A054422 Number of unlabeled asymmetric ternary cacti having n triangles.

1, 1, 0, 3, 10, 54, 222, 1107, 5346, 27399, 142770, 764967, 4170672, 23140813, 130189302, 741650172, 4270501218, 24825326196, 145534796520, 859627488963, 5112003992610, 30586307195304, 184023393204654, 1112800162657899

Offset: 0

Author

Simon Plouffe, Mar 15 2000

Keywords

• nonn
• easy

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=3 of A303913.

Cf. A052393, A054423.

Programs

Mathematica

a[0] = 1; a[n_] := (1/n) Sum[MoebiusMu[n/d] Binomial[3d, d], {d, Divisors[n] } ] - 2 Binomial[3n, n]/(2n + 1);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 24 2018, after Andrew Howroyd *)

PARI

a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n - 2*binomial(3*n, n)/(2*n+1)) \\ Andrew Howroyd, May 02 2018

Formula

a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(3*d, d)) - 2*binomial(3*n, n)/(2*n+1) for n > 0. - Andrew Howroyd, May 02 2018

Extensions

More terms from James Sellers, Mar 16 2000