A035086 -id:A035086 - cp's OEIS frontend

A035082 Number of rooted polygonal cacti (Husimi graphs) with n nodes.

0, 1, 0, 1, 1, 3, 5, 13, 27, 67, 157, 390, 963, 2437, 6186, 15908, 41127, 107148, 280569, 738675, 1953054, 5185364, 13816018, 36934431, 99030038, 266254593, 717652816, 1938831589, 5249221790, 14240130827, 38702218134, 105367669062

Offset: 0

Christian G. Bower, Nov 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71

Andrew Howroyd, Table of n, a(n) for n = 0..500
C. G. Bower, Transforms (2)
F. Harary and R. Z. Norman, The Dissimilarity Characteristic of Husimi Trees, Annals of Mathematics, 58 1953, pp. 134-141.
F. Harary and G. E. Uhlenbeck, On the Number of Husimi Trees, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti
Index entries for sequences related to rooted trees

Cf. A003080, A035083, A035084, A035085, A035086, A035087, A035088.

BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(p))} \\ Andrew Howroyd, Aug 30 2018

Shifts left under transform T where Ta = EULER(BIK(a)-a).

A035088 Number of labeled polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 3, 27, 240, 2985, 42840, 731745, 14243040, 313570845, 7683984000, 207685374435, 6135743053440, 196754537704725, 6805907485977600, 252620143716765825, 10015402456976716800, 422410127508300756825, 18884777200534941696000

Offset: 0

Christian G. Bower, Nov 15 1998

A Husimi tree is a connected graph in which no line lies on more than one cycle [Harary, 1953]. - Jonathan Vos Post, Mar 12 2010

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39. pp. 315-322, 1953.
F. Harary, G. Uhlenbeck (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences 39: 315-322. - From Jonathan Vos Post, Mar 12 2010

Cf. A035082, A035083, A035084, A035085, A035086, A035087.

max = 20; s = 1+InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, max}], x]; a[n_] := SeriesCoefficient[s, n]*(n-1)!; a[0]=1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 27 2016, after Vaclav Kotesovec at A035087 *)

a(n) = A035087(n)/n, n > 0.

cp's OEIS Frontend

A035082 Number of rooted polygonal cacti (Husimi graphs) with n nodes.

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