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Let $l$ be a positive odd integer. Using Cilleruelo's method, we establish anexplicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$,$\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine allvalues of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain valuesof $l$.

A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd