This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A266913 #37 Jun 27 2025 02:25:18
%S A266913 1,1,3,12,5,180,315,2240,567,907200,51975,13305600,289575,80720640,
%T A266913 212837625,3487131648000,2297295,64023737057280,14849255421,
%U A266913 28963119144960000,17717861581875,140500090972200960000,16436269594119375,6204484017332394393600,40639128117328125
%N A266913 Denominator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.
%C A266913 Reference A. Dubickas shows that all the volume integrals are rational with V[d] <= 2^d.
%H A266913 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 A266913 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 A266913 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 A266913 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 A266913 V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],
%t A266913 Table[x[i], {i, 1, d}] \[Element]
%t A266913 Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]]] (* _Lorenz H. Menke, Jr._ *)
%t A266913 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 A266913 v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)
%Y A266913 The numerator sequence is given by A269067.
%Y A266913 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 A266913 nonn,frac
%O A266913 1,3
%A A266913 _Lorenz H. Menke, Jr._, Mar 16 2016
%E A266913 a(11)-a(25) from _Lorenz H. Menke, Jr._, May 10 2018