A094941 a(n) is n! times the coefficient of Pi^floor(n/2) in the volume of an n-dimensional unit ball.
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
Offset: 0
Keywords
Examples
The volume of a sphere is (4/3)*Pi*r^3 so a(3) = 3!*4/3 = 8. G.f. = 1 + 2*x + 2*x^2 + 8*x^3 + 12*x^4 + 64*x^5 + 120*x^6 + 768*x^7 + ...
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.
Crossrefs
Cf. A087299.
Programs
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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 *) -
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))}
Formula
E.g.f.: exp(-x^2)*(1 + 2*Integral_{t=0..x} exp(-t^2) dt).
E.g.f. A(x) satisfies A'(x) = 2+2*x*A(x), A(0)=1.
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)*Integral_{x>=0} (x^n*exp(-Pi*x^2/4)). - 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; corrected by Georg Fischer, Jun 02 2021