A321426 - OEIS

%I #14 Nov 23 2018 03:19:27

%S 0,0,6,810,282660,195192900,235439369550,454833890480970,

%T 1320613138677432600,5490000743915652564600,

%U 31451199565381549069866750,240742295353571264522056037250,2400231508458936741386610203090700,30511229662020079098420585892148047500

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

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

%H Andrew Howroyd, <a href="/A321426/b321426.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. f(x).

%F 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)]

%F 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

%t 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}];

%t 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];

%t seq[13] (* _Jean-François Alcover_, Nov 22 2018, after _Andrew Howroyd_ *)

%o (PARI) \\ here b(n) is A002829

%o 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)!))));

%o 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

%Y Cf. A002829, A321425, A321427.

%K nonn

%O 0,3

%A _R. J. Mathar_, Nov 09 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Nov 09 2018