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A052393
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Number of unlabeled ternary cacti having n triangles.
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4
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1, 1, 3, 6, 19, 57, 258, 1110, 5475, 27429, 143379, 764970, 4173906, 23140816, 130205922, 741650802, 4270593219, 24825326199, 145535320383, 859627488966, 5112006997539, 30586307211945, 184023410798910, 1112800162657902, 6760426635625170
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OFFSET
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0,3
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..200
Index entries for sequences related to cacti
Miklos Bona, Michel Bousquet, Gilbert Labelle, Pierre Leroux, Enumeration of m-ary cacti, Advances in Applied Mathematics, 24 (2000), 22-56 (pdf, dvi).
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FORMULA
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a(n) = (1/n)*(Sum_{d|n} phi(n/d)*binomial(3*d, d)) - 2*binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 02 2018
a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * 2^(2*n + 2) * n^(5/2)). - Vaclav Kotesovec, Jul 17 2017
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MATHEMATICA
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a[n_] := If[n == 0, 1, (Binomial[3*n, n]/(2*n+1) + DivisorSum[n, Binomial[ 3*#, #]*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(3*d, d))/n - 2*binomial(3*n, n)/(2*n+1)) \\ Andrew Howroyd, May 02 2018
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CROSSREFS
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Column k=3 of A303912.
Cf. A054422, A054423.
Sequence in context: A320174 A248603 A332344 * A024607 A186022 A058818
Adjacent sequences: A052390 A052391 A052392 * A052394 A052395 A052396
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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 Jean-François Alcover, Jul 17 2017
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STATUS
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approved
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