we have the sequence of positive real numbers ${a_{j}}$ , such that : $ \frac{1}{j+1}
$\frac{n+2}{n+1}=\prod_{j=1}^{\infty}\left(1-\frac{a_{j}^{2}}{(a_{j}n+a_{j}-1)^{2}}\right)$ 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 $a_{j}$ ?