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%I #21 Nov 23 2018 03:43:34
%S 0,0,6,630,232260,167712300,207994906350,409639268108070,
%T 1206311009131027800,5069191623021896970600,
%U 29288218834810895163954750,225729928889064072869657010750,2263331356064784471285438421502700,28907890013735339531664032407056442500
%N Number of connected labeled almost cubic graphs on 2n nodes.
%C Almost cubic graphs are cubic graphs (A002829) where 2 points have degree 2 and these 2 points are non-adjacent. All other points have degree 3. They are constructed by removing an edge from the cubic graphs.
%H Andrew Howroyd, <a href="/A321425/b321425.txt">Table of n, a(n) for n = 0..100</a>
%H N. C. Wormald, <a href="https://dx.doi.org/10.1112/jlms/s2-20.1.1">Enumeration of labelled graphs II: cubic graphs with a given connectivity</a>, J. Lond Math Soc s2-20 (1979) 1-7, e.g.f. a(x).
%F a(n) = 3*n*A002829(n). [Wormald eq. (2.1)]
%e There is 1 unlabeled almost cubic graph on 4 nodes (the kite, obtained by removing an edge of the tetrahedron K_4). This has 6 = binomial(4,2) labeled versions obtained by selecting two out of 4 labels for the points of degree 2.
%t terms = 14; egf = HypergeometricPFQ[{1/6, 5/6}, {}, 12x/(x^2 + 8x + 4)^(3/2)] Exp[-Log[1/4 x^2 + 2x + 1]/4 - x/3 + (x^2 + 8x + 4)^(3/2)/(24 x) - 1/(3x) - x^2/24 - 1] + O[x]^terms;
%t CoefficientList[egf, x](2 Range[0, terms-1])! 3 Range[0, terms-1] (* _Jean-François Alcover_, Nov 23 2018, from A002829 *)
%o (PARI) b(n) = sum(i=0, 2*n, sum(k=0, min(floor((3*n-i)/3), floor((2*n-i)/2)), sum(j=0, min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2)), ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!)))); \\ A002829
%o vector(20, n, n--; 3*n*b(n)) \\ _Michel Marcus_, Nov 10 2018
%Y Cf. A002829, A321426.
%K nonn
%O 0,3
%A _R. J. Mathar_, Nov 09 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Nov 09 2018
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