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%I #34 May 28 2018 09:16:51
%S 1,1,3,12,5,180,315,2240,567,907200,51975,13305600,289575,80720640,
%T 212837625,3487131648000,2297295,64023737057280,14849255421,
%U 28963119144960000,17717861581875,140500090972200960000,16436269594119375,6204484017332394393600,40639128117328125
%N Denominator 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 numerator sequence is given by A269067.
%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 1,3
%A _Lorenz H. Menke, Jr._, Mar 16 2016
%E a(11)-a(25) from _Lorenz H. Menke, Jr._, May 10 2018
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