OFFSET
0,3
COMMENTS
The 6 beads are just placeholders; only tree nodes are counted.
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 18 2001
STATUS
approved