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A054371
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Number of unlabeled 7-gonal cacti having n polygons.
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4
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1, 1, 1, 4, 33, 300, 3412, 40770, 518043, 6830545, 92909684, 1295151600, 18426823044, 266696759064, 3916798516462, 58253090490630, 875948658280305, 13299481192954961, 203661940884670135, 3142707632566279222, 48829032430870168660, 763383551090733489744
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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 7.
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LINKS
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Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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a[0] = 1;
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);
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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