A286524 Denominator of the volume of the d-th Chern-Vaaler star body.

1, 1, 1, 9, 27, 225, 30375, 7441875, 1071875, 1181472075, 232602314765625, 7296820763203125, 1228777319702643046875, 407740293448065703125, 12646770836979187723815, 13904872587870848957579157123046875, 89477855102948913042021876086806640625, 3865644245069923858264226752294921875, 18203690021177687874093952515006818038631103515625

Offset

0,4

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

Jinyuan Wang, 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

Denominator 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(A286523(n) / a(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[ Denominator[v[d]], {d, 0, 18}]

Prog

(PARI) a(n) = denominator(2^(n+1)*(n+1)^((n-1)\2)*prod(k=1, (n-1)\2, (2*k)^(n-2*k)/(2*k+1)^(n+1-2*k))); \\ Jinyuan Wang, Mar 05 2020

Crossrefs

Cf. A286522, A286523, A288756, A288757, A288758.

Sequence in context: A340235 A217640 A227474 * A057901 A320676 A317027

Adjacent sequences: A286521 A286522 A286523 * A286525 A286526 A286527

Keyword

nonn,frac

Author

Jonathan Sondow, May 26 2017

Status

approved