A266913 Denominator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.

1, 1, 3, 12, 5, 180, 315, 2240, 567, 907200, 51975, 13305600, 289575, 80720640, 212837625, 3487131648000, 2297295, 64023737057280, 14849255421, 28963119144960000, 17717861581875, 140500090972200960000, 16436269594119375, 6204484017332394393600, 40639128117328125

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..25.

R. Chela, Reducible Polynomials, Journal London Math. Soc. 38 (1963), pp 183-188 Eq. 7.

Arturas Dubickas, On the number of reducible polynomials of bounded naive height, Manuscripta Math. 144 (2014), pp 439-456, Eq. 4, 5 & Section 5.

Mathematica Stack Exchange, How to improve or optimize a volume integration over a cuboid

Example

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.

Mathematica

V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],
  Table[x[i], {i, 1, d}] \[Element]
   Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]] (* Lorenz H. Menke, Jr. *)
v[d_] := With[{a = Array[x, d]}, RegionMeasure @ ImplicitRegion[a ∈ Cuboid[-Table[1, d], Table[1, d]] && -1 <= Total[a] <= 1, a]] (* Carl Woll *)
v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)

Crossrefs

The numerator sequence is given by A269067.

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

Sequence in context: A367183 A214401 A009781 * A307027 A304566 A291156

Adjacent sequences: A266910 A266911 A266912 * A266914 A266915 A266916

Keyword

nonn,frac

Author

Lorenz H. Menke, Jr., Mar 16 2016

Extensions

a(11)-a(25) from Lorenz H. Menke, Jr., May 10 2018

Status

approved