A182299 Number of achiral simplicial 4-clusters with n cells.

1, 1, 1, 3, 6, 20, 51, 184, 550, 2009, 6487, 23875, 81724, 302954, 1078409, 4034373, 14771551, 55789188, 208526682, 794933818, 3017839193, 11604938152, 44590911769, 172833268057, 670520982414, 2617397888002, 10234831661388, 40204487779050, 158254659096516, 625142808049902

Offset: 1

Author

Robert A. Russell, Apr 23 2012

Keywords

• nonn
• easy

Comments

This sequence would be H_{4,n} subtracted from twice h_{4,n} in Table 8 of the Hering article if those numbers were correct, but some are not. In addition, the formula in the penultimate line of Table 5 of the article should not have an exponent for the second E_4.

The limit supremum as n approaches infinity of a(n+1)/a(n) is 16(2-sqrt(3)) or about 4.28719. - Robert A. Russell, Oct 21 2014

Examples

			For n=4 the a(4)=3 solutions are the three achiral (there are no chiral) clusters that can be formed from four simplexes in four-space.  One has three attached to a fourth, one has four sharing a common triangle, and the last has neither of these properties.
		

Links

• F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Crossrefs

Cf. A182322, A007175.

Programs

Mathematica

n = 30;
e[d_,t_]:=Sum[Binomial[d k,k]/((d-1)k+1)t^k,{k,0,n}]
CoefficientList[Series[(10e[4,t^2]e[2,e[4,t^2]t]^3t
   +30e[4,t^4]t(1+e[4,t^4]t)
   +20e[1,e[4,t^6] t^2]e[2,e[4,t^6]t^3]t)/60
   -(6(e[2,e[4,t^2]t]-1)^2+6e[4,t^4]^2t^2)/24
   +(4e[4,t^2]^4t^2+8e[1,e[4,t^6]t^2]e[4,t^6]t^2)/24,
   {t,0,n}]/t,t] (* Robert A. Russell, Apr 23 2012 *)