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A217201 Number of simple unlabeled graphs with n nodes of 2 colors whose components are cycles.

%I A217201 #13 Nov 05 2017 11:51:13
%S A217201 1,0,0,4,6,8,23,42,83,166,324,622,1236,2366,4595,8900,17225,33212,
%T A217201 64376,124360,240819,466284,904149,1753782,3407225,6623274,12892131,
%U A217201 25116456,48987833,95633480,186891367,365549578,715661254,1402246154,2749778317,5396266284
%N A217201 Number of simple unlabeled graphs with n nodes of 2 colors whose components are cycles.
%H A217201 Alois P. Heinz, <a href="/A217201/b217201.txt">Table of n, a(n) for n = 0..1000</a>
%F A217201 EULER transform of 0,0,4,6,8,13,30,... A000029.
%p A217201 with (numtheory):
%p A217201 b:= n-> `if`(n<3, 0, add(phi(d)*2^(n/d)/(2*n), d=divisors(n))+
%p A217201     `if`(irem(n, 2)=1, 2^((n-1)/2), 2^(n/2-1)+2^(n/2-2))):
%p A217201 a:= proc(n) option remember; local d, j; `if`(n=0, 1,
%p A217201       add(add(d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
%p A217201     end:
%p A217201 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 27 2012
%t A217201 Needs["Combinatorica`"]
%t A217201 a=Expand[Table[nn=n;CycleIndex[DihedralGroup[nn],s]/.Table[s[i]->2,{i,1,nn}],{n,1,30}]];
%t A217201 nn=30;p=Product[1/(1- x^i)^a[[i]],{i,3,nn}];CoefficientList[Series[p,{x,0,nn}],x]
%t A217201 (* Second program: *)
%t A217201 b[n_] := If[n < 3, 0, Sum[EulerPhi[d]*2^(n/d)/(2*n), {d, Divisors[n]}] +  If[Mod[n, 2] == 1, 2^((n - 1)/2), 2^(n/2 - 1) + 2^(n/2 - 2)]];
%t A217201 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
%t A217201 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 05 2017, after _Alois P. Heinz_ *)
%Y A217201 Cf. A217194, A217093, A005380.
%K A217201 nonn
%O A217201 0,4
%A A217201 _Geoffrey Critzer_, Sep 27 2012