%I M3401 #53 Mar 21 2024 08:31:50
%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 3-clusters with n cells.
%C Also arises in enumeration of stereoisomers of alkane systems.
%C "A simplicial d-cluster may be informally described as being constructed by gluing regular d-simplexes together facet-by-facet, 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 one-to-one correspondence with the combinatorial types of stack polytopes." [Hering et al., 1982] - _Jonathan Vos Post_, Apr 22 2011
%C The Hering article has an error in the 14th term. - _Robert A. Russell_, Apr 11 2012
%C Also same as A027610 with mirror-image not treated as equivalence. - _Brendan McKay_, Mar 08 2014
%C Number of oriented polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. For oriented polyominoes, chiral pairs are counted as two. - _Robert A. Russell_, Mar 20 2024
%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/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Can. J. Math., 26 (1974), 50-67
%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/S0022-2860(97)00025-2">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413-414 (1997), 227-239.
%H F. Hering et al., <a href="http://dx.doi.org/10.1016/0012-365X(82)90121-2">The enumeration of stack polytopes and simplicial clusters</a>, Discrete Math., 40 (1982), 203-217.
%F From _Robert A. Russell_, Mar 20 2024: (Start)
%F a(n) = C(3n,n)/(3*(2n+1)*(2n+2)) + ([0==n mod 2]*C(3n/2,n) + [1==n mod 2]*C((3n-1)/2,(n-1)/2)) / (2n+2) + 2*([1==n mod 3]*C(n,(n-1)/3) + [2==n mod 3]*C(n,(n-2)/3)) / (3n).
%F a(n) = A027610(n) + A371350(n) = 2*A027610(n) - A371351(n) = 2*A371350(n) + A371351(n).
%F a(n) = H(3,n) in Table 8 of Hering link.
%F G.f.: (-8 + 4*G(z) - 2*G(z)^2 + z*G(z)^4 + 6*G(z^2) + 3z*G(z^2)^2 + 8z*G(z^3) + 4z^2*G(z^3)^2)/12, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)
%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.
%Y Cf. A027610 (unoriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A001683(n+2) {3,oo}, A007175 {3,3,3,oo}.
%K nonn,nice,easy
%O 1,4
%A _N. J. A. Sloane_
%E a(14) corrected and additional terms from _Robert A. Russell_, Apr 11 2012