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A058915 Number of graphs with 3 distinct components. 2

%I

%S 2,7,34,181,1266,14106,293756,12362198,1032671168,166176421788,

%T 50672459139597,29105501987344357,31455795559882541775,

%U 64032588337815572241795,246000022800939308314311897

%N Number of graphs with 3 distinct components.

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 48, (2.6.3).

%F G.f.: 1/6*(f(x)^3 - 3*f(x)*f(x^2) + 2*f(x^3)), where f(x) = g(x) - 1 and g(x) is g.f. for connected graphs. Cf. A001349.

%t Needs["Combinatorica`"];max=25;A000088=Table[NumberOfGraphs[n],{n,0,max}];f[x_]=1-Product[1/(1-x^k)^a[k],{k,1,max}];a[0]=a[1]=a[2]=1;coes=CoefficientList[Series[f[x],{x,0,max}],x];sol=First[Solve[Thread[Rest[coes+A000088]== 0]]];cg=Table[a[n],{n,1,max}]/.sol;Take[CoefficientList[CycleIndex[AlternatingGroup[3],s]-CycleIndex[SymmetricGroup[3],s]/.Table[s[j]->Table[Sum[cg[[i]] x^(k*i),{i,1,max}],{k,1,max}][[j]],{j,1,3}],x],{7,max}] (* _Geoffrey Critzer_, Oct 15 2012; after code by _Jean-Fran├žois Alcover_ in A001349 *)

%Y Cf. A001349. Column 3 of A217955.

%K easy,nonn

%O 6,1

%A _Vladeta Jovovic_, Jan 11 2001

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Last modified January 18 14:07 EST 2021. Contains 340254 sequences. (Running on oeis4.)