%I M3401
%S 1,1,1,4,10,40,171,831,4147,21822,117062,642600,3582322,20256885,
%T 115888201,669911568,3907720521,22979343010,136107859377,811430160282,
%U 4866004426320,29337068299728,177738920836446,1081668278379000,6609923004626478,40546403939165805
%N Number of simplicial 3clusters with n cells.
%C Also arises in enumeration of stereoisomers of alkane systems.
%C "A simplicial dcluster may be informally described as being constructed by gluing regular dsimplexes together facetbyfacet, at each stage gluing a new simplex to exactly one facet of a cluster already constructed. The equivalence classes of such clusters under rigid motions are in onetoone correspondence with the combinatorial types of stack polytopes." [Hering et al., 1982]  _Jonathan Vos Post_, Apr 22 2011
%C Hering article has error in the 14th term.  _Robert A. Russell_, Apr 11 2012
%C Also same as A027610 with mirrorimage not treated as equivalence.  _Brendan McKay_, Mar 08 2014
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007173/b007173.txt">Table of n, a(n) for n = 1..200</a>
%H L. W. Beineke and R. E. Pippert <a href="http://dx.doi.org/10.4153/CJM1974006x">Enumerating dissectable polyhedra by their automorphism groups</a>, Can. J. Math., 26 (1974), 5067
%H CombOS  Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>
%H S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, <a href="https://doi.org/10.1016/S00222860(97)000252">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413414 (1997), 227239.
%H F. Hering et al., <a href="http://dx.doi.org/10.1016/0012365X(82)901212">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203217.
%t Table[Binomial[3 n, n]/(3 (2 n + 1) (2 n + 2)) + If[OddQ[n], Binomial[3 (n  1)/2 + 1, n]/(n + 1), Binomial[3 n/2, n]/(n + 1)]/2 + 2 Switch[Mod[n, 3], 0, 0, 1, Binomial[n, (n  1)/3]/n, 2, Binomial[n, (n  2)/3]/n]/3, {n, 1, 30}] (* _Robert A. Russell_, Apr 11 2012 *)
%Y Sum of achiral symmetry types (A047775, A047773, A047760, A047754, A047753, A047751, A047771, A047766 [type N], A047765, A047764) plus twice sum of chiral symmetry types (A047776, A047774, A047762, A047758, A047752, A047769, A047766 [type O]) in Beineke article.
%K nonn,nice,easy
%O 1,4
%A _N. J. A. Sloane_
%E Corrected a(14) and added additional terms.  _Robert A. Russell_, Apr 11 2012
