# Tagged Questions

306 views

### Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$ where $p(n)$ is a polynomial equation. When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
2k views

### sum calculation

I would like to calculate, or bound from above, the following sum $$\sum_{i=0}^n(n-2i)^p{p \choose i},$$ here $p\geq 2$. Any references are very welcome. Thank you.
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### Alternating sum of square roots of binomial coefficients

Let $$c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}.$$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$: ...
Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and f(n,k) = ...