|
| |
|
|
A094941
|
|
n! times coefficient of pi^[n/2] in volume of n-dimensional unit ball.
|
|
0
| |
|
|
1, 2, 2, 8, 12, 64, 120, 768, 1680, 12288, 30240, 245760, 665280, 5898240, 17297280, 165150720, 518918400, 5284823040, 17643225600, 190253629440, 670442572800, 7610145177600, 28158588057600, 334846387814400, 1295295050649600
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| E.g.f. A(x) satisfies A'(x) = 2+2*x*A(x), A(0)=1.
|
|
|
LINKS
| L. Badger, Generating the Measures of n-Balls, Amer. Math. Monthly, 107 (2000), pp. 256-258.
Wikipedia, n-Sphere.
|
|
|
FORMULA
| E.g.f.: exp(-x^2)(1+2*Integral_{t=0..x} exp(-t^2) dt). a(n)=(2n-2)a(n-2), if n>1.
a(n)*a(n+1)=n!*2^(n+1).
a(n)=pi^floor((n+1)/2)*Int(x^n*exp(-pi*x^2/4),x,0,infty). [Paul Barry, Mar 1 2011]
a(n+1) = 2*n*a(n-1); a(2n) = (2n)!/n! = A001813(n) ; a(2n+1) = 2^(2n+1)*n! = 2*A047053(n) = A098560(n) for n>0. - Henry Bottomley, Jun 03 2011
|
|
|
EXAMPLE
| The volume of sphere is 4/3*pi*r^3 so 3!*4/3 = 8 = a(3).
|
|
|
MATHEMATICA
| Table[If[OddQ[n], 2^n ((n - 1)/2)!, 2(n - 1)!/((n/2 - 1)!)], {n, 1, 25}] - Robert A. Russell (russell(AT)post.harvard.edu), May 07 2006
|
|
|
PROG
| (PARI) a(n)=local(A); if(n<0, 0, A=exp(x^2+x*O(x^n)); n!*polcoeff(A*(1+2*intformal(1/A)), n))
|
|
|
CROSSREFS
| Cf. A087299.
Sequence in context: A026537 A089248 A006663 * A002785 A045686 A045677
Adjacent sequences: A094938 A094939 A094940 * A094942 A094943 A094944
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Michael Somos, May 24 2004
|
| |
|
|
