%I #7 May 10 2013 12:44:44
%S 1,15,81,441,2151,9957,43122,174162,666267,2403987,8183601,26281065,
%T 79660856,228180456,618992466,1595081266,3918506466,9211519476,
%U 20797923546,45258309066,95225448306,194283668576,385361919996
%N Number of homeomorphically irreducible general graphs on 6 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.: (6*x^30 - 30*x^29 - 90*x^28 + 898*x^27 - 5703*x^26 + 67854*x^25 - 552925*x^24 + 2795730*x^23 - 9663357*x^22 + 24476292*x^21 - 47540991*x^20 + 73129860*x^19 - 91373250*x^18 + 94675608*x^17 - 82549758*x^16 + 60794764*x^15 - 37293240*x^14 + 18277860*x^13 - 6426742*x^12 + 945252*x^11 + 680499*x^10 - 726250*x^9 + 423825*x^8 - 187536*x^7 + 66981*x^6 - 19092*x^5 + 4065*x^4 - 560*x^3 + 24*x^2 + 6*x - 1)/(x - 1)^21. 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