A286523 Numerator of the volume of the d-th Chern-Vaaler star body.

2, 4, 8, 128, 640, 8192, 1605632, 536870912, 100663296, 137438953472, 32195899484536832, 1152921504606846976, 214035842104995017129984, 75557863725914323419136, 2417851639229258349412352, 2658455991569831745807614120560689152, 16645314084009764791991725029402697793536, 680564733841876926926749214863536422912, 2958953999535335146041291694037024012985750731620352

Offset

0,1

Comments

Chern and Vaaler's estimate of the number M(d,T) of integer polynomials of degree at most d, and of Mahler's measure at most T, is M(d,T) = V(d+1)*T^(d+1) + O(T^d) as T -> infinity, where d is fixed and V(d+1) is the volume of the d-th Chern-Vaaler star body, which is nonconvex and symmetric. For the "monic slice" of the star body, see A288756, A288757, A288758.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..80

S.-J. Chern and J.D. Vaaler, The distribution of values of Mahler's measure, J. Reine. Angew. Math., 540 (2001), 1-47.

A. Dubickas, Review of S.-J. Chern and J.D. Vaaler's "The distribution of values of Mahler's measure", Zentralblatt 0986.11017

Robert Grizzard and Joseph 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 + 1) * (d + 1)^e * Product_{k=1..e}((2*k)^(d - 2*k)/(2*k + 1)^(d + 1 - 2*k)) where e = floor((d-1)/2).

Floor(a(n)/A286524(n)) = A286522(n).

Example

2, 4, 8, 128/9, 640/27, 8192/225, 1605632/30375, 536870912/7441875, 100663296/1071875, ...

Mathematica

v[d_] := (e = Floor[(d - 1)/2]; 2^(d + 1) (d + 1)^e Product[(2 k)^(d - 2 k)/(2 k + 1)^(d + 1 - 2 k), {k, 1, e}]); Table[ Numerator[v[d]], {d, 0, 18}]

Crossrefs

Cf. A286522, A286524, A288756, A288757, A288758.

Sequence in context: A007633 A018777 A130693 * A060815 A200758 A110746

Adjacent sequences: A286520 A286521 A286522 * A286524 A286525 A286526

Keyword

nonn,frac

Author

Jonathan Sondow, May 26 2017

Status

approved