A332648 -id:A332648 - cp's OEIS frontend

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

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

A287891 Number of rooted unlabeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 1, 3, 11, 46, 208, 1002, 5012, 25863, 136519, 733902, 4003475, 22106155, 123313289, 693871975, 3933700703, 22447035938, 128828019447, 743142630614, 4306327193744, 25056121416684, 146325789652514, 857393585946194, 5039223717251954, 29700183601347111, 175496470696059267

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 A332648.

Cf. A003080, A287889, A287890, A287892.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(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)} \\ Andrew Howroyd, Feb 17 2020

a(0) changed and terms a(11) and beyond from Andrew Howroyd, Feb 17 2020

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,])) }

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

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A287891 Number of rooted unlabeled 4-cactus graphs on 3n+1 nodes.

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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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