Infinite sum - generalising Catalan generating function This is a follow up from this question where we proved that
$$ S = \sum_{n\geq 1}\frac{1}{n}\binom{2n}{n+1}2^{-2n} = 1$$
As a matter of facts this can be done using lots of (related) methodologies, including:


*

*Maybe the easiest, using telescopic series by noting that
$$\frac{1}{n}\binom{2n}{n+1}2^{-2n} = 2(b_n-b_{n+1}),\quad \text{with}\quad b_n=\binom{2n}{n}2^{-2n}$$

*Using Gauss Hypergeometric theorem, from the fact that
$$S+1 = {}_2F_1(\frac{1}{2},1,2,1)$$

*Using Taylor / Binomial series, seeing that
$$S= -2\left(\sum_{n\geq 0} \binom{1/2}{n}(-1)^{n}\right)+1$$

*From Catalan number, seeing that this is the generating function for the Catalan numbers taken at value $x=1/4$,
$$S = \frac{1}{2}c(1/4) = \frac{1}{2}\sum_{n=0}^\infty C_n(1/4)^n$$

I want to generalize the results further, showing that
  $$ S_k = \sum_{n\geq k}\frac{k}{n}\binom{2n}{n+k}2^{-2n} = 1$$

I have tried to adapt the proofs for $S$, with no success so far. The fact that I'm no longer looking at the central binomial coefficient is always causing me trouble.


*

*I can't find any telescopic series, I tried adapting $b_n$ or looking at some relation $b_{n+k}-b_n$.

*I failed to reduce my sum to a Taylor series, or to find a hypergeometric function.

*I've tried reasoning by induction. If I can express $S_{k+1}$ from $S_k$, this could work. But I end up with 
$$S_{k+1} = S_k + \sum_{n\geq k}2^{-2n}\binom{2n}{n+k}\frac{n-2k(k+1)}{(n+k+1)n}$$
And proving that this second sum is null seems harder than my original problem (it does go to 0 numerically, so this at least seems correct). 

*I also tried to bound $S_{k+1}$, and to prove that we must have $S_{k+1}\geq S_k$ (or the reverse), but my bounds are not tight enough.



I would appreciate any help or hint, maybe a direction to look at?

For instance I wonder if there are some numbers defined by difference of binomial coefficient, e.g.
$$ T_{n,k} = \binom{2n}{n+k}-\binom{2n}{n+k+1}$$
That would generalize Catalan's number (obtained with $k=1$ here). This might help my induction strategy.

$$S_k = \sum_{n\geq k}\frac{2k(2n-1)!}{(n-k)!(n+k)!}\cdot\frac{1}{4^n}=\sum_{n\geq k}\frac{(2n-1)!\left[(n+k)-(n-k)\right]}{(n-k)!(n+k)!}\cdot\frac{1}{4^n} $$
equals
$$ \sum_{n\geq k}\left[\frac{(2n-1)!}{(n-k)!(n+k-1)!}-\frac{(2n-1)!}{(n-1-k)!(n+k)!}\right]\frac{1}{4^n}$$
or
$$ \frac{1}{ 4^k}\sum_{m\geq 0}\left[\frac{(2m+2k-1)!}{m!(m+2k-1)!}-\frac{(2m+2k-1)!}{(m-1)!(m+2k)!}\right]\frac{1}{4^{m}}$$
or
$$ \frac{1}{4^k}\sum_{m\geq 0}\frac{1}{4^m}[x^m]\left[(1+x)^{2m+2k}\frac{1-x}{1+x}\right]$$
or
$$ \frac{1}{4^k}\sum_{m\geq 0}[x^m]\left[\left(\frac{1+x}{2}\right)^{2m}(1-x)(1+x)^{2k-1}\right]$$
or
$$ \frac{1}{4^k}\sum_{m\geq 0}\operatorname*{Res}_{x=0}\left[\left(\frac{1+x^2}{2x}\right)^{2m}\frac{(1-x^2)(1+x^2)^{2k-1}}{x}\right]$$
which finally boils down to something simple to compute (i.e. the opposite of the residue at infinity of a rational function), by switching $\text{Res}$ and $\sum_{m\geq 0}$. I am pretty sure I messed up some transformations but the approach in itself should be fine.
