A288758 Floor of the volume of the "monic slice" of the d-th Chern-Vaaler star body.

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

Table of n, a(n) for n=1..87.

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

Cf. A286522, A286523, A286524, A288756, A288757.

Sequence in context: A025511 A224367 A308219 * A210936 A140203 A085888

Adjacent sequences: A288755 A288756 A288757 * A288759 A288760 A288761

Keyword

nonn

Author

Jonathan Sondow, Jun 15 2017

Extensions

More terms from Michel Marcus, Jun 17 2017

Status

approved