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A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.

%I A129427 #45 Mar 07 2022 07:33:59
%S A129427 1,2,8,31,140,722,4439,32654,289519,3054067,37584620,527968286,
%T A129427 8308434931,144345554051,2738280739075,56245013793246,
%U A129427 1242596591479816,29366532494796900,739033832149588904,19726887762569763453
%N A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.
%C A129427 a(1)..a(11) computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/
%D A129427 P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.
%H A129427 R. de Mello Koch, S. Ramgoolam, <a href="https://doi.org/10.1103/PhysRevD.85.026007">Strings from Feynman graph counting: Without large N</a>, Phys. Rev. D 85 (2012) 026007
%H A129427 P. A. Morris, <a href="/A003175/a003175.pdf">Letter to N. J. A. Sloane, Mar 02 1971</a>.
%H A129427 R. C. Read, <a href="http://dx.doi.org/10.1112/jlms/s1-34.4.417">The enumeration of locally restricted graphs (I)</a>, J. London Math. Soc. 34 (1959) 417-436. - _Jason Kimberley_, Sep 17 2009
%F A129427 a(n)=N\{S_{2n}[S_3] * S_{3n}[S_2]\}. - _Jason Kimberley_, Sep 17 2009
%o A129427 (Sage)
%o A129427 h = SymmetricFunctions(QQ).homogeneous()
%o A129427 def A129427(n):
%o A129427     X = h([2*n]).plethysm(h([3]))
%o A129427     Y = h([3*n]).plethysm(h([2]))
%o A129427     return X.scalar(Y)
%o A129427 # _Bruce Westbury_, Aug 16 2013
%Y A129427 Column k=3 of A167625.
%Y A129427 Cf. A005967 (connected, inv. Euler trans.), A129416, A129429, A129431, A129433, A129435, A129437, A005638.
%K A129427 nonn
%O A129427 0,2
%A A129427 _Brendan McKay_, Apr 15 2007
%E A129427 Using equation (5.8) of Read 1959, new terms a(12) and a(13) were computed in MAGMA by _Jason Kimberley_, Sep 17 2009
%E A129427 Further terms a(14)-a(16) also computed by _Jason Kimberley_, announced Nov 09 2009
%E A129427 Formula corrected from n vertices to 2n vertices by _Jason Kimberley_, Nov 09 2009
%E A129427 Added a(0). - _N. J. A. Sloane_, Aug 26 2013
%E A129427 a(17)-a(19) from _Sean A. Irvine_, Oct 29 2016