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

%I A138387 #6 Oct 05 2019 09:14:35
%S A138387 1,2,8,23,74,220,674,2011,6038,17980,53547,158907,471225,1394786,
%T A138387 4124929,12185636,35972082,106111713,312835608,921809509,2715058701,
%U A138387 7993741597,23527694230,69228383367,203648980297,598945442071
%N A138387 Numbers of unlabeled graphs with n vertices and 2 unicyclic components.
%C A138387 This sequence is the second row of table T of A137918.
%F A138387 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).
%e A138387 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.
%e A138387 Note that f(4)= 2, f(5) = 5, f(6) = 13, f(7) = 33, f(8) = 89, f(9) = 240 and f(10) = 657.
%t A138387 nmax = 31;
%t A138387 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]];
%t A138387 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]&];
%t A138387 A001429 = seq[nmax];
%t A138387 f[k_] := A001429[[k - 2]];
%t A138387 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];
%t A138387 a /@ Range[6, nmax] (* _Jean-François Alcover_, Oct 05 2019, using _Andrew Howroyd_'s code for A001429 *)
%Y A138387 Cf. A001429, A137918.
%K A138387 easy,nonn
%O A138387 6,2
%A A138387 _Washington Bomfim_, Mar 18 2008