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

Author

Lorenz H. Menke, Jr., Mar 16 2016

Keywords

• nonn
• frac

Comments

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

Examples

			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.
		

Links

• 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

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.

Programs

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 *)

Extensions

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