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

1, 0, 0, 4, 6, 8, 23, 42, 83, 166, 324, 622, 1236, 2366, 4595, 8900, 17225, 33212, 64376, 124360, 240819, 466284, 904149, 1753782, 3407225, 6623274, 12892131, 25116456, 48987833, 95633480, 186891367, 365549578, 715661254, 1402246154, 2749778317, 5396266284

Offset: 0

Author

Geoffrey Critzer, Sep 27 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A217194, A217093, A005380.

Programs

Maple

with (numtheory):
b:= n-> `if`(n<3, 0, add(phi(d)*2^(n/d)/(2*n), d=divisors(n))+
    `if`(irem(n, 2)=1, 2^((n-1)/2), 2^(n/2-1)+2^(n/2-2))):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2012

Mathematica

Needs["Combinatorica`"]
a=Expand[Table[nn=n;CycleIndex[DihedralGroup[nn],s]/.Table[s[i]->2,{i,1,nn}],{n,1,30}]];
nn=30;p=Product[1/(1- x^i)^a[[i]],{i,3,nn}];CoefficientList[Series[p,{x,0,nn}],x]
(* Second program: *)
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)]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 05 2017, after Alois P. Heinz *)

Formula

EULER transform of 0,0,4,6,8,13,30,... A000029.