Let $(a_n)$ be the sequence of minimum values of the expression (depending on $\ell$):

\begin{equation*} a_n=\min \bigg\lbrace ((\ell+1)j-n)!\,(n-\ell j)!\quad \text{for}\; \,\bigg\lceil\frac{n}{\ell+1}\bigg\rceil\le j\le \bigg\lfloor\frac{n}{\ell}\bigg\rfloor \bigg\rbrace % \min \big\lbrace (2j-n)!\,(n- j)!,\; \mathrm{ceil}\,\frac{n}{2}\le j\le n \big\rbrace, .\end{equation*}

My questions is two fold:

Is there a close formula for $(a_n)_{n\ge \ell}$ (for a fixed $\ell$), or a way to characterize its entries? Why?

Let $(a_n)$ be the sequence of minimum values of the expression (depending on $\ell$):

\begin{equation*} a_n=\min \bigg\lbrace ((\ell+1)j-n)!\,(n-\ell j)!\quad \text{for}\; \\,\bigg\lceil\frac{n}{\ell+1}\bigg\rceil\le j\le \bigg\lfloor\frac{n}{\ell}\bigg\rfloor \bigg\rbrace % \min \big\lbrace (2j-n)!\,(n- j)!,\; \mathrm{ceil}\,\frac{n}{2}\le j\le n \big\rbrace, .\end{equation*}

My question is twofold:

Is there a closed formula for $(a_n)_{n\ge \ell}$ (for a fixed $\ell$), or a way to characterize its entries? Why?