A054363 Number of unlabeled 5-ary cacti having n polygons.

1, 1, 5, 15, 85, 510, 4051, 33130, 291925, 2661255, 25059670, 241724380, 2379912355, 23833198140, 242173108050, 2491817151160, 25921371278805, 272256630756265, 2884054952424115, 30784716141936525, 330853932861650870, 3577823885433087690, 38907658120970944700

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=5 of A303912.

Cf. A054364, A054365.

Programs

Mathematica

a[n_] := If[n == 0, 1, (Binomial[5*n, n]/(4*n + 1) + DivisorSum[n, Binomial[5*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 17 2017 *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(5*d, d))/n - 4*binomial(5*n, n)/(4*n+1)) \\ Andrew Howroyd, May 02 2018

Formula

a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(5*d, d)) - 4*binomial(5*n, n)/(4*n+1) for n > 0. - Andrew Howroyd, May 02 2018

a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 17 2017

Extensions

More terms from Jean-François Alcover, Jul 17 2017