All Questions
3
questions with no upvoted or accepted answers
3
votes
0
answers
287
views
$f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce $\exists C>0: f(cn) \geq n$?
Let $f : \Bbb N \to \Bbb N$ be a nondecreasing function that satisfies $f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce that there exists some positive constant $c$ such ...
- 929
2
votes
0
answers
78
views
value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n {n\choose k}{2k\choose 2r} {(\frac{3}{4})}^{(n-k)}({\frac{1}{2}})^{2k-2r}}$ .
The question requires us to find the value of $\frac{\sum_{k=0}^r{n\choose k}{n-2k\choose r-k}}{\sum_{k=r}^n
{n\choose k}{2k\choose 2r} {\left(\frac{3}{4}\right)}^{(n-k)}\left({\frac{1}{2}}\right)^{...
- 97
2
votes
0
answers
60
views
Finding a formula for a sum that involves binomial coefficients
Is there a formula for this sum:
$$ \sum_{j=0}^k {n \choose j} {n \choose k-j} (-2)^j \left(-\frac13 \right)^{k-j} ?$$
It reminds me to Vandermonde's identity; but as you can see there is a slight ...
- 897