C(2n, n) ~=  4^n / sqrt( Rm(n) )

Rm being a rational function of max. degree m+1 (m>0):

  Rm(n) = (pi n^(m+1) + sum(i=0,m) Pi n^i) / (n^m + sum(i=0,m-1) Qi
n^i)

and where the 2m+1 coefficients are defined by simply requiring exact
results for 2m>=n>=0; note that C(0,0)=1 implies that Q0=P0.

Example: m=1
------------

  C(2n, n)  ~=  4^n / sqrt( (pi n^2 + P1 n + P0)/ (n + Q0) )

  P1 = 53 pi - 164
  P0 = 18 pi -  56
  Q0 = P0

with a |rel. error| < 5.0*10^-5 for n>=0.

Error profile:

  n   rel. error        n   rel. error
----------------------------------------
  0    0               12    4.023*10^-5
  1    0               13    3.845*10^-5
  2    0               14    3.678*10^-5
  3    2.923*10^-5     15    3.521*10^-5
  4    4.312*10^-5     16    3.375*10^-5
  5    4.850*10^-5     17    3.239*10^-5
  6    4.983*10^-5     18    3.112*10^-5
  7    4.927*10^-5     19    2.994*10^-5
  8    4.782*10^-5     20    2.884*10^-5
  9    4.600*10^-5     30    2.096*10^-5
 10    4.406*10^-5     40    1.641*10^-5
 11    4.211*10^-5     50    1.346*10^-5

Example: m=2
------------

  C(2n, n)  ~=  4^n / sqrt( (pi n^3 + P2 n^2  + P1 n + P0)/ (n^2 + Q1
n + Q0) )

  P2 =  36928 - 11753 pi
  P1 =  30784 -  9798 pi
  P0 =  6912  -  2200 pi
  Q1 =  11743 -  7475 pi/2
  Q0 =  P0

with a |rel. error| < 2.5*10^-7 for n>=0.

Error profile:

  n   rel. error        n   rel. error
----------------------------------------
  0    0               12   -2.323*10^-7
  1    0               13   -2.390*10^-7
  2    0               14   -2.431*10^-7
  3    0               15   -2.453*10^-7
  4    0               16   -2.459*10^-7
  5   -3.706*10^-8     17   -2.454*10^-7
  6   -8.367*10^-8     18   -2.440*10^-7
  7   -1.263*10^-7     19   -2.420*10^-7
  8   -1.611*10^-7     20   -2.394*10^-7
  9   -1.879*10^-7     30   -2.053*10^-7
 10   -2.078*10^-7     40   -1.740*10^-7
 11   -2.222*10^-7     50   -1.496*10^-7

Best regards,