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
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
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 31 2003
STATUS
approved