A087299 - OEIS

%I #43 Feb 16 2025 08:32:51

%S 1,1,4,6,32,60,384,840,6144,15120,122880,332640,2949120,8648640,

%T 82575360,259459200,2642411520,8821612800,95126814720,335221286400,

%U 3805072588800,14079294028800,167423193907200,647647525324800

%N Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

%D N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292

%H G. C. Greubel, <a href="/A087299/b087299.txt">Table of n, a(n) for n = 0..725</a>

%H Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, <a href="http://dx.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="https://mathworld.wolfram.com/Ball.html">Ball</a>

%F a(n) = 2^n*gamma(n/2+1)*Pi^floor(n/2)/Pi^(n/2), n >= 0. - _Wolfdieter Lang_, Jul 17 2013

%F 0 = a(n)*( 2*a(n+1) - a(n+3) ) + a(n+1)*a(n+2) if n>=0. - _Michael Somos_, Jan 24 2014

%F a(n) = 2*n * a(n-2) if n>=2. - _Michael Somos_, Jan 24 2014

%F a(2*n) = A047053(n). a(2*n + 1) = A000407(n). - _Michael Somos_, Jan 03 2015

%e The volume of sphere (3-ball) is 4/3*Pi*r^3 and circumscribing 3-cube is 2^3*r^3 so ratio is Pi/6 and a(3)=6.

%e G.f. =  1 + x + 4*x^2 + 6*x^3 + 32*x^4 + 60*x^5 + 384*x^6 + 840*x^7 + ...

%t a[ n_] := If[ n < 0, 0, With[ {m = n + 1}, m! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, m}] / 2]]; (* _Michael Somos_, Jan 24 2014 *)

%t Table[2^n*Gamma[n/2 + 1]*Pi^Floor[n/2]/Pi^(n/2), {n,0,50}] (* _G. C. Greubel_, Jan 28 2017 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, n++; A = exp(x^2 + x * O(x^n)); n! * polcoeff( A * (1 + 2 * intformal( 1/A )), n) / 2)}; /* _Michael Somos_, May 25 2004 */

%o (PARI) {a(n) = if( n<2, n>-1, 2*n * a(n-2))}; /* _Michael Somos_, Jan 24 2014 */

%o (PARI) {a(n) = if( n<0, 0, if( n%2, n! / (n\2)!, 2^n * (n\2)!))}; /* _Michael Somos_, Jan 03 2015 */

%Y Cf. A000407, A047053, A072345, A072346.

%K nonn,easy

%O 0,3

%A _Eric W. Weisstein_, Aug 31 2003