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Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ …
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