A321426 Number of connected labeled fairly cubic graphs on 2n nodes.

0, 0, 6, 810, 282660, 195192900, 235439369550, 454833890480970, 1320613138677432600, 5490000743915652564600, 31451199565381549069866750, 240742295353571264522056037250, 2400231508458936741386610203090700, 30511229662020079098420585892148047500

Offset

0,3

Comments

Fairly cubic graphs are cubic graphs (A002829) where 2 points have degree 2. All other points have degree 3.

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. f(x).

Formula

a(n) = A321425(n) + n*(2*n-1)*(2*n-2)*A321427(n-2) + 2*n*(2*n-1)*a(n-1). [Wormald eq (2.3)]

a(n) = 3*n*A002829(n) + 2*n*(2*n-1)*a(n-1) + n*(2*n-1)*(2*n-2)*(2*n-3)*a(n-2). - Andrew Howroyd, Nov 09 2018

Mathematica

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_] := 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];
seq[13] (* Jean-François Alcover, Nov 22 2018, after Andrew Howroyd *)

Prog

(PARI) \\ here b(n) is A002829
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)!))));
seq(n)={my(v=vector(n+1)); for(n=2, n, v[n+1] = 3*n*b(n) + 2*n*(2*n-1)*v[n] + n*(2*n-1)*(2*n-2)*(2*n-3)*v[n-1]); v} \\ Andrew Howroyd, Nov 09 2018

Crossrefs

Cf. A002829, A321425, A321427.

Sequence in context: A006114 A341554 A365511 * A281566 A271637 A249126

Adjacent sequences: A321423 A321424 A321425 * A321427 A321428 A321429

Keyword

nonn

Author

R. J. Mathar, Nov 09 2018

Extensions

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

Status

approved