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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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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.