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%I #31 Jun 08 2021 07:27:15
%S 1,2,2,8,12,64,120,768,1680,12288,30240,245760,665280,5898240,
%T 17297280,165150720,518918400,5284823040,17643225600,190253629440,
%U 670442572800,7610145177600,28158588057600,334846387814400,1295295050649600
%N a(n) is n! times the coefficient of Pi^floor(n/2) in the volume of an n-dimensional unit ball.
%H G. C. Greubel, <a href="/A094941/b094941.txt">Table of n, a(n) for n = 0..730</a>
%H L. Badger, <a href="http://www.jstor.org/stable/2589319">Generating the Measures of n-Balls</a>, Amer. Math. Monthly, 107 (2000), pp. 256-258.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypersphere">n-Sphere</a>.
%F E.g.f.: exp(-x^2)*(1 + 2*Integral_{t=0..x} exp(-t^2) dt).
%F E.g.f. A(x) satisfies A'(x) = 2+2*x*A(x), A(0)=1.
%F a(n) = (2*n - 2) * a(n-2), if n>1.
%F a(n) * a(n+1) = n! * 2^(n+1).
%F a(n) = Pi^floor((n+1)/2)*Integral_{x>=0} (x^n*exp(-Pi*x^2/4)). - _Paul Barry_, Mar 01 2011
%F 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
%F 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
%e The volume of a sphere is (4/3)*Pi*r^3 so a(3) = 3!*4/3 = 8.
%e G.f. = 1 + 2*x + 2*x^2 + 8*x^3 + 12*x^4 + 64*x^5 + 120*x^6 + 768*x^7 + ...
%t 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 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, n}]] (* _Michael Somos_, Jan 24 2014 *)
%t 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 *)
%o (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))}
%Y Cf. A087299.
%K nonn
%O 0,2
%A _Michael Somos_, May 24 2004
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