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%I #21 Jan 09 2024 09:38:48
%S 1,1,0,1,2,8,18,61,170,538,1654,5344,17252,57146,190786,646305,
%T 2209050,7626164,26532732,93013852,328196780,1165060170,4158266282,
%U 14915635376,53745892932,194477856048,706436256598,2575316698792,9419568272632
%N Number of unlabeled asymmetric 2-ary cacti having n polygons.
%H Andrew Howroyd, <a href="/A054358/b054358.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(2*d, d)) - binomial(2*n, n)/(n+1) for n > 0. - _Andrew Howroyd_, May 02 2018
%t a[0] = 1; a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[2#, #]&]/n - Binomial[2n, n]/(n+1);
%t Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1)) \\ _Andrew Howroyd_, May 02 2018
%Y Column k=2 of A303913.
%Y Cf. A002995, A054357.
%K nonn
%O 0,5
%A _Simon Plouffe_
%E More terms from _Vladeta Jovovic_, Oct 04 2007