%I #8 Sep 04 2012 01:25:45
%S 1,1,1,3,2,15,6,105,24,945,120,10395,720,135135,5040,2027025,40320,
%T 34459425,362880,654729075,3628800,13749310575,39916800,316234143225,
%U 479001600,7905853580625,6227020800,213458046676875,87178291200,6190283353629375,1307674368000
%N Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives denominator of C_n.
%C Answer to question of how to extend the sequence 1, 2 r, Pi r^2, 4 Pi r^3 / 3, Pi^2 r^4 / 2, ...
%C Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.
%D Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22; http://www.advancesindifferenceequations.com/content/2012/1/22. - From _N. J. A. Sloane_, Sep 04 2012
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ball.html">Ball</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Four-DimensionalGeometry.html">Four-Dimensional Geometry</a>
%F (n/2)! if n even, n!! if n odd.
%e Sequence of C_n's begins 1, 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, 1/24, 32/945, 1/120, 64/10395, ...
%t f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Denominator[ f[n]], {n, 0, 30} ]
%Y Cf. A072345.
%Y Cf. A001147.
%K nonn,frac
%O 0,4
%A _N. J. A. Sloane_, Jul 31 2002