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%I #9 Jul 22 2018 10:39:24
%S 1,3,9,21,60,135,282,537,945,1561,2451,3693,5378,7611,10512,14217,
%T 18879,24669,31777,40413,50808,63215,77910,95193,115389,138849,165951,
%U 197101,232734,273315,319340,371337,429867,495525,568941,650781,741748
%N Number of homeomorphically irreducible general graphs on 3 labeled node and with n edges.
%C A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
%H V. Jovovic, <a href="/A060576/a060576.pdf">Generating functions for homeomorphically irreducible general graphs on n labeled nodes</a>
%H V. Jovovic, <a href="/A060576/a060576_rec.pdf">Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges</a>
%F G.f.: - (8*x^9 - 36*x^8 + 66*x^7 - 70*x^6 + 51*x^5 - 24*x^4 + 8*x^3 - 6*x^2 + 3*x - 1)/(x - 1)^6. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
%t CoefficientList[Series[-(8x^9-36x^8+66x^7-70x^6+51x^5-24x^4+8x^3-6x^2+3x-1)/(x-1)^6,{x,0,40}],x] (* _Harvey P. Dale_, Jul 22 2018 *)
%Y Cf. A003514, A060516, A060533-A060537, A060576-A060581.
%K easy,nonn
%O 0,2
%A _Vladeta Jovovic_, Apr 03 2001
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