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%I #17 Sep 20 2017 16:22:54
%S 1,1,3,8,24,74,245,815,2796,9707,34186,121562,436298,1577310,5740299,
%T 21008777,77279892,285544700,1059332082,3944254118,14734260864,
%U 55207053787,207421476390,781283558998,2949675307082,11160264942376,42309912978708,160700303600030
%N 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.
%C 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
%H Alois P. Heinz, <a href="/A046342/b046342.txt">Table of n, a(n) for n = 0..1000</a>
%F Plug g.f. for A000108, 1/2*(1-(1-4*x)^(1/2))/x, into cycle index for dihedral group D_6.
%F Cycle index for D_6: 1/6*Z[1]^3+1/2*Z[1]*Z[2]+1/3*Z[3].
%F a(n) = Sum_{j=0..3} A275431(n,j). - _Alois P. Heinz_, Sep 20 2017
%t 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 *)
%Y See A058855 (a 6-bead analog) for details.
%Y Cf. A000108, A058855, A056711, A275431.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 19 2001