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A269067 Numerator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1. 1

%I #31 May 28 2018 09:17:04

%S 2,3,16,115,88,5887,19328,259723,124952,381773117,41931328,

%T 20646903199,866732192,467168310097,2386873693184,75920439315929441,

%U 97261697912,5278968781483042969,2387693641959232,9093099984535515162569,10872995484706511008,168702835448329388944396777,38650653745373963289088

%N Numerator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.

%C Reference A. Dubickas shows that all the volume integrals are rational with V[d] <= 2^d.

%H R. Chela, <a href="http://jlms.oxfordjournals.org/content/s1-38/1/183.extract">Reducible Polynomials</a>, Journal London Math. Soc. 38 (1963), pp 183-188 Eq. 7.

%H Arturas Dubickas, <a href="http://dx.doi.org/10.1007/s00229-014-0657-y">On the number of reducible polynomials of bounded naive height</a>, Manuscripta Math. 144 (2014), pp 439-456, Eq. 4, 5 & Section 5.

%H Mathematica Stack Exchange, <a href="https://mathematica.stackexchange.com/questions/172777/how-to-improve-or-optimize-a-volume-integration-over-a-cuboid">How to improve or optimize a volume integration over a cuboid</a>

%e For d = 3 the volume is 16/3, for each volume we have V[1] = 2, V[2] = 3, V[3] = 16/3, V[4] = 115/12, V[5] = 88/5, V[6] = 5887/180, V[7] = 19328/315, V[8] = 259723/2240, V[9] = 124952/567, V[10] = 381773117/907200, etc.

%t V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],

%t Table[x[i], {i, 1, d}] \[Element]

%t Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]] (* Lorenz H. Menke, Jr. *)

%t v[d_] := With[{a = Array[x,d]}, RegionMeasure @ ImplicitRegion[ a ∈ Cuboid[-Table[1, d], Table[1, d]] && -1 <= Total[a] <= 1, a]] (* Carl Woll *)

%t v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)

%Y The denominator sequence is given by A266913.

%Y Cf. A199832 The rational coefficient of the leading coefficient of the empirical rows duplicates these volume integrals in sequence. This is not a proof.

%K nonn,frac

%O 1,1

%A _Lorenz H. Menke, Jr._, Feb 19 2016

%E a(11)-a(23) from _Lorenz H. Menke, Jr._, May 10 2018

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