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A288757
Denominator of the volume of the "monic slice" of the d-th Chern-Vaaler star body.
5
1, 1, 3, 9, 135, 2025, 70875, 2480625, 781396875, 246140015625, 170575030828125, 118208496363890625, 354980114580763546875, 1066005284086032931265625, 6859744003093621912694296875, 44142452659907457008187800390625, 4828963608730576259410704423732421875
OFFSET
1,3
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
Denominator 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(A288756(n)/a(n)) = A288758(n).
EXAMPLE
Fractions begin: 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[ Denominator[ vol[d]], {d, 1, 17}]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Jonathan Sondow, Jun 15 2017
STATUS
approved