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 $\sum_{n\geq1} > \binom{2n}{n}^{-2}$?

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