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
Examples
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 + ...
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
Programs
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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 *) -
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 */
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