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%I #19 Jan 09 2024 09:38:18
%S 1,1,0,15,110,1095,10326,107056,1149126,12845166,147817170,1743640902,
%T 20987930112,256987965373,3192889862238,40171643818920,
%U 510996955110550,6563060603543652,85017386945583408,1109744672540225361,14585261031466256370,192882251992018723947
%N Number of unlabeled asymmetric 6-ary cacti having n polygons.
%H Andrew Howroyd, <a href="/A054367/b054367.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(6*d, d)) - 5*binomial(6*n, n)/(5*n+1) for n > 0. - _Andrew Howroyd_, May 02 2018
%t a[0] = 1;
%t a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[6#, #]&]/n - 5 Binomial[6n, n]/(5n + 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(6*d, d))/n - 5*binomial(6*n, n)/(5*n+1)) \\ _Andrew Howroyd_, May 02 2018
%Y Column k=6 of A303913.
%Y Cf. A054366, A054368.
%K nonn
%O 0,4
%A _Simon Plouffe_
%E Terms a(11) and beyond from _Andrew Howroyd_, May 02 2018
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