# Closed form expression for sum of first m terms of a combinatorial number

by JTHM
Tags: algorithms, combinatorics
 P: 1 Dear Physics Forums denizens, 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 (r1 choose k)+(r2 choose k-1)+(r3 choose k-2)...(rk choose 1). For every t and s where t and s are non-negative integers such that trs (this is just true by the definition of a combinatorial number). For example, for k=5, we can express the number 36 as (7 choose 5)+(6 choose 4)+(2 choose 3)+(1 choose 2)+(0 choose 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.
 P: 962 If I correctly understand your problem and the wiki page which says in the second or third paragraph "Indeed a greedy algorithm finds the k-combination corresponding to N: take ck maximal with (ck,k)<=n, then take ck − 1 maximal with (ck-1,k-1), and so forth." (with my apologies for mashing the notation into ascii) When a mathematician is reduced to writing the word "algorithm" I always assume that means there is no known explicit equation giving the result. If you can find a closed form solution giving the first term or, even better, giving the first n of k terms then there might be a chance that this could be used to find the closed form for the sum of those terms. But personally I wouldn't hold out much hope.

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