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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
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
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jun 15 2017
EXTENSIONS
More terms from Michel Marcus, Jun 17 2017
STATUS
approved