Sum of first m terms of a combinatorial number

Dear Math Help Forum,

I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through *m*-th terms of a combinatorial number. For those of you unfamiliar with combinatorial numbers, here's some useful reading: http://en.wikipedia.org/wiki/Combina..._number_system

Basically, the idea is this: for any non-negative integers *k*, and *b*, we can express the value of *b* as a sum of *k* terms of the form $\displaystyle \binom{r_1}{k}+\binom{r_2}{k-1}+\binom{r_3}{k-2}... \binom{r_k}{1}$. For every *t* and *s* where *t* and *s* are non-negative integers such that $\displaystyle t<s \leq k$, it will be the case that $\displaystyle r_t>r_s$ (this is just true by the definition of a combinatorial number).

For example, for $\displaystyle k=5$, we can express the number 36 as $\displaystyle \binom{7}{5}+\binom{6}{4}+\binom{2}{3}+\binom{1}{2 }+\binom{0}{1}$.

Now, the sum of all five of these terms will be 36. But suppose I just want, say, the sum of the first two terms, four terms, or any arbitrary number of terms, and I don't want to exhaustively find every term and add all of them up. The question, then is this: given *k*, *b*, and *m*, where *k* is the total number of terms in the combinatorial number, *b* is the value of the combinatorial number, and m is the number of terms (starting with the first term) that we want to sum, what is the closed-form expression for the sum of those terms?

Admittedly, I am not certain that a closed-form expression even exists. If you can think of a reason why there might not be a closed-form expression for the above, please share it. In the eventuality that there is no closed-form expression, if you can think of a fast algorithm to find such a sum--something faster than just adding the terms individually--that would be helpful, too.

Re: Sum of first m terms of a combinatorial number

Hello,

I came across something like this during my degree and I know of a way to show that any positive integer $\displaystyle m$ can be written uniquely in the form

$\displaystyle {m_r\choose r} +{m_{r-1}\choose r-1}+...+{m_s\choose s},$

where $\displaystyle m_r>m_{r-1}>\cdots>m_s>0$ and $\displaystyle r\geqslant s\geqslant 1$.

The fact that each integer can be written uniquely in such a way implies to me that no such closed form expression could exist, since a closed expression could be re-written in many different ways. I don't know if this helps at all, but I can try to reproduce the proof if you'd like.