A binomial sum

[Karol PENSON]

 Maple does it too:

> sum((-1)^m*binomial(n+m,m),m=0..n);

   (-n - 1)       (n + 1)
  2         - (-1)           * binomial(2 n + 1, n + 1)* hypergeom([1, 2 n 
+ 2], [n + 2], -1)

> seq(round(evalf(2^(-n-1)-(-1)^(n+1)*binomial(2*n+1,n+1)*hypergeom([1, 
2*n+2],[n+2],-1))),n=1..12);

  -1, 4, -13, 46, -166, 610, -2269, 8518, -32206, 122464, -467842,

        1794196

Cheers, Karol A. Penson




On Mon, 19 Apr 2004, Eric W. Weisstein wrote:

> Date: Mon, 19 Apr 2004 15:30:31 -0500 (CDT)
> From: Eric W. Weisstein <eww at wolfram.com>
> To: Henry Gould <gould at math.wvu.edu>
> Cc: pin <pin at myway.com>, Sequence Fans Mailing List <seqfan at ext.jussieu.fr>
> Subject: Re: A binomial sum
> 
> On Mon, 19 Apr 2004, Henry Gould wrote:
> 
> > Well the original question actually comes down to this:
> > 
> > Can we find a closed formula for the binomial coefficient summation
> > sum(m=0,n,(-1)^m*binomial(n+m,m))?
> > The answer is NO.
> 
> Doesn't the following qualify?
> 
> Mathematica 5.0 for Mac OS X
> Copyright 1988-2003 Wolfram Research, Inc.
>  -- Terminal graphics initialized -- 
> 
> In[1]:= FullSimplify[Sum[(-1)^m Binomial[m + n, m], {m, 0, n}]]//InputForm
> 
> Out[1]//InputForm= 
> (2^(-n) + ((-1)^n*Gamma[3 + 2*n]*Hypergeometric2F1Regularized[1, 2 + 2*n, 
>      2 + n, -1])/Gamma[2 + n])/2
> 
> In[2]:= Table[{Sum[(-1)^m Binomial[m + n, m], {m, 0, n}],
>         (2^(-n) + ((-1)^n*Gamma[3 + 2*n]*Hypergeometric2F1Regularized[1, 2 + 2*n, 
>      2 + n, -1])/Gamma[2 + n])/2},
>         {n,0,10}]
> 
> Out[2]= {{1, 1}, {-1, -1}, {4, 4}, {-13, -13}, {46, 46}, {-166, -166}, 
> >    {610, 610}, {-2269, -2269}, {8518, 8518}, {-32206, -32206}, 
> >    {122464, 122464}}
> 
> In[3]:= Subtract@@@%
> 
> Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
> 
> Cheers,
> -Eric
> 

-- 
_________________________________________________________________________
Karol A. PENSON
Universite Paris 6              |  Internet : penson at lptl.jussieu.fr.
Lab. Physique Theorique des     |
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