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%I #44 Apr 21 2021 04:38:43
%S 1,1,11,211,187,5899,3524083,28603,13845523,35907769,61759507,
%T 19402663051,858927089957,1899225207389,231506711118139,
%U 31829999554600097,16467609235037969,326500242053339,13505534128658829949,2757127523531006233,7914666027558189047
%N Numerator of the 2n-th raw moment for distribution of distances between two points picked at random in the interior of a unit cube.
%H Robert G. Wilson v, <a href="/A160693/b160693.txt">Table of n, a(n) for n = 0..100</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubeLinePicking.html">Cube Line Picking</a>
%F 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
%e 1, 1/2, 11/30, 211/630, 187/525, 5899/13860, 3524083/6306300, 28603/36036, ...
%t 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 +
%t 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 *)
%Y Cf. A160694.
%K nonn,frac
%O 0,3
%A _Eric W. Weisstein_, May 24 2009
%E Edited and a(9) added by _Max Alekseyev_, Feb 16 2012
%E a(10) - a(20) from _Robert G. Wilson v_, Nov 06 2013
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