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%I #11 Sep 13 2015 20:58:42
%S 1,1,4,8,22,52,142,362,973,2574,6935,18643,50573,137401,375306,
%T 1027898,2825831,7790055,21539352,59706865,165921896,462127857,
%U 1289901083,3607567539,10108555623,28374358327,79777757405,224653284863
%N Number of 6-bead necklaces where each bead is an unlabeled rooted tree, by total number of nodes.
%C The 6 beads are just placeholders; only tree nodes are counted.
%F Plug g.f. for A000081, 1+x+x^2+2*x^3+4*x^4+ ... into cycle index for dihedral group D_12.
%F 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.
%e 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.
%t 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 *)
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 18 2001
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