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

6,2

Comments

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

Links

Table of n, a(n) for n=6..31.

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).

Example

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.

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 *)

Crossrefs

Cf. A001429, A137918.

Sequence in context: A304304 A072842 A303861 * A354176 A007346 A062247

Adjacent sequences: A138384 A138385 A138386 * A138388 A138389 A138390

Keyword

easy,nonn

Author

Washington Bomfim, Mar 18 2008

Status

approved