A288756 - OEIS

%I #14 Jun 16 2017 02:51:37

%S 2,4,16,64,1024,16384,524288,16777216,4294967296,1099511627776,

%T 562949953421312,288230376151711744,590295810358705651712,

%U 1208925819614629174706176,4951760157141521099596496896,20282409603651670423947251286016,1329227995784915872903807060280344576

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

%C 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.

%H S.-J. Chern and J.D. Vaaler, <a href="https://doi.org/10.1515/crll.2001.084">The distribution of values of Mahler's measure</a>, J. Reine. Angew. Math., 540 (2001), 1-47.

%H R, Grizzard and J. Gunther, <a href="https://arxiv.org/abs/1609.08720">Slicing the stars: counting algebraic numbers, integers, and units by degree and height</a>, arXiv:1609.08720 [math.NT] 2016.

%F 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).

%F Floor(a(n)/A288757(n)) = A288758(n).

%e 2, 4, 16/3, 64/9, 1024/135, 16384/2025, 524288/70875, 16777216/2480625, 4294967296/781396875, 1099511627776/246140015625, ...

%t 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}]

%Y Cf. A286522, A286523, A286524, A288757, A288758.

%K nonn,frac

%O 1,1

%A _Jonathan Sondow_, Jun 15 2017