A054366 Number of unlabeled 6-ary cacti having n polygons.

1, 1, 6, 21, 146, 1101, 10632, 107062, 1151802, 12845442, 147845706, 1743640908, 20988257544, 256987965379, 3192893716320, 40171643847696, 510997002280522, 6563060603543658, 85017387536789916, 1109744672540225367, 14585261039005676046

Offset

0,3

Links

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

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

Index entries for sequences related to cacti

Formula

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(6*d, d)) - 5*binomial(6*n, n)/(5*n+1) for n > 0. - Andrew Howroyd, May 02 2018

a(n) ~ sqrt(3) * 6^(6*n) / (sqrt(Pi) * n^(5/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 17 2017

Mathematica

a[n_] := If[n == 0, 1, (Binomial[6*n, n]/(5 n + 1) + DivisorSum[n, Binomial[6*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 17 2017 *)

Prog

(PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(6*d, d))/n - 5*binomial(6*n, n)/(5*n+1)) \\ Andrew Howroyd, May 02 2018

Crossrefs

Column k=6 of A303912.

Cf. A054367, A054368.

Sequence in context: A093775 A318103 A058821 * A304264 A210443 A179768

Adjacent sequences: A054363 A054364 A054365 * A054367 A054368 A054369

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from Jean-François Alcover, Jul 17 2017

Status

approved