# Does this sum involving the central binomial coefficient have a closed form expression? [duplicate]

Since Lehmer we know that $$\sum_{n\geq1} \binom{2n}{n}^{-1} = \frac{2\pi\sqrt3}{27} + \frac43,$$ which is due to the identity $$\sum_{n\geq1} x^n \binom{2n}{n}^{-1} = \int_0^1 \frac{x(1-t)}{(1-xt(1-t))^2} dt.$$ Substituting different values of $$x$$ above gives a variety of remarkable formulae.

However, I have not found similar expressions for any $$\sum_{n\geq1} \binom{2n}{n}^{-k}$$ with $$k>1$$. In particular, I am interested in $$\sum_{n\geq1} \binom{2n}{n}^{-2}$$. Unfortunately I cannot get a better ''closed form'' expression for this sum than the hypergeometric series $$\frac14{}_3F_2(1,2,2;\frac32,\frac32;\frac{1}{16})$$.

Does a closed form expression exist for $$\sum_{n\geq1} > \binom{2n}{n}^{-2}$$?

• Seems related math.stackexchange.com/questions/2833496/… – Sil Mar 29 at 22:03
• Function.wolfram.com gives no closed form to that 3F2. – User Mar 30 at 4:27
• @Sil Yes, it does. Thank you. – Klangen Mar 30 at 9:22
• @Sil: wow, I did not remember posting the same qeustion 2 years ago. Strange! – Klangen Mar 30 at 9:25