For questions about recurrence relations, convergence tests, and identifying sequences

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6
votes
3answers
139 views
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A finite sum over $\pm 1$ vectors

What is a nice way to show $$\sum_{u\in\{-1,+1\}^N}|\sum_{i=1}^Nu_i|=N\cdot\binom{N}{\frac{N}{2}}$$ when $N$ is even? Could there be a short inductive proof?
3
votes
0answers
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How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
4
votes
0answers
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How find this sequence recursive relations

Question: Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...