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A007099 Number of labeled trivalent (or cubic) 2-connected graphs with 2n nodes.

%I A007099 M5344 #28 Dec 26 2021 21:38:25
%S A007099 0,1,70,19320,11052720,11408720400,19285018552800,49792044478176000,
%T A007099 186348919238786304000,970566620767088881536000,
%U A007099 6808941648018137282054400000,62642603299257346706851910400000
%N A007099 Number of labeled trivalent (or cubic) 2-connected graphs with 2n nodes.
%D A007099 R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%D A007099 R. W. Robinson, personal communication.
%D A007099 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A007099 R. W. Robinson, Computer print-out, no date. Gives first 29 terms.
%D A007099 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007099 R. W. Robinson, <a href="/A007099/b007099.txt">Table of n, a(n) for n = 1..29</a> (corrected by Michel Marcus, Jan 19 2019)
%H A007099 G.-B. Chae, E. M. Palmer, R. W. Robinson, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.011">Counting labeled general cubic graphs</a>, Discr. Math. 307 (2007) 2979-2992, eqs. (23) and (24).
%H A007099 R. W. Robinson, <a href="/A002829/a002829.pdf">Cubic labeled graphs, computer print-out, n.d.</a>
%F A007099 a(n) = (2*n)! * (s(n) - 2*s(n-1)) / (3*n*2^n) where s(1)=0, s(2)=1, and s(n) = 3*n*s(n-1) + 2*s(n-2) + (3*n-1) * Sum_{i=2..n-3} s(i) * s(n-1-i). - _Sean A. Irvine_, Oct 11 2017
%p A007099 s := proc(n)
%p A007099     option remember;
%p A007099     if n = 1 then
%p A007099         0;
%p A007099     elif n = 2 then
%p A007099         1;
%p A007099     else
%p A007099         3*n*procname(n-1)+2*procname(n-2)+(3*n-1)*add(procname(i)*procname(n-1-i),i=2..n-3) ;
%p A007099     end if;
%p A007099 end proc:
%p A007099 A007099 := proc(n)
%p A007099     if n = 1 then
%p A007099         0;
%p A007099     elif n = 2 then
%p A007099         1;
%p A007099     else
%p A007099         (2*n)!/3/n/2^n*(s(n)-2*s(n-1)) ;
%p A007099     end if;
%p A007099 end proc: # _R. J. Mathar_, Nov 08 2018
%t A007099 s[n_] := s[n] = If[n <= 2, n - 1, 3 n s[n - 1] + 2 s[n - 2] + (3 n - 1) Sum[s[i] s[n - 1 - i], {i, 2, n - 3}]]; Array[Floor[(2 #)!*(s[#] - 2 s[# - 1])/(3 # 2^#)] &, 12] (* _Michael De Vlieger_, Oct 11 2017 *)
%Y A007099 Cf. A002829, A004109.
%K A007099 nonn
%O A007099 1,3
%A A007099 _N. J. A. Sloane_