A058855 Number of 6-bead necklaces where each bead is an unlabeled rooted tree, by total number of nodes.

1, 1, 4, 8, 22, 52, 142, 362, 973, 2574, 6935, 18643, 50573, 137401, 375306, 1027898, 2825831, 7790055, 21539352, 59706865, 165921896, 462127857, 1289901083, 3607567539, 10108555623, 28374358327, 79777757405, 224653284863

Offset

0,3

Comments

The 6 beads are just placeholders; only tree nodes are counted.

Links

Table of n, a(n) for n=0..27.

Formula

Plug g.f. for A000081, 1+x+x^2+2*x^3+4*x^4+ ... into cycle index for dihedral group D_12.

Cycle index for D_12 is 1/12*Z[1]^6+1/6*Z[6]+1/4*Z[1]^2*Z[2]^2+1/6*Z[3]^2+1/3*Z[2]^3.

Example

a(3) = 8 since the 3 nodes may be arranged around the necklace as 111000, 110100, 101010, 210000, 201000, 200100, 300000 and in the latter arrangement there are two possible trees that can be used because A000081(3)=2.

Mathematica

nn=20; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x]-x Product[1/(1-x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; t=Prepend[Table[a[n], {n, 1, nn}]/.sol//Flatten, 1]; Drop[CoefficientList[Series[CycleIndex[DihedralGroup[6], s]/.Table[s[i]->Sum[t[[k]]x^((k-1) i), {k, 1, nn-1}], {i, 1, 6}], {x, 0, nn}], x], -2]  (* Geoffrey Critzer, Feb 22 2013 *)

Crossrefs

Sequence in context: A000639 A190795 A052528 * A297339 A290138 A266922

Adjacent sequences: A058852 A058853 A058854 * A058856 A058857 A058858

Keyword

nonn

Author

N. J. A. Sloane, Jan 18 2001

Status

approved