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A286524
Denominator of the volume of the d-th Chern-Vaaler star body.
6
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
S.-J. Chern and J.D. Vaaler, The distribution of values of Mahler's measure, J. Reine. Angew. Math., 540 (2001), 1-47.
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
KEYWORD
nonn,frac
AUTHOR
Jonathan Sondow, May 26 2017
STATUS
approved