# an infinite system of infinite products

we have the sequence of positive real numbers ${a_{j}}$ , such that : $$\frac{1}{j+1}<a_{j}<\frac{1}{j} \;\;\;\;,\;j\geqslant 1$$ and we have, for every integer $n\geqslant 1$, the equality :

$$\frac{n+2}{n+1}=\prod_{j=1}^{\infty}\left(1-\frac{a_{j}^{2}}{(a_{j}n+a_{j}-1)^{2}}\right)$$ furthermore, the infinite product : $$\prod_{j=1}^{\infty}(1+a_{j})e^{-a_{j}}$$ is convergent. in fact, there is an entire function defined as: $$f(x)=C\prod_{j=1}^{\infty}(1+xa_{j})e^{-xa_{j}}$$ such that, at negative integers: $$f(-n)=K(-1)^{n}n!$$ $C$ and $K$ being constants. can we prove that such a sequence exists ? how can we solve for the numbers $a_{j}$ ?

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