# Funny double infinite sum

I was playing with a modified version of Pascal’s triangle (with ${n \choose k}^{-1}$ instead of $n \choose k$ everywhere) and this infinite sum popped out:

The partial sums seem to approach $\alpha \approx 1.317...$

Does a closed form for $\alpha$ exist?

It is not difficult to de-nest such double series.

Lemma 1. For any $k\geq 2$, we have

Proof: it is enough to exploit Euler’s beta function and a geometric series, since:

In particular, we have:

The latter is not an elementary integral but an exponential integral.

At last, a huge WELCOME to MSE.