A046342 Number of 3-bead necklaces where each bead is a planted trivalent plane tree [or anything else enumerated by the Catalan numbers], by total number of nodes.

1, 1, 3, 8, 24, 74, 245, 815, 2796, 9707, 34186, 121562, 436298, 1577310, 5740299, 21008777, 77279892, 285544700, 1059332082, 3944254118, 14734260864, 55207053787, 207421476390, 781283558998, 2949675307082, 11160264942376, 42309912978708, 160700303600030

Offset: 0

Author

N. J. A. Sloane, Jan 19 2001

Keywords

• nonn

Comments

With offset = 3, a(n) is the number of forests having exactly three rooted plane trees with n total nodes. - Geoffrey Critzer, Feb 22 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

See A058855 (a 6-bead analog) for details.

Cf. A000108, A058855, A056711, A275431.

Programs

Mathematica

nn=30;Drop[CoefficientList[Series[ CycleIndex[SymmetricGroup[3],s]/.Table[s[i]->(1-(1-4x^i)^(1/2))/2,{i,1,nn}],{x,0,nn}],x],3] (* Geoffrey Critzer, Feb 22 2013 *)

Formula

Plug g.f. for A000108, 1/2*(1-(1-4*x)^(1/2))/x, into cycle index for dihedral group D_6.

Cycle index for D_6: 1/6*Z[1]^3+1/2*Z[1]*Z[2]+1/3*Z[3].

a(n) = Sum_{j=0..3} A275431(n,j). - Alois P. Heinz, Sep 20 2017