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%I #23 Jun 17 2017 08:41:57
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Floor of the volume of the "monic slice" of the d-th Chern-Vaaler star body.
%C 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.
%C Unimodal: increases from 2 to a maximum of 8, then decreases to 0 = a(d) for d >= 15 (conjectured).
%H S.-J. Chern and J.D. Vaaler, <a href="https://doi.org/10.1515/crll.2001.084">The distribution of values of Mahler's measure</a>, J. Reine. Angew. Math., 540 (2001), 1-47.
%H R, Grizzard and J. Gunther, <a href="https://arxiv.org/abs/1609.08720">Slicing the stars: counting algebraic numbers, integers, and units by degree and height</a>, arXiv:1609.08720 [math.NT], 2016.
%F 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).
%F a(n) = floor(A288756(n)/A288757(n)).
%e Floor of 2, 4, 16/3, 64/9, 1024/135, 16384/2025, 524288/70875, 16777216/2480625, 4294967296/781396875, 1099511627776/246140015625
%t 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}]
%o (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
%Y Cf. A286522, A286523, A286524, A288756, A288757.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Jun 15 2017
%E More terms from _Michel Marcus_, Jun 17 2017
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