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 A054368 Number of unlabeled 6-gonal cacti having n polygons. 4
 1, 1, 1, 4, 25, 187, 1772, 17880, 191967, 2141232, 24640989, 290610414, 3498042924, 42831369777, 532148952720, 6695274478834, 85166167050949, 1093843440166718, 14169564589464986, 184957445502335682, 2430876839834279341, 32147041999684759275, 427520786795342624432 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also, the number of noncrossing partitions up to rotation composed of n blocks of size 6. - Andrew Howroyd, May 04 2018 LINKS 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). FORMULA 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 MAPLE 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 MATHEMATICA a[0] = 1; 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); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *) PROG (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 CROSSREFS Column k=6 of A303694. Cf. A054366, A054367. Sequence in context: A051820 A246539 A166697 * A135147 A221382 A215791 Adjacent sequences:  A054365 A054366 A054367 * A054369 A054370 A054371 KEYWORD nonn AUTHOR EXTENSIONS More terms from Zerinvary Lajos, Dec 01 2006 Terms a(21) and beyond from Andrew Howroyd, May 04 2018 STATUS approved

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Last modified March 30 09:22 EDT 2020. Contains 333125 sequences. (Running on oeis4.)