A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

1, 1, 4, 6, 32, 60, 384, 840, 6144, 15120, 122880, 332640, 2949120, 8648640, 82575360, 259459200, 2642411520, 8821612800, 95126814720, 335221286400, 3805072588800, 14079294028800, 167423193907200, 647647525324800

Offset

0,3

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..725

Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From N. J. A. Sloane, Sep 04 2012

Eric Weisstein's World of Mathematics, Ball

Formula

a(n) = 2^n*gamma(n/2+1)*Pi^floor(n/2)/Pi^(n/2), n >= 0. - Wolfdieter Lang, Jul 17 2013

0 = a(n)*( 2*a(n+1) - a(n+3) ) + a(n+1)*a(n+2) if n>=0. - Michael Somos, Jan 24 2014

a(n) = 2*n * a(n-2) if n>=2. - Michael Somos, Jan 24 2014

a(2*n) = A047053(n). a(2*n + 1) = A000407(n). - Michael Somos, Jan 03 2015

Example

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

Mathematica

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

Prog

(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 */
(PARI) {a(n) = if( n<2, n>-1, 2*n * a(n-2))}; /* Michael Somos, Jan 24 2014 */
(PARI) {a(n) = if( n<0, 0, if( n%2, n! / (n\2)!, 2^n * (n\2)!))}; /* Michael Somos, Jan 03 2015 */

Crossrefs

Cf. A000407, A047053, A072345, A072346.

Sequence in context: A068402 A078250 A239224 * A229712 A331513 A164127

Adjacent sequences: A087296 A087297 A087298 * A087300 A087301 A087302

Keyword

nonn,easy

Author

Eric W. Weisstein, Aug 31 2003

Status

approved