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

1, 2, 4, 9, 20, 49, 118, 300, 765, 1998, 5255, 14027, 37670, 102095, 278262, 763022, 2101905, 5816142, 16153148, 45017423, 125836711, 352723949, 991143727, 2791422887, 7877935985, 22275473767, 63096075118, 179012076933

Offset: 1

Author

Christian G. Bower, Feb 12 2002

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A217781.

Programs

Mathematica

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 *)

PARI

\\ TreeGf gives gf of A000081
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)}
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

Formula

"DIK" transform of A000081.

a(n) = A000081(n) + A027852(n) + A000226(n) + A000368(n) + ... [Geoffrey Critzer, Mar 24 2013]