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A054371 Number of unlabeled 7-gonal cacti having n polygons.

%I A054371 #24 Jan 09 2024 09:38:36
%S A054371 1,1,1,4,33,300,3412,40770,518043,6830545,92909684,1295151600,
%T A054371 18426823044,266696759064,3916798516462,58253090490630,
%U A054371 875948658280305,13299481192954961,203661940884670135,3142707632566279222,48829032430870168660,763383551090733489744
%N A054371 Number of unlabeled 7-gonal cacti having n polygons.
%C A054371 Also, the number of noncrossing partitions up to rotation composed of n blocks of size 7.
%H A054371 Andrew Howroyd, <a href="/A054371/b054371.txt">Table of n, a(n) for n = 0..200</a>
%H A054371 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 A054371 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F A054371 a(n) = ((Sum_{d|n} phi(n/d)*binomial(7*d, d)) + (Sum_{d|gcd(n-1, 7)} phi(d)*binomial(7*n/d, (n-1)/d)))/(7*n) - binomial(7*n, n)/(6*n+1) for n > 0. - _Andrew Howroyd_, May 04 2018
%p A054371 with(combinat): with(numtheory): m := 7: for p from 2 to 27 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 A054371 a[0] = 1;
%t A054371 a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[7#, #]&] + DivisorSum[GCD[n - 1, 7], EulerPhi[#] Binomial[7n/#, (n-1)/#]&])/(7n) - Binomial[7n, n]/(6 n + 1);
%t A054371 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%o A054371 (PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(7*d, d)) + sumdiv(gcd(n-1, 7), d, eulerphi(d)*binomial(7*n/d, (n-1)/d)))/(7*n) - binomial(7*n, n)/(6*n+1))} \\ _Andrew Howroyd_, May 04 2018
%Y A054371 Column k=7 of A303694.
%Y A054371 Cf. A054369, A054370.
%K A054371 nonn
%O A054371 0,4
%A A054371 _Simon Plouffe_
%E A054371 More terms from _Zerinvary Lajos_, Dec 01 2006
%E A054371 Terms a(20) and beyond from _Andrew Howroyd_, May 04 2018