cp's OEIS Frontend

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

%I A182299 #19 Jan 11 2020 18:34:08
%S A182299 1,1,1,3,6,20,51,184,550,2009,6487,23875,81724,302954,1078409,4034373,
%T A182299 14771551,55789188,208526682,794933818,3017839193,11604938152,
%U A182299 44590911769,172833268057,670520982414,2617397888002,10234831661388,40204487779050,158254659096516,625142808049902
%N A182299 Number of achiral simplicial 4-clusters with n cells.
%C A182299 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.
%C A182299 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
%H A182299 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.
%e A182299 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.
%t A182299 n = 30;
%t A182299 e[d_,t_]:=Sum[Binomial[d k,k]/((d-1)k+1)t^k,{k,0,n}]
%t A182299 CoefficientList[Series[(10e[4,t^2]e[2,e[4,t^2]t]^3t
%t A182299    +30e[4,t^4]t(1+e[4,t^4]t)
%t A182299    +20e[1,e[4,t^6] t^2]e[2,e[4,t^6]t^3]t)/60
%t A182299    -(6(e[2,e[4,t^2]t]-1)^2+6e[4,t^4]^2t^2)/24
%t A182299    +(4e[4,t^2]^4t^2+8e[1,e[4,t^6]t^2]e[4,t^6]t^2)/24,
%t A182299    {t,0,n}]/t,t] (* _Robert A. Russell_, Apr 23 2012 *)
%Y A182299 Cf. A182322, A007175.
%K A182299 nonn,easy
%O A182299 1,4
%A A182299 _Robert A. Russell_, Apr 23 2012