A093585 Numerators of even raw moments in the distribution of a triangle picked at random from points on the circumference of a unit circle.

1, 3, 45, 105, 17325, 189189, 1072071, 6235515, 4732755885, 56968357875, 1387749197835, 1066617152055, 211545735157575, 2639940564777075, 33133947904855125, 208964764786619655, 677672732203007541165, 8617464674207794857375, 219904931871450765064125, 703573951001456048889375, 144457803619618955957966475

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Circle Triangle Picking

Formula

Conjecture: the moments are sqrt(3)*Gamma(n+2/3)*Gamma(n+1/3)*(27/16)^n/(2*Pi*(n!)^2). - Robert Israel, Jan 01 2018

Example

1, 3/(2*Pi), 3/8, 35/(32*Pi), 45/128, 3003/(2560*Pi), ...

Maple

M := n -> int(int( ( 2*sin(u/2)*sin(v/2)*sin((u-v)/2) )^(2*n), u=0..Pi), v=0..2*Pi) / 2 / Pi^2; # Max Alekseyev, Jun 18 2011

Mathematica

a[n_] := Integrate[ Integrate[ (2*Sin[u/2]*Sin[v/2]* Sin[(u-v)/2])^(2*n), {u, 0, Pi}], {v, 0, 2*Pi}]/(2*Pi^2) // Numerator; Table[ Print[an = a[n]]; an, {n, 0, 20}] (* Jean-François Alcover, Nov 09 2012, after Max Alekseyev *)

Crossrefs

Cf. A093583, A093584, A093586.

Sequence in context: A075320 A071968 A170921 * A062270 A069955 A289193

Adjacent sequences: A093582 A093583 A093584 * A093586 A093587 A093588

Keyword

nonn,frac

Author

Eric W. Weisstein, Apr 01 2004

Extensions

More terms from Max Alekseyev, Jun 18 2011

Status

approved