All Questions

Fix an integer $n\geq1$. QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that $$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...

Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$. Define the proportion $$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...

I'm trying to find the exact asymptotics of a sum: $$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$ as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...