A114291 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.

0, 1, 7, 24, 62, 141, 287, 561, 1035, 1886, 3319, 5838, 10030, 17323, 29395, 50291, 84795, 144374, 242641, 412126, 691522, 1173151, 1966929, 3334931, 5589311, 9474106, 15875699, 26906538, 45083426, 76404103, 128014623, 216944163

Offset

1,3

References

B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.

Links

Table of n, a(n) for n=1..32.

Éric Fusy, Counting d-polytopes with d+3 vertices, arXiv:math/0511466 [math.CO], 2005.

Éric Fusy, Counting d-polytopes with d+3 vertices, Electron. J. Comb. 13 (2006), no. 1, research paper R23, 25 pp.

E. K. Lloyd, The number of d-polytopes with d+3 vertices, Mathematika 17 (1970), 120-132.

Index entries for linear recurrences with constant coefficients, signature (2, 6, -14, -12, 38, 8, -54, 5, 44, -12, -20, 8, 4, -2).

Formula

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).

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000943, A114289, A114290.

Sequence in context: A014205 A002969 A029585 * A211382 A217746 A211381

Adjacent sequences: A114288 A114289 A114290 * A114292 A114293 A114294

Keyword

nonn

Author

Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005

Status

approved