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A054371 Number of unlabeled 7-gonal cacti having n polygons. 4

%I

%S 1,1,1,4,33,300,3412,40770,518043,6830545,92909684,1295151600,

%T 18426823044,266696759064,3916798516462,58253090490630,

%U 875948658280305,13299481192954961,203661940884670135,3142707632566279222,48829032430870168660,763383551090733489744

%N Number of unlabeled 7-gonal cacti having n polygons.

%C Also, the number of noncrossing partitions up to rotation composed of n blocks of size 7.

%H Andrew Howroyd, <a href="/A054371/b054371.txt">Table of n, a(n) for n = 0..200</a>

%H Miklos Bona, Michel Bousquet, Gilbert Labelle and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (<a href="http://www.lacim.uqam.ca/~leroux/articles/cactus.pdf">pdf</a>, <a href="http://www.math.ufl.edu/~bona/cactusJCTA.dvi">dvi</a>).

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>

%F 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 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 a[0] = 1;

%t 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 Table[a[n], {n, 0, 21}] (* _Jean-Fran├žois Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)

%o (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 Column k=7 of A303694.

%Y Cf. A054369, A054370.

%K nonn

%O 0,4

%A _Simon Plouffe_

%E More terms from _Zerinvary Lajos_, Dec 01 2006

%E Terms a(20) and beyond from _Andrew Howroyd_, May 04 2018

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Last modified April 7 10:46 EDT 2020. Contains 333301 sequences. (Running on oeis4.)