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%I #15 Nov 10 2016 12:05:16
%S 1,6,3,10,48,84,182,372,699,1222,2007,3132,4688,6780,9528,13068,17553,
%T 23154,30061,38484,48654,60824,75270,92292,112215,135390,162195,
%U 193036,228348,268596,314276,365916,424077,489354,562377,643812,734362,834768,945810
%N Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 4 labeled nodes.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
%H Colin Barker, <a href="/A060534/b060534.txt">Table of n, a(n) for n = 0..1000</a>
%H Vladeta Jovovic, <a href="/A060533/a060533.pdf">Generating functions for homeomorphically irreducible multigraphs on n labeled nodes</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: - (4*x^12 - 12*x^11 + 6*x^10 + 50*x^9 - 180*x^8 + 282*x^7 - 208*x^6 + 30*x^5 + 72*x^4 - 62*x^3 + 18*x^2 - 1)/((x - 1)^6).
%F E.g.f. for homeomorphically irreducible multigraphs 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, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
%F From _Colin Barker_, Nov 10 2016: (Start)
%F a(n) = 60 + 48*(1+n) - 12*(1+n)*(2+n) + (1+n)*(2+n)*(3+n)*(4+n)*(5+n)/120 for n>6.
%F a(n) = 6*a(n-1)- 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>12.
%F (End)
%o (PARI) Vec(-(4*x^12-12*x^11+6*x^10+50*x^9-180*x^8+282*x^7-208*x^6+30*x^5+72*x^4-62*x^3+18*x^2-1)/((x-1)^6) + O(x^40)) \\ _Colin Barker_, Nov 10 2016
%Y Cf. A003514, A060516, A060533, A060535-A060537.
%K nonn,easy
%O 0,2
%A _Vladeta Jovovic_, Apr 01 2001
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