All Questions

$\mu=1+\epsilon$ where $\epsilon>0$ holds. 1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$ This quantity can be ...

Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$) $$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{...

Denote $$T(m)=\sum_{1\leq n_m\leq n_{m-1}\leq\dots\leq n_2\leq n_1\leq m}\prod_{i=1}^{m}\binom{n_i}{n_{i+1}}.$$ Is there a name for this kind of summation and is there a good estimate for $\ln T(m)$ ...

Define $S(k,2n)=\sum_{i=-k}^k\binom{2n}{n+i}$ at every $k\in\{0,\dots,n\}$. We know $$\ln(S(\gamma\mbox{ } n^\gamma,2n))\asymp(2n\ln2-\frac12\ln\pi-\frac12\ln n)$$ at $\gamma\rightarrow0$ and $$\ln(S(...

Binomial coefficients have a well known asymptotics (https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas) given by $$\binom nk\sim\binom{n}{\frac{n}{2}} e^{-d^2/(2n)} \...