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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 In case you are interested: If my derivation is correct, the above formula is the exact probability that an ordered partition of $n$ elements into $k+1$ pieces does not contain a summand $\geq m$. The formula was derived in two different ways. By simply counting intervals and by counting points on a truncated $n$-simplex.

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    @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

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