4 votes

How to find the coefficient of $x^k$ in the expression $\prod_{p=2}^n (1+xp) $

From representation $$\prod_{p=2}^n (1+xp) = (-x)^{n+1} (-1/x)_{n+1} (1+x)^{-1}= \sum_{i\geq 0} s_1(n+1,i) (-x)^{n+1-i}\cdot \sum_{j\geq0} (-x)^j, $$ it follows that the coefficient of $x^{k-1}$ in ...
Max Alekseyev's user avatar
3 votes

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

Fix a smallish integer $m$, and assume for simplicity that $n$ is a multiple of $m$. Break $\{1,\dots,n\}$ into $n/m$ consecutive intervals of $m$ consecutive integers each. Each such interval must ...
Greg Martin's user avatar
  • 12.7k
3 votes

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

As a supplement of @Jorge Zuniga's answer, we can obtain more similar series which are $\pi$ and $\log 2$ related via the same idea, such as: $$ \begin{eqnarray} \log2&=&\sum_{n=0}^{\infty}\...
xiaoshuchong's user avatar
2 votes

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

Let $r_3(N)$ denote the maximum cardinality of a $3$-AP subset of $\{1,\dots,N\}$. It clear that $T(N)\ge 2^{r_3(N)}$. Meanwhile, it was proved by Balogh, Liu, and Sharifzadeh that we have $T(N) \le 2^...
Zach Hunter's user avatar
  • 3,375

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