# A closed-form expression for a sequence of integers

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

\begin{equation*} a_n=\arg\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 closed-form expression for $(a_n)_{n\ge \ell}$ (for a fixed $\ell$), or a way to characterize its entries? Why?

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So, $l$ is fixed in the definition of $a_n$, and only $j$ varies? –  Thomas Andrews Aug 8 '12 at 18:22
Yes, that is correct. –  daniel birmajer Aug 8 '12 at 18:59
You mean closeD formula. –  Mariano Suárez-Alvarez Aug 9 '12 at 16:40
@dbir The values you give for $a_n$ above look more like values of $j$. Did you mean "arg min" rather than "min"? Are there possibly ties? If so, how are you defining $j$ to be chosen? –  Erick Wong Aug 9 '12 at 16:45
Also posted, without acknowledgement here or there, to MO: mathoverflow.net/questions/104367/… –  Gerry Myerson Aug 9 '12 at 23:48