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 A288756 Numerator of the volume of the "monic slice" of the d-th Chern-Vaaler star body. 5
 2, 4, 16, 64, 1024, 16384, 524288, 16777216, 4294967296, 1099511627776, 562949953421312, 288230376151711744, 590295810358705651712, 1208925819614629174706176, 4951760157141521099596496896, 20282409603651670423947251286016, 1329227995784915872903807060280344576 (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. 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 Numerator of 2^(d - e) * (e!)^-1 * Product_{j = 1..e} (2*j/(2*j + 1))^(d - 2 j)) * Sum_{j = 0..e} ((-1)^j * (d - 2*j)^e * binomial(e, j)), where e = floor((d-1)/2). Floor(a(n)/A288757(n)) = A288758(n). EXAMPLE 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[ Numerator[ vol[d]], {d, 1, 17}] CROSSREFS Cf. A286522, A286523, A286524, A288757, A288758. Sequence in context: A060656 A271234 A061286 * A019279 A061652 A278913 Adjacent sequences:  A288753 A288754 A288755 * A288757 A288758 A288759 KEYWORD nonn,frac AUTHOR Jonathan Sondow, Jun 15 2017 STATUS approved

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)