I'm interested in finding a nice closed form expression for the sum $\sum_{k=1}^n\frac{1}{k}\binom{n}{k}$. I've tried using the Binomial Theorem to get
\begin{align*}
\sum_{k=1}^n\frac{1}{k}\binom{n}{k}x^k & =\int_0^1\frac{(1+x)^n-1}{x} \, dx\\
&=\int_1^2 (1+u+\cdots+u^{n-1}) \, du
\end{align*}
using the substitution $u=1+x$ but I can't quite to simplify this integral either. I have also not been able to come up with a combinatorial approach, which may not exist since the summation and its terms are in general not integers. Any help in evaluating this sum would be appreciated, thanks!
Answer
In the question, the problem is in "nice closed form expression" since
$$\sum_{k=1}^n\frac{1}{k}\binom{n}{k}x^k=n x \, _3F_2(1,1,1-n;2,2;-x)$$ where appears an hypergoemetric function.
So, let us forget the $x$ and compute for a few values of $n$
$$S_n=\sum_{k=1}^n\frac{1}{k}\binom{n}{k}$$
$$\left\{1,\frac{5}{2},\frac{29}{6},\frac{103}{12},\frac{887}{60},\frac{1517}{60},\frac{18239}{420}\right\}$$ which are
$$\left\{1,\frac{5}{2},\frac{29}{6},\frac{206}{24},\frac{1774}{120},\frac{18204}{720},\frac{218868}{5040}\right\}$$ The denominators are clearly $n!$ and the numerators corresponds to sequence A103213 in OEIS.
As you will see in the link is that, for large $n$
$$S_n\approx \frac{2^{n+1}} n$$ It is also given that
$$S_n=-H_n-\Re(B_2(n+1,0))$$ where appear the harmonic number and the real part of the incomplete beta function.
Update
Concerning the asymptotic behavior, it seems that it could be slightly improved using
$$S_n\approx 2^{n+1} n^{\frac{1}{4 n}-1}$$
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