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%I #19 Jan 09 2024 09:38:56
%S 1,1,0,3,10,54,222,1107,5346,27399,142770,764967,4170672,23140813,
%T 130189302,741650172,4270501218,24825326196,145534796520,859627488963,
%U 5112003992610,30586307195304,184023393204654,1112800162657899
%N Number of unlabeled asymmetric ternary cacti having n triangles.
%H Andrew Howroyd, <a href="/A054422/b054422.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(3*d, d)) - 2*binomial(3*n, n)/(2*n+1) for n > 0. - _Andrew Howroyd_, May 02 2018
%t a[0] = 1; a[n_] := (1/n) Sum[MoebiusMu[n/d] Binomial[3d, d], {d, Divisors[n] } ] - 2 Binomial[3n, n]/(2n + 1);
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jul 24 2018, after _Andrew Howroyd_ *)
%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n - 2*binomial(3*n, n)/(2*n+1)) \\ _Andrew Howroyd_, May 02 2018
%Y Column k=3 of A303913.
%Y Cf. A052393, A054423.
%K nonn,easy
%O 0,4
%A _Simon Plouffe_, Mar 15 2000
%E More terms from _James A. Sellers_, Mar 16 2000
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