%I #57 Mar 07 2023 07:40:50

%S 1,1,15,735,76545,13835745,3859590735,1539272109375,831766748637825,

%T 585243816844111425,520038240188935042575,569585968715180280038175,

%U 753960950911045074462890625,1186626209895384011075327630625,2190213762744801162239116550679375

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

%C Also the number of 3-uniform hypertrees spanning 2n + 1 labeled vertices. - _Gus Wiseman_, Jan 12 2019

%C Number of rank n+1 simple series-parallel matroids on [2n+1]. - _Matt Larson_, Mar 06 2023

%H Alois P. Heinz, <a href="/A034941/b034941.txt">Table of n, a(n) for n = 0..200</a>

%H Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.

%H Luis Ferroni and Matt Larson, <a href="https://arxiv.org/abs/2303.02253">Kazhdan-Lusztig polynomials of braid matroids</a>, arXiv:2303.02253 [math.CO], 2023.

%H Katie Gedeon, N. Proudfoot, and B. Young, <a href="http://arxiv.org/abs/1611.07474">Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures</a>, arXiv preprint arXiv:1611.07474 [math.CO], 2016-2017.

%H Nicholas Proudfoot and Ben Young, <a href="https://arxiv.org/abs/1704.04510">Configuration spaces, FS^op-modules, and Kazhdan-Lusztig polynomials of braid matroids</a>, arXiv:1704.04510 [math.RT], 2017.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CactusGraph.html">Cactus Graph</a>

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>

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

%F 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

%e a(3) = 5!! * 7^2 = (1*3*5) * 49 = 735.

%e From _Gus Wiseman_, Jan 12 2019: (Start)

%e The a(2) = 15 3-uniform hypertrees:

%e {{1,2,3},{1,4,5}}

%e {{1,2,3},{2,4,5}}

%e {{1,2,3},{3,4,5}}

%e {{1,2,4},{1,3,5}}

%e {{1,2,4},{2,3,5}}

%e {{1,2,4},{3,4,5}}

%e {{1,2,5},{1,3,4}}

%e {{1,2,5},{2,3,4}}

%e {{1,2,5},{3,4,5}}

%e {{1,3,4},{2,3,5}}

%e {{1,3,4},{2,4,5}}

%e {{1,3,5},{2,3,4}}

%e {{1,3,5},{2,4,5}}

%e {{1,4,5},{2,3,4}}

%e {{1,4,5},{2,3,5}}

%e 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:

%e 105 X {{1,2,7},{3,4,7},{5,6,7}}

%e 630 X {{1,2,6},{3,4,7},{5,6,7}}

%e (End)

%t Table[(2n+1)^(n-1)(2n)!/(2^n n!), {n, 0, 14}] (* _Jean-François Alcover_, Nov 06 2018 *)

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

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

%K nonn

%O 0,3

%A _Christian G. Bower_, Oct 15 1998

%E Typo in a(10) corrected and more terms from _Alois P. Heinz_, Jun 23 2017