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With $S(0)=0$ one has $$ S(n)=-\sum_{k=1}^n\frac{F_k}{k!} $$ where $F_k$ are the Fubini numbers (also known as ordered Bell numbers). The proof is contained in my comments above, given that the expone …

For brevity, rewrite the recursion as$$f_k(n)=\sum_{1\leqslant i\leqslant\frac n2}\binom nif_k(i)f_k(n-i).$$Now divide it by $n!k^n$ and rewrite like this:$$\frac{f_k(n)}{n!k^n}=\frac1{n!}\sum_{1\leqs …

For $n\geqslant0$ let $F_n(t)=\sum_{m\in\mathbb Z}a_{n,m}t^m$, where we are going to define $a_{n,m}$ for negative $m$ in such a way that $a_{n+1,m}=\frac{a_{n,m-1}+a_{n,m+1}}2$ for all $n\geqslant0$ …

This is an extension of the accepted answer. Using it, and ideas from some other papers, I managed to obtain an alternative expression for the generating function which I find interesting. As explain …

The following answer depends on what I mentioned in a comment — that the minimum is attained on $\lceil n/2\rceil$ for $n>13$; I don't know how to prove it. For $n\geqslant3$, $$ f(n)=\frac1{3\times4^ …

Decided to do a separate answer as there is a subtle point involved which is not mentioned in my comments to the answer by @Max Starting from the generating function by Max Alexeyev $$ \sum_{m,n\geqsl …

To illustrate the suggestion of Richard Stanley about positivity of real parts of zeroes, here are the zeroes of $Q_{20}$. The pattern seems to be the same for all of them. Another empirical observ …