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%I #20 Mar 03 2024 14:24:06
%S 0,1,7,24,62,141,287,561,1035,1886,3319,5838,10030,17323,29395,50291,
%T 84795,144374,242641,412126,691522,1173151,1966929,3334931,5589311,
%U 9474106,15875699,26906538,45083426,76404103,128014623,216944163
%N Number of combinatorial types of achiral n-dimensional polytopes with n+3 vertices, where a polytope is achiral if one of its geometric realizations has a reflection-symmetry.
%D B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
%H Éric Fusy, <a href="https://arxiv.org/abs/math/0511466">Counting d-polytopes with d+3 vertices</a>, arXiv:math/0511466 [math.CO], 2005.
%H Éric Fusy, <a href="https://doi.org/10.37236/1049">Counting d-polytopes with d+3 vertices</a>, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.
%H E. K. Lloyd, <a href="http://dx.doi.org/10.1112/S0025579300002795">The number of d-polytopes with d+3 vertices</a>, Mathematika 17 (1970), 120-132.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2).
%F G.f.: (2*x^11+4*x^10-2*x^9-15*x^8-5*x^7+23*x^6+15*x^5-17*x^4 -14*x^3 +4*x^2+5*x+1) *x^2 / ((-1+x)^5*(2*x^6-4*x^4+4*x^2-1)*(x+1)^3).
%t LinearRecurrence[{2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2}, {0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323}, 32] (* _Jean-François Alcover_, Dec 14 2018 *)
%o (PARI) concat(0, Vec((2*x^11+4*x^10-2*x^9-15*x^8-5*x^7+23*x^6+15*x^5 -17*x^4-14*x^3+4*x^2 +5*x+1)*x^2/ (-1+x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^3 + O(x^50))) \\ _Michel Marcus_, Dec 12 2014
%Y Cf. A000943, A114289, A114290.
%K nonn
%O 1,3
%A Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
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