%I #38 Sep 25 2024 23:26:45
%S 1,2,1,4,1,8,1,16,1,32,1,64,1,128,1,256,1,512,1,1024,1,2048,1,4096,1,
%T 8192,1,16384,1,32768,1,65536,1,131072,1,262144,1,524288,1,1048576,1,
%U 2097152,1,4194304,1,8388608,1,16777216,1,33554432,1,67108864,1,134217728,1,268435456
%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 numerator 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). - see A072478/A072479.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.
%H N. Cakic, D. Letic, and B. Davidovic, <a href="https://doi.org/10.1155/2010/364292">The Hyperspherical functions of a derivative</a>, Abstr. Appl. Anal. (2010).
%H Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, <a href="https://doi.org/10.1186/1687-1847-2012-22">Orthogonal and diagonal dimension fluxes of hyperspherical function</a>, Advances in Difference Equations 2012, 2012:22. - 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>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).
%F 1 if n even, 2^((n+1)/2) if n odd.
%F a(n) = 3*a(n-2)-2*a(n-4). G.f.: (1+2*x-2*x^2-2*x^3)/((1-x)*(1+x)*(1-2*x^2)). [_Colin Barker_, Sep 04 2012]
%F a(n) = 2^((n+1)/2)*(1-(-1)^n)/2+(1+(-1)^n)/2. - _Wesley Ivan Hurt_, Jan 10 2017
%F E.g.f.: sqrt(2)*sinh(sqrt(2)*x) + cosh(x). - _Ilya Gutkovskiy_, Mar 16 2017
%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, ...
%p seq(seq(k^n, k=1..2), n=1..28); # _Zerinvary Lajos_, Jun 29 2007
%t f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 55} ]
%t Riffle[2^Range[30],1,{1,-1,2}] (* or *) LinearRecurrence[{0,3,0,-2},{1,2,1,4},60] (* _Harvey P. Dale_, Oct 16 2013 *)
%t CoefficientList[ Series[(-2x^3 - 2x^2 + 2x + 1)/(2x^4 - 3x^2 + 1), {x, 0, 56}],
%t x] (* _Robert G. Wilson v_, Jul 31 2018 *)
%o (Magma) [2^((n+1) div 2)*(1-(-1)^n)/2+(1+(-1)^n)/2 : n in [0..100]]; // _Wesley Ivan Hurt_, Jan 10 2017
%o (Python)
%o def A072345(n): return 2<<(n>>1) if n&1 else 1 # _Chai Wah Wu_, Sep 24 2024
%Y Cf. A072346, A072478, A072479.
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_, Jul 31 2002