A054369 Number of unlabeled 7-ary cacti having n polygons.

1, 1, 7, 28, 231, 2100, 23884, 285390, 3626295, 47813815, 650367788, 9066061200, 128987761308, 1866877313448, 27417589615234, 407771633434368, 6131640607962135, 93096368350684727, 1425633586192690945, 21998953427963954554, 341803227016091180620

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

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

Crossrefs

Column k=7 of A303912.

Cf. A054370, A054371.

Programs

Mathematica

a[n_] := If[n == 0, 1, (Binomial[7*n, n]/(6 n + 1) + DivisorSum[n, Binomial[7*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 17 2017 *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(7*d, d))/n - 6*binomial(7*n, n)/(6*n+1)) \\ Andrew Howroyd, May 02 2018

Formula

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(7*d, d)) - 6*binomial(7*n, n)/(6*n+1) for n > 0. - Andrew Howroyd, May 02 2018

a(n) ~ 7^(7*n + 1/2) / (2 * sqrt(3*Pi) * n^(5/2) * 6^(6*n + 1)). - Vaclav Kotesovec, Jul 17 2017

Extensions

More terms from Jean-François Alcover, Jul 17 2017