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%I A054368 #24 Jan 09 2024 09:38:23
%S A054368 1,1,1,4,25,187,1772,17880,191967,2141232,24640989,290610414,
%T A054368 3498042924,42831369777,532148952720,6695274478834,85166167050949,
%U A054368 1093843440166718,14169564589464986,184957445502335682,2430876839834279341,32147041999684759275,427520786795342624432
%N A054368 Number of unlabeled 6-gonal cacti having n polygons.
%C A054368 Also, the number of noncrossing partitions up to rotation composed of n blocks of size 6. - _Andrew Howroyd_, May 04 2018
%H A054368 Andrew Howroyd, <a href="/A054368/b054368.txt">Table of n, a(n) for n = 0..200</a>
%H A054368 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 A054368 <a href="/https/oeis.org/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F A054368 a(n) = ((Sum_{d|n} phi(n/d)*binomial(6*d, d)) + (Sum_{d|gcd(n-1, 6)} phi(d)*binomial(6*n/d, (n-1)/d)))/(6*n) - binomial(6*n, n)/(5*n+1) for n > 0. - _Andrew Howroyd_, May 04 2018
%p A054368 with(combinat): with(numtheory): m := 6: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od: # _Zerinvary Lajos_, Dec 01 2006
%t A054368 a[0] = 1;
%t A054368 a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[6#, #]&] + DivisorSum[GCD[n - 1, 6], EulerPhi[#] Binomial[6n/#, (n-1)/#]&])/(6n) - Binomial[6n, n]/(5 n + 1);
%t A054368 Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A054368 (PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(6*d, d)) + sumdiv(gcd(n-1, 6), d, eulerphi(d)*binomial(6*n/d, (n-1)/d)))/(6*n) - binomial(6*n, n)/(5*n+1))} \\ _Andrew Howroyd_, May 04 2018
%Y A054368 Column k=6 of A303694.
%Y A054368 Cf. A054366, A054367.
%K A054368 nonn
%O A054368 0,4
%A A054368 _Simon Plouffe_
%E A054368 More terms from _Zerinvary Lajos_, Dec 01 2006
%E A054368 Terms a(21) and beyond from _Andrew Howroyd_, May 04 2018

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