Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$? I am interested in finding a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here?
Thank you.

In Mathematica Sum[Binomial[n, k] (k - 1)/k, {k, 2, n}] returns:
-1 + 2^n - n HypergeometricPFQ[{1, 1, 1 - n}, {2, 2}, -1]
In traditional form: $-1 + 2^n - n * {}_{3}F_2(1, 1, 1 - n; 2, 2; -1)$
