A052395 Number of unlabeled asymmetric 4-ary cacti having n polygons.

1, 1, 0, 6, 28, 193, 1140, 7688, 52364, 373560, 2732836, 20506254, 156899748, 1221179922, 9642327324, 77092881840, 623120435820, 5085013101160, 41850590485164, 347060754685884, 2897800074184240, 24344668688255109, 205667186830447412, 1746375819789491992

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

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

Crossrefs

Column k=4 of A303913.

Cf. A052394, A054362.

Programs

Mathematica

a[0] = 1;
a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[4#, #]&]/n - 3 Binomial[4n, n]/(3n + 1);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(4*d, d))/n - 3*binomial(4*n, n)/(3*n+1)) \\ Andrew Howroyd, Apr 30 2018

Formula

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(4*d, d)) - 3*binomial(4*n, n)/(3*n+1) for n > 0. - Andrew Howroyd, Apr 30 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, May 02 2018