Quirinus Quirrell Posted January 8, 2013 Share Posted January 8, 2013 (edited) Dear Science Forums members, 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_systemBasically, 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 t<s, it will be the case that rt>rs (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. Edited January 8, 2013 by Quirinus Quirrell Link to comment Share on other sites More sharing options...
Time And Space Posted January 13, 2013 Share Posted January 13, 2013 sum of all binomial co-efficiants=2n Link to comment Share on other sites More sharing options...
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