%I #18 Jan 09 2024 09:39:52

%S 1,1,0,10,60,505,3876,33125,290700,2661100,25049020,241724375,

%T 2379812100,23833198135,242172147380,2491817140380,25921361665100,

%U 272256630756260,2884054853862540,30784716141936520,330853931834416520,3577823885432126890,38907658110093347780

%N Number of unlabeled asymmetric 5-ary cacti having n polygons.

%H Andrew Howroyd, <a href="/A054364/b054364.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} mu(n/d)*binomial(5*d, d)) - 4*binomial(5*n, n)/(4*n+1) for n > 0. - _Andrew Howroyd_, May 02 2018

%t a[0] = 1;

%t a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[5#, #]&]/n - 4 Binomial[5n, n]/(4n+1);

%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)

%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(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 A303913.

%Y Cf. A054363, A054365.

%K nonn

%O 0,4

%A _Simon Plouffe_

%E Terms a(13) and beyond from _Andrew Howroyd_, May 02 2018