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A052393 Number of unlabeled ternary cacti having n triangles. 4
1, 1, 3, 6, 19, 57, 258, 1110, 5475, 27429, 143379, 764970, 4173906, 23140816, 130205922, 741650802, 4270593219, 24825326199, 145535320383, 859627488966, 5112006997539, 30586307211945, 184023410798910, 1112800162657902, 6760426635625170 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for sequences related to cacti

Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (pdf, dvi).

FORMULA

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(3*d, d)) - 2*binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 02 2018

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * 2^(2*n + 2) * n^(5/2)). - Vaclav Kotesovec, Jul 17 2017

MATHEMATICA

a[n_] := If[n == 0, 1, (Binomial[3*n, n]/(2*n+1) + DivisorSum[n, Binomial[ 3*#, #]*EulerPhi[n/#]*Boole[# < n]&])/n]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 17 2017 *)

PROG

(PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(3*d, d))/n - 2*binomial(3*n, n)/(2*n+1)) \\ Andrew Howroyd, May 02 2018

CROSSREFS

Column k=3 of A303912.

Cf. A054422, A054423.

Sequence in context: A248603 A332344 A345244 * A024607 A186022 A058818

Adjacent sequences:  A052390 A052391 A052392 * A052394 A052395 A052396

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Jean-François Alcover, Jul 17 2017

STATUS

approved

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Last modified October 24 17:43 EDT 2021. Contains 348233 sequences. (Running on oeis4.)