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A068051 Number of n-node connected graphs with one cycle, possibly of length 1 or 2.

%I A068051 #20 Feb 10 2025 05:47:04
%S A068051 1,2,4,9,20,49,118,300,765,1998,5255,14027,37670,102095,278262,763022,
%T A068051 2101905,5816142,16153148,45017423,125836711,352723949,991143727,
%U A068051 2791422887,7877935985,22275473767,63096075118,179012076933
%N A068051 Number of n-node connected graphs with one cycle, possibly of length 1 or 2.
%H A068051 Andrew Howroyd, <a href="/A068051/b068051.txt">Table of n, a(n) for n = 1..200</a>
%H A068051 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A068051 "DIK" transform of A000081.
%F A068051 a(n) = A000081(n) + A027852(n) + A000226(n) + A000368(n) + ...  [_Geoffrey Critzer_, Mar 24 2013]
%t A068051 nn=20;t[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0==Series[t[x]-x Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}],x];b=Table[a[n],{n,1,nn}]/.sol//Flatten;Map[Total,Drop[Transpose[Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[b[[i]]x^(i*k),{i,1,nn}],{k,1,nn}][[j]],{j,1,n}],x],nn],{n,1,nn}]],1]]  (* _Geoffrey Critzer_, Mar 24 2013 *)
%o A068051 (PARI) \\ TreeGf gives gf of A000081
%o A068051 TreeGf(N)={my(A=vector(N, j, 1)); for(n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o A068051 seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec((sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2)/2)} \\ _Andrew Howroyd_, Jun 20 2018
%Y A068051 Cf. A217781.
%K A068051 nonn
%O A068051 1,2
%A A068051 _Christian G. Bower_, Feb 12 2002