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 A054371 Number of unlabeled 7-gonal cacti having n polygons. 4
 1, 1, 1, 4, 33, 300, 3412, 40770, 518043, 6830545, 92909684, 1295151600, 18426823044, 266696759064, 3916798516462, 58253090490630, 875948658280305, 13299481192954961, 203661940884670135, 3142707632566279222, 48829032430870168660, 763383551090733489744 (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 7. 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(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 MAPLE 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 MATHEMATICA 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); Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *) PROG (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 CROSSREFS Column k=7 of A303694. Cf. A054369, A054370. Sequence in context: A264830 A237872 A123780 * A028576 A093185 A198006 Adjacent sequences: A054368 A054369 A054370 * A054372 A054373 A054374 KEYWORD nonn AUTHOR EXTENSIONS More terms from Zerinvary Lajos, Dec 01 2006 Terms a(20) and beyond from Andrew Howroyd, May 04 2018 STATUS approved

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Last modified January 29 21:02 EST 2023. Contains 359931 sequences. (Running on oeis4.)