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

0,3

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

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

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 *)

Crossrefs

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

Cf. A000108, A058855, A056711, A275431.

Sequence in context: A275856 A303607 A281872 * A238977 A182453 A047087

Adjacent sequences: A046339 A046340 A046341 * A046343 A046344 A046345

Keyword

nonn

Author

N. J. A. Sloane, Jan 19 2001

Status

approved