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%I #20 Jan 09 2024 09:39:43
%S 1,1,0,6,28,193,1140,7688,52364,373560,2732836,20506254,156899748,
%T 1221179922,9642327324,77092881840,623120435820,5085013101160,
%U 41850590485164,347060754685884,2897800074184240,24344668688255109,205667186830447412,1746375819789491992
%N Number of unlabeled asymmetric 4-ary cacti having n polygons.
%H Andrew Howroyd, <a href="/A052395/b052395.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(4*d, d)) - 3*binomial(4*n, n)/(3*n+1) for n > 0. - _Andrew Howroyd_, Apr 30 2018
%t a[0] = 1;
%t a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[4#, #]&]/n - 3 Binomial[4n, n]/(3n + 1);
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jun 29 2018, after _Andrew Howroyd_ *)
%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(4*d, d))/n - 3*binomial(4*n, n)/(3*n+1)) \\ _Andrew Howroyd_, Apr 30 2018
%Y Column k=4 of A303913.
%Y Cf. A052394, A054362.
%K nonn
%O 0,4
%A _Simon Plouffe_
%E Terms a(13) and beyond from _Andrew Howroyd_, May 02 2018
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