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 A288758 Floor of the volume of the "monic slice" of the d-th Chern-Vaaler star body. 5
 2, 4, 5, 7, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The "monic slice" corresponds to integer polynomials of degree at most d, and of Mahler's measure at most 1. See Grizzard and Gunther (2016) section 2.1. For the volume of the d-th Chern-Vaaler star body, see A286522, A286523, A286524. Unimodal: increases from 2 to a maximum of 8, then decreases to 0 = a(d) for d >= 15 (conjectured). LINKS S.-J. Chern and J.D. Vaaler, The distribution of values of Mahler's measure, J. Reine. Angew. Math., 540 (2001), 1-47. R, Grizzard and J. Gunther, Slicing the stars: counting algebraic numbers, integers, and units by degree and height, arXiv:1609.08720 [math.NT], 2016. FORMULA a(d) = floor of 2^(d - e) * (e!)^-1 * Product_{j = 1..e} (2*j/(2*j + 1))^(d - 2*j) * Sum_{j = 1..e} ((-1)^j * (d - 2*j)^e * binomial(e, j)), where e = floor((d-1)/2). a(n) = floor(A288756(n)/A288757(n)). EXAMPLE Floor of 2, 4, 16/3, 64/9, 1024/135, 16384/2025, 524288/70875, 16777216/2480625, 4294967296/781396875, 1099511627776/246140015625 MATHEMATICA vol[d_] := (e = Floor[(d - 1)/2];  2^(d - e) (e!)^-1 Product[(2 j/(2 j + 1))^(d - 2 j), {j, 1,  e}]  Sum[(-1)^j (d - 2 j)^e Binomial[e, j], {j, 0, e}]); Table[ Floor[ vol[d]], {d, 1, 17}] PROG (PARI) a(d) = my(e=(d-1)\2); floor(2^(d - e) * (e!)^(-1) * prod(j=1, e, (2*j/(2*j + 1))^(d - 2*j)) * sum(j=0, e, (-1)^j * (d - 2*j)^e * binomial(e, j))); \\ Michel Marcus, Jun 17 2017 CROSSREFS Cf. A286522, A286523, A286524, A288756, A288757. Sequence in context: A025511 A224367 A308219 * A210936 A140203 A085888 Adjacent sequences:  A288755 A288756 A288757 * A288759 A288760 A288761 KEYWORD nonn AUTHOR Jonathan Sondow, Jun 15 2017 EXTENSIONS More terms from Michel Marcus, Jun 17 2017 STATUS approved

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Last modified November 29 15:46 EST 2021. Contains 349416 sequences. (Running on oeis4.)