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A054365
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Number of unlabeled 5-gonal cacti having n polygons.
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5
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1, 1, 1, 3, 17, 102, 811, 6626, 58385, 532251, 5011934, 48344880, 475982471, 4766639628, 48434621610, 498363430232, 5184274255789, 54451326151253, 576810990484823, 6156943228387305, 66170786572330174, 715564777086617766
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OFFSET
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0,4
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COMMENTS
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Also, the number of noncrossing partitions up to rotation composed of n blocks of size 5. - Andrew Howroyd, May 04 2018
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..200
Miklos Bona, Michel Bousquet, Gilbert Labelle and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (pdf, dvi).
Luke Lippstreu, Jorge Mago, Marcus Spradlin, Anastasia Volovich, Weak Separation, Positivity and Extremal Yangian Invariants, arXiv:1906.11034 [hep-th], 2019.
Index entries for sequences related to cacti
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FORMULA
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a(n) = ((Sum_{d|n} phi(n/d)*binomial(5*d, d)) + (Sum_{d|gcd(n-1, 5)} phi(d)*binomial(5*n/d, (n-1)/d)))/(5*n) - binomial(5*n, n)/(4*n+1) for n > 0. - Andrew Howroyd, May 04 2018
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MAPLE
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with(combinat): with(numtheory): m := 5: 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
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MATHEMATICA
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a[0] = 1;
a[n_] := (DivisorSum[n, EulerPhi[n/#] Binomial[5#, #]&] + DivisorSum[GCD[n - 1, 5], EulerPhi[#] Binomial[5n/#, (n-1)/#]&])/(5n) - Binomial[5n, n]/ (4n+1);
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)
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PROG
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(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(5*d, d)) + sumdiv(gcd(n-1, 5), d, eulerphi(d)*binomial(5*n/d, (n-1)/d)))/(5*n) - binomial(5*n, n)/(4*n+1))} \\ Andrew Howroyd, May 04 2018
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CROSSREFS
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Column k=5 of A303694.
Cf. A054363, A054364, A303871.
Sequence in context: A074565 A339565 A241768 * A116886 A163064 A020069
Adjacent sequences: A054362 A054363 A054364 * A054366 A054367 A054368
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KEYWORD
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nonn
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AUTHOR
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Simon Plouffe
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EXTENSIONS
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More terms from Zerinvary Lajos, Dec 01 2006
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STATUS
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approved
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