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A114290
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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.
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2
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0, 1, 7, 38, 170, 617, 1979, 5859, 16571, 45516, 123159, 330736, 885780, 2372305, 6362965, 17102719, 46078541, 124440388, 336829857, 913658780, 2483217288, 6761405513, 18441239903, 50375429081, 137807555515, 377492301876
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OFFSET
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1,3
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REFERENCES
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B. Grünbaum, Convex Polytopes, Springer-Verlag, 2003, Second edition prepared by V. Kaibel, V. Klee and G. M. Ziegler, p. 121a.
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LINKS
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Éric Fusy (eric.fusy(AT)inria.fr), Nov 21 2005
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STATUS
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approved
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