A288757 Denominator of the volume of the "monic slice" of the d-th Chern-Vaaler star body.

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

Table of n, a(n) for n=1..17.

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

Cf. A286522, A286523, A286524, A288756, A288758.

Sequence in context: A141143 A368092 A163402 * A289900 A235062 A082707

Adjacent sequences: A288754 A288755 A288756 * A288758 A288759 A288760

Keyword

nonn,frac

Author

Jonathan Sondow, Jun 15 2017

Status

approved