I have come across the following alternating sum, as part of a larger calculation, that I would like to be able to compute efficiently. Can anyone help me simplify this: $p(n,k,m)=\frac{\sum_{i=1}^{\lfloor\frac{n}{m}\rfloor}(-1)^i\binom{n-im}{k}\binom{k+1}{i}}{\binom{n}{k}}$ I need to be able to calculate this accurately for small values (about 1-1000) of all the parameters. If an exact simplification is not possible, a good approximation will also helpful. The following constraints apply: $k
Alternating sum of products of binomial coefficients
3
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combinatorics
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0@Sasha I need to calculate these in great numbers within a C program, so I am looking for a formula, that is computationally more affordable. Unfortunately, precomputation is not an option. – 2012-05-18