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

%I A321427 #13 Nov 23 2018 15:04:28
%S A321427 0,0,30,5670,2543940,2147121900,3060711804150,6822508357214550,
%T A321427 22450423357516354200,104310014134397398727400,
%U A321427 660475190873012530467201750,5537072793132139084007288856750,60005787711473418534665255077267500,823803200874542135657355819087997282500
%N A321427 Number of connected labeled closely cubic graphs on 2n+1 nodes.
%C A321427 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).
%H A321427 Andrew Howroyd, <a href="/A321427/b321427.txt">Table of n, a(n) for n = 0..100</a>
%H A321427 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. See e.g.f. c(x).
%F A321427 a(n) = (2*n+1)*A321426(n). [Wormald eq. (2.2)]
%t A321427 nmax = 13;
%t A321427 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 A321427 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];
%t A321427 a[n_] := (2n+1) seq[nmax][[n+1]];
%t A321427 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 23 2018, after _Andrew Howroyd_ *)
%Y A321427 Cf. A002829, A321425, A321426.
%K A321427 nonn
%O A321427 0,3
%A A321427 _R. J. Mathar_, Nov 09 2018
%E A321427 Terms a(10) and beyond from _Andrew Howroyd_, Nov 09 2018