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][1] 


  [1]: https://mathoverflow.net/questions/128835?sort=newest#sort-top