A003081 -id:A003081 - cp's OEIS frontend

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

Christian G. Bower, Oct 15 1998

nonn

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

			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)

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

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

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

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

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

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

A003080 Number of rooted triangular cacti with 2n+1 nodes (n triangles).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 111, 345, 1105, 3624, 12099, 41000, 140647, 487440, 1704115, 6002600, 21282235, 75890812, 272000538, 979310627, 3540297130, 12845634348, 46764904745, 170767429511, 625314778963, 2295635155206, 8447553316546, 31153444946778, 115122389065883

Offset: 0

N. J. A. Sloane

a(n) is also the number of isomorphism classes of Fano Bott manifolds of complex dimension n (see [Cho-Lee-Masuda-Park]). - Eunjeong Lee, Jun 29 2021

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 305, (4.2.34).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.20).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
Yunhyung Cho, Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, On the enumeration of Fano Bott manifolds, arXiv:2106.12788 [math.AG], 2021. See Table 1 p. 8.
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)
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti

Column k=3 of A332648.

Cf. A003081, A034940, A034941.

terms = 30;
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}];
DeleteCases[CoefficientList[A[x], x], 0] (* Jean-François Alcover, Sep 02 2018 *)

a(n)=b(2n+1). b shifts left under transform T where Tb = EULER(E_2(b)). E_2(b) has g.f. (B(x^2)+B(x)^2)/2.

a(n) ~ c * d^n / n^(3/2), where d = 3.90053254788870206167147120260433375638561926371844809... and c = 0.4861961460367182791173441493565088408563977498871021... - Vaclav Kotesovec, Jul 01 2021

Sequence extended by Paul Zimmermann, Mar 15 1996

Additional comments from Christian G. Bower

A332649 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 6, 1, 1, 1, 1, 3, 7, 8, 11, 1, 1, 1, 1, 4, 8, 25, 19, 23, 1, 1, 1, 1, 4, 13, 31, 88, 48, 47, 1, 1, 1, 1, 5, 14, 67, 132, 366, 126, 106, 1, 1, 1, 1, 5, 20, 80, 372, 636, 1583, 355, 235, 1

Offset: 0

Andrew Howroyd, Feb 18 2020

The number of nodes will be n*(k-1) + 1.

			Array begins:
======================================================
n\k | 1  2   3    4    5     6     7     8      9
----+-------------------------------------------------
  0 | 1  1   1    1    1     1     1     1      1 ...
  1 | 1  1   1    1    1     1     1     1      1 ...
  2 | 1  1   1    1    1     1     1     1      1 ...
  3 | 1  2   2    3    3     4     4     5      5 ...
  4 | 1  3   4    7    8    13    14    20     22 ...
  5 | 1  6   8   25   31    67    80   143    165 ...
  6 | 1 11  19   88  132   372   504  1093   1391 ...
  7 | 1 23  48  366  636  2419  3659  9722  13485 ...
  8 | 1 47 126 1583 3280 16551 28254 91391 138728 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Wikipedia, Cactus graph
Index entries for sequences related to cacti

Columns k=1..4 are A000012, A000055(n+1), A003081, A287892.

Cf. A303694, A332648, A332650, A332651.

\\ here R(n,k) is column k+1 of A332648.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec((g^k + g^(k%2)*subst(g^(k\2), x, x^2))/2))); concat([1], v)}
U(n,k)={my(p=Ser(R(n,k-1))); my(g(d)=subst(p + O(x*x^(n\d)), x, x^d)); Vec(g(1) + x*sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - x*(g(1)^k)/2 + x*if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)}
T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n, k, Col(U(n,k+1)))))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) }

A287892 Number of unrooted unlabeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 1, 1, 3, 7, 25, 88, 366, 1583, 7336, 34982, 172384, 867638, 4452029, 23194392, 122462546, 653957197, 3527218134, 19192275883, 105248481503, 581223149532, 3230039198628, 18053111982952, 101426901301489, 572554846192811, 3246191706162233, 18478844801342495

Offset: 0

N. J. A. Sloane, Jun 21 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.

Column k=4 of A332649.

Cf. A003081, A287889, A287890, A287891.

\\ Here G(n) is A287891 as vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
G(n)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec(g*(g^2 + subst(g, x, x^2))/2))); concat([1], v)}
seq(n)={my(p=Ser(G(n))); my(g(d)=subst(p,x,x^d)); Vec(g(1) + x*(2*g(4) + 3*g(2)^2 - 2*g(1)^2*g(2) - 3*g(1)^4)/8)} \\ Andrew Howroyd, Feb 18 2020

G.f.: g(x) + x*(2*g(x^4) + 3*g(x^2)^2 - 2*g(x)^2*g(x^2) - 3*g(x)^4)/8 where g(x) is the g.f. of A287891.

a(0) changed and terms a(12) and beyond from Andrew Howroyd, Feb 18 2020

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

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A003080 Number of rooted triangular cacti with 2n+1 nodes (n triangles).

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A332649 Array read by antidiagonals: T(n,k) is the number of unlabeled k-gonal cacti having n polygons.

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A287892 Number of unrooted unlabeled 4-cactus graphs on 3n+1 nodes.

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