David W. Cantrell:
> I'm not running the most recent version of Mathematica and so I
> checked with Wolfram Alpha. Asked to compute
>
> Sum[(1/i)*Binomial[2n-i-1,i-1],{i,1,n}]
>
> it returns
>
> (-1 + Hypergeometric2F1[1/2 - n, -n, 1 - 2 n, -4])/(2 n)
Mathematica 7.0 returns
(-1 + Hypergeometric2F1[-n, 1/2 - n, 1 - 2 n, -4])/(2 n)