cp's OEIS Frontend

A114290 Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.

%I A114290 #16 Sep 16 2020 05:02:13
%S A114290 0,1,7,38,170,617,1979,5859,16571,45516,123159,330736,885780,2372305,
%T A114290 6362965,17102719,46078541,124440388,336829857,913658780,2483217288,
%U A114290 6761405513,18441239903,50375429081,137807555515,377492301876
%N A114290 Number of oriented n-dimensional polytopes with n+3 vertices, meaning that two polytopes are identified if they have the same combinatorial type and there exists an orientation-preserving homeomorphism mapping the first polytope to the second polytope.
%D A114290 B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
%H A114290 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 A114290 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 A114290 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.
%p A114290 N:=30: with(numtheory): G:=-ln(1-2*x^3/(1-2*x)^2): H:=-log(1-2*x)+ln(1-x): K:=-(x^10+3*x^9-3*x^8-7*x^7+4*x^6+4*x^5+4*x^4+3*x^3-2*x^2+1)*x/(1-x)^5/(x+1)^3: series(1/(x^3-x^4)*(1/2*sum(phi(2*r+1)/(2*r+1)*subs(x=x^(2*r+1),G),r=0..N)+sum(phi(r)/r*subs(x=x^r,H),r=1..N)+K),x,N);
%t A114290 terms = 26;
%t A114290 G[x_] = -Log[1 - 2 (x^3/(1 - 2 x)^2)];
%t A114290 H[x_] = -Log[1 - 2 x] + Log[1 - x];
%t A114290 K[x_] = -(x^10 + 3 x^9 - 3 x^8 - 7 x^7 + 4 x^6 + 4 x^5 + 4 x^4 + 3 x^3 - 2 x^2 + 1) x/(1 - x)^5/(x + 1)^3;
%t A114290 1/(x^3 - x^4) (1/2 Sum[EulerPhi[2 r + 1]/(2 r + 1) G[x^(2 r + 1)], {r, 0, terms+3}] + Sum[EulerPhi[r]/r H[x^r], {r, 1, terms+3}] + K[x]) + O[x]^(terms+2) // CoefficientList[#, x]& // Rest // Most // Round (* _Jean-François Alcover_, Dec 14 2018 *)
%Y A114290 Cf. A000943, A114289, A114291.
%K A114290 nonn
%O A114290 1,3
%A A114290 Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005