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A052394
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Number of unlabeled 4-ary cacti having n polygons.
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4
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1, 1, 4, 10, 44, 197, 1228, 7692, 52828, 373636, 2735952, 20506258, 156922676, 1221179926, 9642496488, 77092885016, 623121750844, 5085013101164, 41850600967984, 347060754685888, 2897800158952304, 24344668688424333, 205667187527660076, 1746375819789491996, 14898241072028602276
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OFFSET
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0,3
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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) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(4*d, d)) - 3*binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, May 02 2018
a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 17 2017
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MATHEMATICA
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a[n_] := If[n == 0, 1, (Binomial[4*n, n]/(3*n + 1) + DivisorSum[n, Binomial[4*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jul 17 2017 *)
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PROG
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(PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(4*d, d))/n - 3*binomial(4*n, n)/(3*n+1)) \\ Andrew Howroyd, May 02 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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