Let 
$$a\_{n,k}=\sum\_{s\_i \geq 1 \atop \sum\_{i=1}^{n-k} s\_i \leq n} \frac{2^{n}}{(2(n-\sum\_{i=1}^{n-k} s\_i)+1)!\prod\_{i=1}^{n-k} (2s\_i)! }$$
for $0 \leq k \leq n-1$. Prove for $1 \leq k \leq n-1$ that
$$b\_{n,k}=\sum\_{l=1}^k (-1)^{k-l} \sum\_{s\_i \geq 1 \atop \sum\_{i=1}^l s\_i =k} \prod\_{i=1}^l a\_{n,s\_i}>0.$$
Motivation and alternative formulation can be found [here](https://mathoverflow.net/questions/128835?sort=newest#sort-top) 



positive expression
Found the answer on page 102 of
<http://web.iitd.ac.in/~maz088121/chui.pdf>


