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 A058855 Number of 6-bead necklaces where each bead is an unlabeled rooted tree, by total number of nodes. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The 6 beads are just placeholders; only tree nodes are counted. LINKS 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^6+1/6*Z+1/4*Z^2*Z^2+1/6*Z^2+1/3*Z^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, 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

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Last modified February 18 03:33 EST 2020. Contains 332006 sequences. (Running on oeis4.)