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%I #7 May 10 2013 12:44:44
%S 1,10,40,185,765,2845,10220,33885,105185,305465,830811,2119875,
%T 5091525,11565505,24977315,51552005,102175360,195301015,361365695,
%U 649360880,1136438375,1941722170,3245874555,5318438260,8555568895,13531506921
%N Number of homeomorphically irreducible general graphs on 5 labeled nodes 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.: - (5*x^22 - 20*x^21 + 23*x^20 - 815*x^19 + 8110*x^18 - 37255*x^17 + 104890*x^16 - 204469*x^15 + 296720*x^14 - 337455*x^13 + 310150*x^12 - 229885*x^11 + 131054*x^10 - 50485*x^9 + 6490*x^8 + 7255*x^7 - 6730*x^6 + 3242*x^5 - 995*x^4 + 180*x^3 - 5*x^2 - 5*x + 1)/(x - 1)^15. 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!.
%Y Cf. A003514, A060516, A060533-A060537, A060576-A060581.
%K easy,nonn
%O 0,2
%A _Vladeta Jovovic_, Apr 03 2001
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