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
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.
MAPLE
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);
MATHEMATICA
terms = 26;
G[x_] = -Log[1 - 2 (x^3/(1 - 2 x)^2)];
H[x_] = -Log[1 - 2 x] + Log[1 - x];
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;
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
STATUS
approved