Let $$a_{n,k}=\sum_{s_i \geq 1 \atop \sum_{i=1}^{nk} s_i \leq n} \frac{2^{n}}{(2(n\sum_{i=1}^{nk} s_i)+1)!\prod_{i=1}^{nk} (2s_i)! }$$ for $0 \leq k \leq n1$. Prove for $1 \leq k \leq n1$ that $$b_{n,k}=\sum_{l=1}^k (1)^{kl} \sum_{s_i \geq 1 \atop \sum_{i=1}^l s_i =k} \prod_{i=1}^l a_{n,s_i}>0.$$ Motivation and alternative formulation can be found here
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Found the answer on page 102 of http://web.iitd.ac.in/~maz088121/chui.pdf 

