenzo Librandi, Feb 19 2020

Crossrefs

Cf. A000272, A000665, A003081, A030019, A035053, A125791, A302374, A320444, A323292, A323298.

Sequence in context: A209461 A209523 A117812 * A199227 A055683 A196465

Adjacent sequences: A034938 A034939 A034940 * A034942 A034943 A034944

Keyword

nonn

Author

Christian G. Bower, Oct 15 1998

Extensions

Typo in a(10) corrected and more terms from Alois P. Heinz, Jun 23 2017

Status

approved