A332649 -id:A332649 - cp's OEIS frontend

A003081 Number of triangular cacti with 2n+1 nodes (n triangles).

1, 1, 1, 2, 4, 8, 19, 48, 126, 355, 1037, 3124, 9676, 30604, 98473, 321572, 1063146, 3552563, 11982142, 40746208, 139573646, 481232759, 1669024720, 5819537836, 20390462732, 71762924354, 253601229046, 899586777908, 3202234779826, 11435967528286, 40964243249727

Offset: 0

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 306, (4.2.35).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.21).
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.
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)
Index entries for sequences related to cacti

Column k=3 of A332649.

Cf. A003080, A034940, A034941.

terms = 31;
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}];
g[x_] = (A[x] /. x^k_ -> x^((k - 1)/2)) - x + 1;
g[x] + x((g[x^3] - g[x]^3)/3) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2020, after Andrew Howroyd *)

a(n)=b(2n+1). A003080(n)=c(2n+1).
G.f.: B(x)=C(x)+(C(x^3)-C(x)^3)/3.
G.f.: g(x) + x*(g(x^3) - g(x)^3)/3 where g(x) is the g.f. of A003080. - Andrew Howroyd, Feb 18 2020

Extended with formula by Christian G. Bower, 10/98

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

A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 3, 5, 9, 1, 1, 1, 3, 11, 13, 20, 1, 1, 1, 4, 13, 46, 37, 48, 1, 1, 1, 4, 22, 62, 208, 111, 115, 1, 1, 1, 5, 25, 140, 333, 1002, 345, 286, 1, 1, 1, 5, 37, 176, 985, 1894, 5012, 1105, 719, 1, 1, 1, 6, 41, 319, 1397, 7374, 11258, 25863, 3624, 1842, 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
----+-------------------------------------------------
  0 | 1   1    1     1     1      1      1       1 ...
  1 | 1   1    1     1     1      1      1       1 ...
  2 | 1   2    2     3     3      4      4       5 ...
  3 | 1   4    5    11    13     22     25      37 ...
  4 | 1   9   13    46    62    140    176     319 ...
  5 | 1  20   37   208   333    985   1397    3059 ...
  6 | 1  48  111  1002  1894   7374  11757   31195 ...
  7 | 1 115  345  5012 11258  57577 103376  331991 ...
  8 | 1 286 1105 25863 68990 463670 937179 3643790 ...
  ...

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, A000081(n+1), A003080, A287891.

Cf. A303694, A332649.

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)}
T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n,k,Col(R(n,k)))))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) }

A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.

Original entry on oeis.org

1, 1, 2, 4, 10, 30, 105, 400, 1654, 7229, 32944, 154749, 744973, 3655993, 18232812, 92162974, 471301437, 2434542190, 12687850499, 66646225443, 352548333438, 1876770716627, 10048289587337, 54079948967654, 292447643655469, 1588388448970674, 8661869330014601

Offset: 1

Andrew Howroyd, Feb 18 2020

nonn

			a(3) = 2 because there are two cacti on 5 nodes which are a pentagon and 2 triangles joined at a node.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Cactus graph
Index entries for sequences related to cacti

Cf. A000083, A035085, A091487, A332649, A332651.

\\ Here UCacti gives number of unrooted cacti with restricted polygons.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
RCacti(u)={my(v=[1]); while(#v<#u, my(g=x*Ser(v), g2=subst(g,x,x^2) + O(x^2*x^#v), r=sum(k=1, #u-1, my(c=u[k+1]); if(c, c*(g^k + g^(k%2)*g2^(k\2))))/2 + O(x^#u)); v=concat([1], EulerT(Vec(r, 1-serprec(r, x))))); v}
UCacti(u)={my(p=x*Ser(RCacti(u))); my(g(d)=subst(p + O(x*x^(#u\d)), x, x^d)); Vec(g(1) + sum(k=1, #u, my(c=u[k]); if(c, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - (g(1)^k)/2 + if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)))}
seq(n)={my(v=UCacti(vector(2*n-1, i, i>2 && isprime(i)))); vector(n, i, v[2*i-1])}

A332651 Number of polygonal cacti on n unlabeled nodes with every polygon having an even number of edges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 4, 2, 7, 9, 14, 26, 48, 71, 154, 243, 478, 894, 1631, 3149, 6062, 11295, 22469, 42900, 83528, 164829, 321012, 632960, 1255613, 2472803, 4928140, 9808439, 19533534, 39134059, 78345317, 157177556, 316398963, 636790282, 1284910954

Offset: 0

Andrew Howroyd, Feb 18 2020

nonn

Bridges are disallowed.

			a(6) = 1 corresponding with a hexagon.
a(7) = 1 corresponding with two quadrilaterals joined at a node.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Cactus graph
Index entries for sequences related to cacti

Cf. A000083, A035085, A091487, A332649, A332650.

\\ See A332650 for UCacti.
seq(n)={concat([1], UCacti(vector(n, i, i>2&&i%2==0)))}

cp's OEIS Frontend

A003081 Number of triangular cacti with 2n+1 nodes (n triangles).

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

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A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.

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A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.

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A332651 Number of polygonal cacti on n unlabeled nodes with every polygon having an even number of edges.

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PARI