|
%I #22 Jan 09 2024 09:39:48
%S 1,1,5,15,85,510,4051,33130,291925,2661255,25059670,241724380,
%T 2379912355,23833198140,242173108050,2491817151160,25921371278805,
%U 272256630756265,2884054952424115,30784716141936525,330853932861650870,3577823885433087690,38907658120970944700
%N Number of unlabeled 5-ary cacti having n polygons.
%H Andrew Howroyd, <a href="/A054363/b054363.txt">Table of n, a(n) for n = 0..200</a>
%H Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, <a href="https://doi.org/10.1006/aama.1999.0665">Enumeration of m-ary cacti</a>, Advances in Applied Mathematics, 24 (2000), 22-56.
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F 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
%F a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 7/2)). - _Vaclav Kotesovec_, Jul 17 2017
%t 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 *)
%o (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
%Y Column k=5 of A303912.
%Y Cf. A054364, A054365.
%K nonn
%O 0,3
%A _Simon Plouffe_
%E More terms from _Jean-François Alcover_, Jul 17 2017
|
