Evaluating the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer

How to evaluate the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer?

For $k=1$, the series does not converge.
When $k=2$, I can prove that:
$$\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}}=\frac{\pi}{3\sqrt{3}}$$
Usually, this can be proven by differentiating $\sum_{n=1}^\infty \frac{x^{2n}}{n^2 \binom{2n}{n}}=2(\arcsin{\frac{x}{2}})^2$, but I have an alternative proof.


Using the result of:
$$\int_0^\infty \frac{x^ndx}{(x+1)^{y+n+1}}=\frac{1}{y \binom{y+n}{n}} \tag1$$
, which can be easily proved.
I can substitute $y=n$ to obtain
$$\int_0^\infty \frac{x^ndx}{(x+1)^{2n+1}}=\frac{1}{n \binom{2n}{n}}$$
$$\sum_{n=1}^\infty\int_0^\infty \frac{x^ndx}{(x+1)^{2n+1}}=\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}}$$
Therefore,
\begin{align}
\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}} & = \int_0^\infty \frac{xdx}{(x+1)(x^2+x+1)} \\
& = \lim_{L\to \infty} \frac{1}{2}\ln(x^2+x+1)-\ln(x+1)+\frac{\tan^{-1}(\frac{2x+1}{\sqrt{3}})}{\sqrt{3}}\large{|_0^L} \\
&= \frac{\pi}{3\sqrt{3}}
\end{align}


Now for $k=3$, I tried to substitute $y=2n$ into $(1)$:
$$\int_0^\infty \frac{x^ndx}{(x+1)^{3n+1}}=\frac{1}{2n \binom{3n}{n}}$$
$$\sum_{n=1}^\infty\int_0^\infty \frac{x^ndx}{(x+1)^{3n+1}}=\sum_{n=1}^\infty\frac{1}{2n \binom{3n}{n}}$$
So we can have
$$\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}=\int_0^\infty\frac{2xdx}{(x+1)(x^3+3x^2+2x+1)}$$
However, by partial fraction $$\frac{2xdx}{(x+1)(x^3+3x^2+2x+1)}=-\frac{2}{1+x}+\frac{2x^2+4x+2}{x^3+3x^2+2x+1}$$
The left part does not seem to converge.


Feeling frustrated, Wolfram Alpha plays its part. It spits out these results:
$$\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}=\frac{1}{3}{}_3F_2\left(\left.\begin{array}{c} 1,1,\frac{3}{2}\\ \frac{4}{3}, \frac{5}{3} \end{array}\right| \frac{4}{27}\right)$$
$$\sum_{n=1}^\infty\frac{1}{n \binom{4n}{n}}=\frac{1}{4}{}_4F_3\left(\left.\begin{array}{c} 1,1,\frac{4}{3},\frac{5}{3}\\ \frac{5}{4}, \frac{6}{4}, \frac{7}{4} \end{array}\right| \frac{27}{256}\right)$$
However, I am not very familiar with hypergeometric function.
The pattern suggests that $$\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}}=\frac{1}{2}{}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)=\frac{\pi}{3\sqrt{3}}$$
Thus, $${}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)=\frac{2\pi}{3\sqrt{3}}$$


Arriving these results, I have the following questions:

How can ${}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)$ be expressed into this simple elementary form?
How can we arrive to the result for $\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$ given by Wolfram Alpha?
Ultimately, can we evaluate $\sum_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ for all integers k $\ge$ $2$?

Solutions Collecting From Web of "Evaluating the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer"

The exact value of $\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$

We have already found that $$\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}=\int_0^\infty\frac{2xdx}{(x+1)(x^3+3x^2+2x+1)}$$ if we can evaluate the integral, we are done.
First of all, we need to find the roots $\{l,m,n\}$ of $x^3+3x^2+2x+1$ by Cardano’s method. We can obtain: $$l=-\frac{\sqrt[3]{\frac{1}{2} \left(9-\sqrt{69}\right)}}{3^{2/3}}-\sqrt[3]{\frac{2}{3 \left(9-\sqrt{69}\right)}}-1$$ $$m=\frac{\left(1+i \sqrt{3}\right) \sqrt[3]{\frac{1}{2} \left(9-\sqrt{69}\right)}}{2\ 3^{2/3}}+\frac{1-i \sqrt{3}}{2^{2/3} \sqrt[3]{3 \left(9-\sqrt{69}\right)}}-1$$ $$n=\frac{\left(1-i \sqrt{3}\right) \sqrt[3]{\frac{1}{2} \left(9-\sqrt{69}\right)}}{2\ 3^{2/3}}+\frac{1+i \sqrt{3}}{2^{2/3} \sqrt[3]{3 \left(9-\sqrt{69}\right)}}-1$$ Now we need to break down the integrand. Let $a=-l$, $b=-(m+n)$ and $c=mn$ . Then, $x^3+3x^2+2x+1=(x+a)(x^2+bx+c)$ .
By partial fraction,
\begin{align}
\frac{2x}{(x+1)(x^3+3x^2+2x+1)} & = \frac{\alpha}{1+x}+\frac{\beta}{a+x}+\frac{\gamma+\delta x}{x^2+bx+c} \\
& = \frac{\alpha}{1+x}+\frac{\beta}{a+x}+\frac{2x\delta+b\delta}{2x^2+2bx+2c}+\frac{-b\delta+2\gamma}{2x^2+2bx+2c}
\end{align}
where $\alpha=\frac{2}{(1-b+c)(1-a)}$, $\beta=\frac{2a}{(a^2-ab+c)(1-a)}$, $\gamma=\frac{2(1+a-b)c}{(a^2-ab+c)(1-b+c)}$ and $\delta=\frac{2a-2c}{(a^2-ab+c)(1-b+c)}$.
Now we can integrate $\frac{2x}{(x+1)(x^3+3x^2+2x+1)}$ . Since $\int_0^\infty \frac{dx}{x^2+2ax+b}
=\frac{1}{\sqrt{b-a^2}}\left(\frac{\pi}{2}-\arctan\left(\frac{a}{\sqrt{b-a^2}}\right)\!\right)$, we can have
\begin{align}
\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}} & = \int_0^\infty\frac{2xdx}{(x+1)(x^3+3x^2+2x+1)} \\
& = \int_0^\infty (\frac{\alpha}{1+x}+\frac{\beta}{a+x}+\frac{2x\delta+b\delta}{2x^2+2bx+2c}+\frac{-b\delta+2\gamma}{2x^2+2bx+2c})dx
\end{align}

$$\sum_{n=1}^\infty\frac{1}{n \binom{3n}{n}}= -\beta\ln{a}-\frac{1}{2}\delta\ln{c}+(-\arctan(\frac{b}{\sqrt{-b^2+4c}})+\frac{\pi}{2})(\frac{2\gamma-b\delta}{\sqrt{-b^2+4c}})$$


The exact value of ${}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)$

Since $$\frac{(\beta)_k}{(\gamma)_k}=\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}\int_0^1 t^{\beta-1+k} (1-t)^{\gamma-\beta-1}dt$$ for non-negative integer k, and by the binomial theorem, $$\sum_{k=0}^\infty \frac{(\alpha)_k}{k!}(zt)^k=(1-zt)^{-\alpha}$$ where $0 \le t \le 1$, $-1 \lt z \lt 1$, we have: $${}_2F_1\left(\left.\begin{array}{c} \alpha,\beta\\ \gamma \end{array}\right| z\right)=\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}\int_0^1 t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-zt)^{-\alpha}dt$$ So, $${}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)=\frac{\Gamma(\frac{3}{2})}{\Gamma(\frac{1}{2})}\int_0^1 \frac{dt}{\sqrt{1-t}(1-t/4)}$$ Since $\Gamma(z+1)=z\Gamma(z)$ and the integral can be easily calculated, we finally obtain

$${}_2F_1\left(\left.\begin{array}{c} 1,1\\ \frac{3}{2} \end{array}\right| \frac{1}{4}\right)=\frac{2\pi}{3\sqrt{3}}$$

I’m late for this party but the appearance of the plastic constant’s minpoly, or $x^3-x-1=0$, got me interested.

The binomial sum can be expressed as a concise finite sum of logarithms. For $k>1$,

$$\sum_{n=1}^\infty\frac1{n\binom{kn}n} =\sum_{n=1}^k \frac{\ln(1-x_n)}{k-(k-1)x_n}=\int_1^\infty\frac1{x(x^k-x+1)}$$

and where the $x_n$, naturally enough, are the roots of $x^k-x+1=0$. Example, for $k=3$,
$$A=\sum_{n=1}^\infty\frac1{n\binom{3n}n} = \frac{\ln(1-x_1)}{3-2x_1}+ \frac{\ln(1-x_2)}{3-2x_2}+ \frac{\ln(1-x_3)}{3-2x_3}=0.371216\dots$$
and the $x_n$ are the three roots of $x^3-x+1=0$, one of which is the negated plastic constant $x\approx-1.32472$. It was pointed out that equivalently,
$$3A ={_3F_2}\left(\frac32,1,1;\ \frac43,\frac53;\ \frac4{27}\right)$$
Interestingly, the plastic constant also appears in a similar generalized hypergeometric function,
$$2B ={_3F_2}\left(\frac12,1,1;\ \frac43,\frac53;\ \frac4{27}\right)$$
discussed by Reshetnikov in this post.

For the most general case,

$\sum\limits_{n=1}^\infty\dfrac{1}{n\binom{kn}{n}}$

$=\sum\limits_{n=1}^\infty\dfrac{\Gamma(n+1)\Gamma((k-1)n+1)}{n\Gamma(kn+1)}$

$=\sum\limits_{n=1}^\infty\dfrac{\Gamma(n)\Gamma((k-1)n+1)}{\Gamma(kn+1)}$

$=\sum\limits_{n=0}^\infty\dfrac{\Gamma(n+1)\Gamma((k-1)n+k)}{\Gamma(kn+k+1)}$

$=~_3\Psi_1\left[\begin{matrix}(1,1)~~(1,1)~~(k,k-1)\\(k+1,k)\end{matrix};1\right]$ (according to http://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function)

Similar techniques are discussed by Borwein and Girgensohn in http://dx.doi.org/10.1007/s00010-005-2774-x , entitled “Evaluations of binomial series” of 2005.

Again, this is not going to be a full answer. However since my approach is slightly different from what was presented here so far and since my result is similar to the result given by Tito Piezas III I will post my answer.
\begin{eqnarray}
&&S_k:=\sum\limits_{n=1}^\infty \frac{1}{n \binom{ k n}{n}}=\\
&&\sum\limits_{n=1}^\infty \frac{k n+1}{n} \cdot \int\limits_0^1 t^n \cdot (1-t)^{(k-1) n} dt=\\
&&\int\limits_0^1 \left[ k \frac{t\cdot (1-t)^{k-1}}{1-t \cdot (1-t)^{k-1}} – \log(1-t \cdot (1-t)^{k-1})\right] dt=\\
&&-\int\limits_0^1 t \cdot \frac{ (1-t)^{k-3} (k t-1) \left(t (1-t)^k+k (t-1)+t-1\right)}{\left(t (1-t)^{k-1}-1\right)^2} dt=\\
&&\int\limits_0^1 \frac{(u-1) (k (u-1)+1) u^{k-2} \left(-u^{k-1}+u^k+k+1\right)}{\left(-u^{k-1}+u^k+1\right)^2}du=\\
&&\int\limits_1^\infty \left(\frac{\left(k^2-1\right) (v-1)}{v \left(v^k-v+1\right)}-\frac{k (k+1) (v-1)}{v^2 \left(v^k-v+1\right)}-\frac{k^2 (v-1)^2}{v^2 \left(v^k-v+1\right)^2}+\frac{(k-1) k (v-1)^2}{v \left(v^k-v+1\right)^2}\right) dv=\\
&&\int\limits_1^\infty \frac{(1-v) (k-(k-1) v) \left((k+1) v^k-v+1\right)}{v^2 \left(v^k-v+1\right)^2} dv
\end{eqnarray}
Let us discuss the steps from the very top to the bottom.
In the second line we used the integral representation of the beta function and in the third line we did the sum in question. In the fourth line we integrated by parts to get rid of the logarithm.In the fifth line we substituted for $u:=1-t$ and finally in the sixth and seventh lines we substituted for $v:=1/u$ and simplified the integrand.

Now, oddly enough the equation in the last line is similar to that given by Tito Piezas III. This would suggest that maybe the integrands of both results are the same, which they are not, as I have checked. It would be interesting to simplify my result further and bring it to the form given by Tito Piezas III.

Update: Note that:
\begin{equation}
\frac{(1+k) v^k-v+1}{v^2(1-v+v^k)^2}= \frac{d}{d v} \left[\frac{1}{v \cdot(1-v+v^k)} \right] + \frac{1}{v (1-v+v^k)^2}
\end{equation}
Inserting this into the last line above and integrating by parts we get:
\begin{eqnarray}
S_k&=&\int\limits_1^{\infty} \frac{1}{v \cdot (1-v+v^k)} dv + \int\limits_1^\infty \frac{(k-(k-1)v)}{v \cdot (1-v+v^k)^2} \cdot (-2 v^k-1+v) dv\\
&=&\int\limits_1^{\infty} \frac{1}{v \cdot (1-v+v^k)} dv + \left.\left( \frac{1-v}{1-v+v^k} + \log[\frac{1}{v^k}-\frac{1}{v^{k-1}}+1]\right)\right|_{1}^\infty\\
&=&\int\limits_1^{\infty} \frac{1}{v \cdot (1-v+v^k)} dv+0\\
&=&\int\limits_1^{\infty} \sum\limits_{p=1}^k \frac{1}{v\cdot(v-\xi_p)} \cdot \prod\limits_{q=1,q\neq p}^k\frac{1}{\xi_p-\xi_q)}dv=\\
&=&-\sum\limits_{p=1}^k \frac{\log(1-\xi_p)}{\xi_p} \cdot \prod\limits_{q=1,q\neq p}^k\frac{1}{(\xi_p-\xi_q)}
\end{eqnarray}
Here $\left\{ \xi_p \right\}_{p=1}^k$ are roots of the polynomial $v^k-v+1$.

As a final comment we note that in general the following holds:
\begin{equation}
\sum\limits_{n=1}^\infty \frac{x^n}{n} \cdot \frac{1}{\binom{k\cdot n}{n}} =x \cdot \int\limits_1^\infty \frac{1}{v\cdot(v^k+x-x v)} dv
\end{equation}
where again the integral on the right hand side can be expressed through roots of the polynomial in the denominator.