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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. 2
1, 1, 11, 211, 187, 5899, 3524083, 28603, 13845523, 35907769, 61759507, 19402663051, 858927089957, 1899225207389, 231506711118139, 31829999554600097, 16467609235037969, 326500242053339, 13505534128658829949, 2757127523531006233, 7914666027558189047 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)