A321427 Number of connected labeled closely cubic graphs on 2n+1 nodes.

0, 0, 30, 5670, 2543940, 2147121900, 3060711804150, 6822508357214550, 22450423357516354200, 104310014134397398727400, 660475190873012530467201750, 5537072793132139084007288856750, 60005787711473418534665255077267500, 823803200874542135657355819087997282500

Offset

0,3

Comments

Closely cubic graphs are cubic graphs (A002829) where 1 point has degree 2. All other points have degree 3. They are constructed by removing a point from the fairly cubic graphs (A321426).

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. See e.g.f. c(x).

Formula

a(n) = (2*n+1)*A321426(n). [Wormald eq. (2.2)]

Mathematica

nmax = 13;
b[n_] := Sum[Sum[Sum[((-1)^(i + j) (2n)! (2(3n - i - 2j - 3k))!)/(2^(5n - i - 2j - 4k) 3^(2n - i - 2j - k)(3n - i - 2j - 3k)! i! j! k! (2n - i - 2j - 2k)!), {j, 0, Min[Floor[(3n - i - 3k)/2], Floor[(2n - i - 2k)/2]]}], {k, 0, Min[Floor[(3n - i)/3], Floor[(2n - i)/2]]}], {i, 0, 2n}];
seq[n_] := seq[n] = Module[{v = Table[0, {n + 1}]}, For[k = 2, k <= n, k++, v[[k + 1]] = 3k b[k] + 2k(2k - 1) v[[k]] + k(2k - 1)(2k - 2)(2k - 3)v[[k - 1]]]; v];
a[n_] := (2n+1) seq[nmax][[n+1]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 23 2018, after Andrew Howroyd *)

Crossrefs

Cf. A002829, A321425, A321426.

Sequence in context: A204975 A204702 A206647 * A050984 A169686 A184889

Adjacent sequences: A321424 A321425 A321426 * A321428 A321429 A321430

Keyword

nonn

Author

R. J. Mathar, Nov 09 2018

Extensions

Terms a(10) and beyond from Andrew Howroyd, Nov 09 2018

Status

approved