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
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
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