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

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