A binomial sum

[Eric W. Weisstein]

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