Voozh

A072346

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 denominator of C_n.

1, 1, 1, 3, 2, 15, 6, 105, 24, 945, 120, 10395, 720, 135135, 5040, 2027025, 40320, 34459425, 362880, 654729075, 3628800, 13749310575, 39916800, 316234143225, 479001600, 7905853580625, 6227020800, 213458046676875, 87178291200, 6190283353629375, 1307674368000

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OFFSET

0,4

COMMENTS

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, ...

Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1).

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9, Eq. 17.

Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22; http://www.advancesindifferenceequations.com/content/2012/1/22. - From N. J. A. Sloane, Sep 04 2012

LINKS

Table of n, a(n) for n=0..30.

Eric Weisstein's World of Mathematics, Hypersphere

Eric Weisstein's World of Mathematics, Ball

Eric Weisstein's World of Mathematics, Four-Dimensional Geometry

FORMULA

(n/2)! if n even, n!! if n odd.

EXAMPLE

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, ...

MATHEMATICA

f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Denominator[ f[n]], {n, 0, 30} ]

CROSSREFS