I have a sum $$\sum_{m=0}^{n}\sum_{j=0}^{k}(-1)^j{k \choose j}{n-mj \choose k}$$ that comes up when counting compositions. Now the trouble is, if I would interpret it literally and for $n<mj$ counted with the value of the negative binomial coefficient, it has a very nice closed form, namely $$\frac{m^{n+1}-1}{m-1}$$ However, the context the formula comes from does not allow me to do that, instead I have to count any term with $n<mj$ as zero. Is there any way to deal with this difficulty and still get a closed form for the sum ?


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