A138387 Numbers of unlabeled graphs with n vertices and 2 unicyclic components.

1, 2, 8, 23, 74, 220, 674, 2011, 6038, 17980, 53547, 158907, 471225, 1394786, 4124929, 12185636, 35972082, 106111713, 312835608, 921809509, 2715058701, 7993741597, 23527694230, 69228383367, 203648980297, 598945442071

Offset: 6

Author

Washington Bomfim, Mar 18 2008

Keywords

• easy
• nonn

Comments

This sequence is the second row of table T of A137918.

Examples

			a(13) = 2,011, since n is odd and the partitions are 3+10, 4+9, 5+8 and 6+7. This gives 657 + 480 + 445 + 429 graphs.
Note that f(4)= 2, f(5) = 5, f(6) = 13, f(7) = 33, f(8) = 89, f(9) = 240 and f(10) = 657.
		

Crossrefs

Cf. A001429, A137918.

Programs

Mathematica

nmax = 31;
TreeGf[nn_] := Module[{A}, A = Table[1, {nn}]; For[n = 1, n <= nn - 1, n++, A[[n + 1]] = 1/n * Sum[Sum[ d*A[[d]], {d, Divisors[k]}]*A[[n - k + 1]], {k, 1, n}]]; x A.x^Range[0, nn - 1]];
seq[n_] := Module[{t, g}, If[n < 3, {}, t = TreeGf[n - 2]; g[e_] := Normal[t + O[x]^(Quotient[n, e] + 1)] /. x -> x^e  + O[x]^(n + 1); Sum[Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2], {k, 3, n}]]/2 // Drop[CoefficientList[#, x], 3]&];
A001429 = seq[nmax];
f[k_] := A001429[[k - 2]];
a[n_] := If[OddQ[n], Sum[f[i] * f[n - i], {i, 3, (n - 1)/2}], Sum[f[i] * f[n - i], {i, 3, n/2 - 1 }] + (f[n/2] + 1)*f[n/2]/2];
a /@ Range[6, nmax] (* Jean-François Alcover, Oct 05 2019, using Andrew Howroyd's code for A001429 *)

Formula

For n odd, a(n) = Sum(3 <= i <= (n-1)/2){f(i) * f(n-i)}; for n even, a(n) = Sum(3 <= i <= n/2 - 1){f(i) * f(n-i)} + (f(n/2)+1)*f(n/2)/2, where f(k) is A001429(k).