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A321425 Number of connected labeled almost cubic graphs on 2n nodes. 3
0, 0, 6, 630, 232260, 167712300, 207994906350, 409639268108070, 1206311009131027800, 5069191623021896970600, 29288218834810895163954750, 225729928889064072869657010750, 2263331356064784471285438421502700, 28907890013735339531664032407056442500 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

N. C. Wormald, Enumeration of labelled graphs II: cubic graphs with a given connectivity, J. Lond Math Soc s2-20 (1979) 1-7, e.g.f. a(x).

FORMULA

a(n) = 3*n*A002829(n). [Wormald eq. (2.1)]

EXAMPLE

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.

MATHEMATICA

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;

CoefficientList[egf, x](2 Range[0, terms-1])! 3 Range[0, terms-1] (* Jean-Fran├žois Alcover, Nov 23 2018, from A002829 *)

PROG

(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

vector(20, n, n--; 3*n*b(n)) \\ Michel Marcus, Nov 10 2018

CROSSREFS

Cf. A002829, A321426.

Sequence in context: A265116 A132928 A055317 * A220858 A337652 A222743

Adjacent sequences:  A321422 A321423 A321424 * A321426 A321427 A321428

KEYWORD

nonn

AUTHOR

R. J. Mathar, Nov 09 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Nov 09 2018

STATUS

approved

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Last modified August 9 05:53 EDT 2022. Contains 356016 sequences. (Running on oeis4.)