login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A094941 n! times coefficient of Pi^[n/2] in volume of n-dimensional unit ball. 1
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; text; internal format)
OFFSET

0,2

COMMENTS

E.g.f. A(x) satisfies A'(x) = 2+2*x*A(x), A(0)=1.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..730

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) = (2*n - 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 01 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

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

EXAMPLE

The volume of sphere is 4/3*Pi*r^3 so 3!*4/3 = 8 = a(3).

G.f. = 1 + 2*x + 2*x^2 + 8*x^3 + 12*x^4 + 64*x^5 + 120*x^6 + 768*x^7 + ...

MATHEMATICA

Join[{1}, Table[If[OddQ[n], 2^n ((n - 1)/2)!, 2(n - 1)!/((n/2 - 1)!)], {n, 1, 25}]] (* Robert A. Russell, May 07 2006 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, n}]] (* Michael Somos, Jan 24 2014 *)

a[ n_] := If[ n < 1, Boole[n == 0], If[ OddQ[n], 2^n ((n - 1)/2)!, 2 (n - 1)! / ((n/2 - 1)!)]] (* Michael Somos, Jan 24 2014 *)

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 A301603 A292038

Adjacent sequences:  A094938 A094939 A094940 * A094942 A094943 A094944

KEYWORD

nonn

AUTHOR

Michael Somos, May 24 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 27 15:14 EST 2020. Contains 331295 sequences. (Running on oeis4.)