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A327728 Number of unlabeled multigraphs with loops allowed and n edges covering three vertices.

%I A327728 #27 Oct 24 2019 16:24:02
%S A327728 0,2,8,19,40,77,132,217,340,510,742,1054,1456,1976,2634,3453,4464,
%T A327728 5703,7194,8987,11120,13636,16588,20036,24024,28630,33916,39951,46816,
%U A327728 54601,63376,73253,84324,96690,110466,125778,142728,161468,182126,204841,229768,257075,286902,319447,354880,393384
%N A327728 Number of unlabeled multigraphs with loops allowed and n edges covering three vertices.
%H A327728 Andrew Howroyd, <a href="/A327728/b327728.txt">Table of n, a(n) for n = 1..1000</a>
%H A327728 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
%F A327728 a(n) = A050531(n) - A002620(n+2).
%F A327728 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n > 12.
%F A327728 G.f.: x^2*(2 + 4*x + x^2 - 2*x^3 + x^6)/((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).
%e A327728 a(2) = 2 since three vertices may be covered with two edges in 2 ways: the path graph P(3) or an edge plus a loop.
%o A327728 (PARI) concat([0], Vec((2 + 4*x + x^2 - 2*x^3 + x^6)/((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^40)))
%Y A327728 Column k=3 of A327615.
%Y A327728 Cf. A002620, A050531.
%K A327728 nonn,easy
%O A327728 1,2
%A A327728 _Andrew Howroyd_, Oct 23 2019