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A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

%I A093526 #22 Feb 16 2025 08:32:53
%S A093526 1,1,5,7,42,22,429,715,4862,8398,58786,52003,742900,1337220,646323,
%T A093526 17678835,129644790,79606450,1767263190,328206021,8155422340,
%U A093526 45741281820,343059613650,107492012277,4861946401452,9183676536076
%N A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.
%H A093526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiskLinePicking.html">Disk Line Picking</a>
%F A093526 a(k) = Numerator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
%F A093526 a(n) = denominator((n+1)/C(n+1)). - _Paul Barry_, Nov 17 2004
%F A093526 a(n) = A195686(n+1) / (n+2). - _Peter Luschny_, Oct 06 2011
%e A093526 1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...
%p A093526 A195686 := n -> binomial(2*n,n) / igcd(n,binomial(2*n,n));
%p A093526 A093526 := n -> A195686(n+1)/(n+2); # _Peter Luschny_, Oct 06 2011
%t A093526 a[n_] := Binomial[2(n+1), n+1]/((n+2) GCD[n+1, Binomial[2(n+1), n+1]]);
%t A093526 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 30 2018, after _Peter Luschny_ *)
%o A093526 (PARI) a(n) = denominator((n+1)*(n+2)/binomial(2*n+2, n+1)); \\ _Michel Marcus_, Jul 30 2018
%Y A093526 Cf. A093070, A093527, A093528, A093529.
%Y A093526 Cf. A098512, A000108.
%K A093526 nonn,frac
%O A093526 0,3
%A A093526 _Eric W. Weisstein_, Mar 30 2004