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%I #16 Jan 01 2018 17:52:06
%S 1,3,45,105,17325,189189,1072071,6235515,4732755885,56968357875,
%T 1387749197835,1066617152055,211545735157575,2639940564777075,
%U 33133947904855125,208964764786619655,677672732203007541165,8617464674207794857375,219904931871450765064125,703573951001456048889375,144457803619618955957966475
%N Numerators of even raw moments in the distribution of a triangle picked at random from points on the circumference of a unit circle.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircleTrianglePicking.html">Circle Triangle Picking</a>
%F 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
%e 1, 3/(2*Pi), 3/8, 35/(32*Pi), 45/128, 3003/(2560*Pi), ...
%p 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
%t 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_ *)
%Y Cf. A093583, A093584, A093586.
%K nonn,frac
%O 0,2
%A _Eric W. Weisstein_, Apr 01 2004
%E More terms from _Max Alekseyev_, Jun 18 2011
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