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One standard method is generating functions. Set $A(t)=\sum_{n=0}^\infty a_n t^n$ and $B(t)=\sum_{i=1}^\infty b_i t^i$. Then $$A(t)=a_0+B(t)A(t)+\sum_n f(n)t^n$$ so that $$A(t)=(1-B(t))^{-1}\left(a_0+ …

It's half the sum of the same thing from $0$ to $2n$, which in turn is easily related to the variance of the number of heads in a sequence of $2n$ tosses of a fair coin.

These are related to Stirling numbers. These don't have a "closed form". Your $R(k,n)=k!S(n,k)$ where $S(n,k)$ is the Stirling number of the second kind as defined at http://en.wikipedia.org/wiki/Stir …

Obviously, as you know, writing down something like $(-3)!$ is absurd and meaningless. But I would contend that absurd and meaningless as an expression like $$\frac{(-3)!}{(-6)!}$$ is, that it still e …

Rather obviously yes. Let $A$ be the algebra over the field $K$ generated by elements $a_1,a_2,\ldots,$ with $a_i$ in dimension $i$ and with $a_ia_j=0$ for all $i$ and $j$. This is an incredibly unin …

See this paper by Canfield and McKay. As the title suggests it focuses on the asymptotic enumeration, but it has lots of useful references. Added A simple arithmetic construction that realizes all pos …

I shouldn't expect there to be exact results; compare the similar problem with matrices with entries $\pm1$. For an $n$-by-$n$ matrix with entries $\pm1$ one gets an upper bound for the determinant of …