> "sa...@kt-algorithms.com" <sa...@kt-algorithms.com> wrote:
> > An interesting type of approximation to the central binomial
> > coefficient is of the form:

> >   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.

> Hello,Knud!

> Yes, your form of approximation is nice.

> FWIW, here are simple upper and lower bounds for C(2n, n).

> Upper bound: 4^n / sqrt(pi(n + 1/4))                             (Ub)

> Lower bound: Ub * (1 - 1/(8(n + 1/4))^2)                         (Lb)

> Your approximations are precise for small n, while the relative errors of
> my bounds at n = 0 are roughly 13% and -15%, resp. But for large n, those
> bounds make reasonably good approximations. For example, at n = 50, their
> relative errors are about 6*10^-6 and -4*10^-10, resp.

> Best regards,
> David W. Cantrell

> > 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,

> >KnudThomsen- Hide quoted text -

> - Show quoted text -