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A093527 Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

%I A093527 #38 Feb 16 2025 08:32:53
%S A093527 1,1,3,2,5,1,7,4,9,5,11,3,13,7,1,8,17,3,19,1,7,11,23,2,25,13,27,1,29,
%T A093527 15,31,16,11,17,5,9,37,19,39,2,41,1,43,11,1,23,47,4,49,25,17,13,53,9,
%U A093527 55,7,19,29,59,5,61,31,21,32,13,1,67,17,23,7,71,2,73,37,5,19,1,13,79
%N A093527 Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.
%H A093527 Stefano Spezia, <a href="/A093527/b093527.txt">Table of n, a(n) for n = 0..10000</a>
%H A093527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiskLinePicking.html">Disk Line Picking</a>
%F A093527 a(k) = Denominator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
%F A093527 From _Paul Barry_, Sep 11 2004: (Start)
%F A093527 a(n) = numerator((n+1)(n+2)/binomial(2(n+1), n+1));
%F A093527 a(n) = numerator(2*binomial(n+2, 2)/binomial(2(n+1), n+1)). (End)
%F A093527 a(n) = numerator((n+1)/C(n+1)). - _Paul Barry_, Nov 17 2004
%F A093527 a(n) = denominator(binomial(2n, n)/n). - _Enrique Pérez Herrero_, Oct 05 2011
%F A093527 a(n) = n/gcd(n,binomial(2n,n)). - _Peter Luschny_, Oct 05 2011
%F A093527 a(n) = denominator((n + 2)*binomial(2*n+3, n+1)/((n + 1)*(2*n + 3))). - _Stefano Spezia_, Aug 06 2022
%e A093527 1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...
%p A093527 A093527 := n -> n / igcd(n,binomial(2*n,n)): # _Peter Luschny_, Oct 05 2011
%t A093527 A093527[n_]:=Denominator[Binomial[2n,n]/n]; Array[A093527,200] (* _Enrique Pérez Herrero_, Oct 05 2011 *)
%Y A093527 Cf. A093070, A093526, A093528, A093529.
%Y A093527 Second column of A098505.
%Y A093527 Cf. A000108.
%K A093527 nonn,frac
%O A093527 0,3
%A A093527 _Eric W. Weisstein_, Mar 30 2004