%I #21 Sep 16 2020 05:02:00

%S 0,1,7,31,116,379,1133,3210,8803,23701,63239,168287,447905,1194814,

%T 3196180,8576505,23081668,62292381,168536249,457035453,1241954405,

%U 3381289332,9221603416,25189382006,68906572413,188750887991

%N Number of combinatorial types of n-dimensional polytopes with n+3 vertices.

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

%H Lukas Finschi, <a href="http://dx.doi.org/10.3929/ethz-a-004255224">A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids</a>, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001. See Table 7.5.

%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 Aleksandr Maksimenko, <a href="https://arxiv.org/abs/1904.03638">2-neighborly 0/1-polytopes of dimension 7</a>, arXiv:1904.03638 [math.CO], 2019.

%p N:=60: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-ln(1-2*x)+ln(1-x): K:=-1/2*x*(x-8*x^3-1+5*x^2-7*x^4+2*x^6+5*x^8-9*x^7+19*x^5-14*x^9+x^10+19*x^11-5*x^12+4*x^14-8*x^13)/(1-x)^5/(2*x^6-4*x^4+4*x^2-1)/(x+1)^2: series(1/(x^3-x^4)*(1/4*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+1/2*sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);

%t terms = 26;

%t G[x_] = -Log[1 - 2(x^3/(1 - 2x)^2)];

%t H[x_] = -Log[1 - 2x] + Log[1 - x];

%t K[x_] = -1/2 x (x - 8x^3 - 1 + 5x^2 - 7x^4 + 2x^6 + 5x^8 - 9x^7 + 19x^5 - 14x^9 + x^10 + 19x^11 - 5x^12 + 4x^14 - 8x^13)/(1-x)^5/(2x^6 - 4x^4 + 4x^2 - 1)/(x+1)^2;

%t 1/(x^3 - x^4) (1/4 Sum[EulerPhi[2r + 1]/(2r + 1) G[x^(2r + 1)], {r, 0, terms+2}] + 1/2 Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+2}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* _Jean-François Alcover_, Dec 14 2018 *)

%Y Cf. A000943, A114290, A114291.

%K nonn

%O 1,3

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