A052394 Number of unlabeled 4-ary cacti having n polygons.

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

Offset

0,3

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.

Index entries for sequences related to cacti

Formula

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

Mathematica

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 *)

Prog

(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

Crossrefs

Column k=4 of A303912.

Cf. A054362, A052395.

Sequence in context: A149225 A122648 A058819 * A197627 A066229 A149226

Adjacent sequences: A052391 A052392 A052393 * A052395 A052396 A052397

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from Jean-François Alcover, Jul 17 2017

Status

approved