A288756 Numerator of the volume of the "monic slice" of the d-th Chern-Vaaler star body.

2, 4, 16, 64, 1024, 16384, 524288, 16777216, 4294967296, 1099511627776, 562949953421312, 288230376151711744, 590295810358705651712, 1208925819614629174706176, 4951760157141521099596496896, 20282409603651670423947251286016, 1329227995784915872903807060280344576

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

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

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