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Consider the two sequences $$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$ and $$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$ QUESTION. Is this true? $$\frac{a(n)}{b(n)}\longrightar …

This MO question prompted me to ask: What is the second order asymptotic growth/decay rate for the sum $$\sum_{k=0}^n\frac1{\binom{n}k}$$ as $n\rightarrow\infty$?

Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix. There has been some study (e.g. s …

Take two positive integers $m$ and $n$ and consider the rational function $$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$ and the corresponding Taylor expansion as $$G_{m,n}(x,t)=u_0 …

Consider all $n\times n$ binary (entries are either $0$ or $1$) matrices, denoted $\mathcal{B}_n$. Define the $X$-ray sequence of $A=(a_{ij})\in\mathcal{B}$ by $X(A)=x(1)x(2)\cdots x(2n-1)$ where $x(k …

Consider the generating function of "partitions with distinct parts" $$\sum_nQ(n)x^n=\prod_k(1+x^k).$$ It's known that $$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}}x^{j(3j- …

Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)$. …

Consider the binary partitions of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function $$\sum_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{2^n}}.$$ A result of De Bruijn …

I stumbled on the following problem, if you can see a way through it. Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$. QUESTION. For $x\rightarrow0$, does there exist …

I am finding the following first order estimate. Question. As $y\rightarrow\infty$, $$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\, \sim\,\,\frac{\pi}4\log^2y.$$ Is it true?

Given an integer $n\geq2$, consider the following integral $$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$ QUESTION. Is this true? It appears to be so. $$\lim_{n\ …

Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$. I would like to estimate the following alternating sum. QUESTION. Is this true? …

I brought up a couple of combinatorial and number-theoretic items with this MO question. Now, I shall inquire on growth estimates. Recall $$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left( 4\c …

Define the sequence given by the finite sum $$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$ Questions. (1) Is $0<a_n<1$? (2) Does the limit $\lim_{n\rightarrow\infty}a_n$ …

The number of Young tableau of size $n$ is given by $g_1(n)=\sum_{\lambda\in\mathbb{Y}_n}\dim\lambda$ which has known asymptotics. …