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

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

0,9

Comments

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

Links

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

Example

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 ...
  ...

Prog

(PARI)
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, ])) }

Crossrefs

Columns k=1..4 are A000012, A000081(n+1), A003080, A287891.

Cf. A303694, A332649.

Sequence in context: A287216 A145515 A267383 * A272896 A188919 A026519

Adjacent sequences: A332645 A332646 A332647 * A332649 A332650 A332651

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 18 2020

Status

approved