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

1,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

C. G. Bower, Transforms (2)

Formula

"DIK" transform of A000081.

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 *)

Prog

(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

Crossrefs

Cf. A217781.

Sequence in context: A286983 A289971 A093637 * A032289 A006648 A128496

Adjacent sequences: A068048 A068049 A068050 * A068052 A068053 A068054

Keyword

nonn

Author

Christian G. Bower, Feb 12 2002

Status

approved