A160693 Numerator of the 2n-th raw moment for distribution of distances between two points picked at random in the interior of a unit cube.

1, 1, 11, 211, 187, 5899, 3524083, 28603, 13845523, 35907769, 61759507, 19402663051, 858927089957, 1899225207389, 231506711118139, 31829999554600097, 16467609235037969, 326500242053339, 13505534128658829949, 2757127523531006233, 7914666027558189047

Offset

0,3

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..100

Eric Weisstein's World of Mathematics, Cube Line Picking

Formula

Conjecture: a(n)/A160694(n) = n!*Sum_{i=0..n,j=0..n-i} 1/((i+1)!(j+1)!(n-i-j+1)!(2i+1)(2j+1)(2(n-i-j)+1)). - Jérôme Houdayer, Nov 23 2016

Example

1, 1/2, 11/30, 211/630, 187/525, 5899/13860, 3524083/6306300, 28603/36036, ...

Mathematica

p1[n_] := 4 Pi/(2 n + 3) + 8/(2 n + 5) - 1/(2 n + 6) - 3 Pi/(n + 2); p2[n_] := (6 ((2^(n + 1) (-I Hypergeometric2F1[1/2, n + 1, n + 2, 2] + I Hypergeometric2F1[-1/2, n + 1, n + 2, 2] + n Pi + Pi))/(n + 1) + (I Sqrt[Pi] Gamma[n + 2])/Gamma[n + 5/2]))/(n + 2) + (1 - 2^(n + 1))/(2 n +
  2) + (3 Pi (2^(n + 1) - 1))/(n + 1) + (3 (2^(n + 2) - 1))/(n + 2) + (2^(n + 3) - 1)/(n + 3) - (8 Pi (2^(n + 3/2) - 1))/(2 n + 3) + 4 Beta[1/2, -n - 3/2, 3/2] + 8 Beta[1/2, -n - 5/2, 3/2]; p3[n_] := Integrate[(-x^(2 n + 1)) (x^4 - 8 Sqrt[x^2 - 2] x^2 - 6 Pi x^2 + 6 x^2 - 8 Sqrt[x^2 - 2] - 16 x ArcTan[x Sqrt[x^2 - 2]] + 24 (x^2 + 1) ArcTan[Sqrt[x^2 - 2]] + 16 x ArcCsc[Sqrt[2 - 2/x^2]] - 6 Pi + 5), {x, Sqrt[2], Sqrt[3]}]; a[n_] := p1[n] + p2[n] + p3[n] // FullSimplify // Numerator; Table[an = a[n]; Print[an]; an, {n, 0, 20}] (* Jean-François Alcover, Dec 26 2012, after Eric W. Weisstein, updated Dec 08 2016 *)

Crossrefs

Cf. A160694.

Sequence in context: A157691 A196944 A112386 * A167442 A249974 A124991

Adjacent sequences: A160690 A160691 A160692 * A160694 A160695 A160696

Keyword

nonn,frac

Author

Eric W. Weisstein, May 24 2009

Extensions

Edited and a(9) added by Max Alekseyev, Feb 16 2012

a(10) - a(20) from Robert G. Wilson v, Nov 06 2013

Status

approved