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 . For every t and s where t and s are non-negative integers such that , it will be the case that (this is just true by the definition of a combinatorial number).
For example, for , we can express the number 36 as .
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
I came across something like this during my degree and I know of a way to show that any positive integer can be written uniquely in the form
where and .
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.