# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
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### Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
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### A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$H_n=\sum_{k=1}^{n}\frac{1}{k},$$ and the generalized harmonic numbers are defined by $$H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}.$$ Recently, I have found ...
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### Is this formula for $\zeta(15)$ true?

Apery gave, \begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned} J. Borwein and D. Bradley found this can be generalized to $\zeta(4n+3)$...
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### Simplifying an expression involving binomial coefficients

Consider $$\prod_{j=0}^{i-1}\binom{n-2j}{2}\left[1+\sum_{k=0}^{N-1}{\left( \prod_{l=0}^{k} \dfrac{N-l}{M-l}\right)\left(1 + \sum_{m=k+1}^{N-1} \prod_{p=k+1}^m \dfrac{N-p}{M-p}\right)+1}\right]$$ ...
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### How fast does $\binom{n}{k}$ grow when $k \le n/2$?

How fast does $\binom{n}{k}$, $n$ fixed, grow when $k \le n/2$? Especially, I'm interested in the growth of the "inverse" of binomial coefficient $B_n(x) := \min \{k:\binom{n}{k} \ge x\}$. EDIT: ...
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### p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a $p$-...
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I noticed that when $n$ is an odd prime, the following congruence $$\binom{n-j-1}{k}+\binom{k+j}{k} \equiv 0 \pmod{n}$$ holds for $0 \le j \le \frac{(n-k)}2$ and odd values of $k$ such that $0 < k ... 0answers 53 views ### An Identity with Binomials and Harmonic Numbers Let$m,n,p$positive integers with$m\geq n$and$H_m=1+1/2+1/3+\cdots+1/m$the$m-$ith Harmonic Number with$H_0:=0$. Show that for the values of$m,p,n$for which the denominators do not vanish, ... 0answers 63 views ### A combinatorial proof for a bound on diagonal Ramsey numbers I wish to prove$R(p,p)\leq\frac{2^{2p-2}}{\sqrt{p}}$combinatorially. I have proved this algebraically through the definition of the binomial coefficient but I would much prefer a proof from ... 0answers 65 views ### How to evaluate:$\int_0^1x^{n-1}(1-x)^{n+1}dx$How can I evaluate the following integral? ($n \in R$,$n>0$) $$\int_0^1x^{n-1}(1-x)^{n+1}dx$$ I was solving the following problem (as practice) in school: Prove that the sum of$n+1$terms of ... 0answers 40 views ### How to prove this$p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$Let$p$be a prime number and$g\in \mathbb{Z}[x]$. Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every$k \geq 0$. Fix an integer$k$. Write the integer-valued ... 0answers 90 views ### Square of hockey stick identity:$\sum_{i=r}^n{i \choose r}^2$Evaluate$\sum_{i=r}^n{i \choose r}^2$where$n,r\in \mathbb{N},n>r$. This looks like the hockey stick identity but I can't find a way to evaluate it without a computer. Can someone help me out? 0answers 49 views ### Summing the binomial pmf over$n$, part 2 After the great answers I got to this question, I tried summing a similar-looking series using the same strategies ($k \geq 0, \alpha > 1, p \in (0,1)$): $$\sum_{n=k}^{\infty} {n \choose k} p^k (... 0answers 56 views ### Can a certain polynomial have all its coefficients in some basis divisible by a prime p? I fix n\in\mathbf{N}^{*} and n elements \alpha_1,\ldots,\alpha_n in \mathbf{N}^{*}. Consider the polynomial$$Q(T)=\prod\limits_{1\leq i \leq n} \prod\limits_{0\leq j \leq \alpha_i -1} (\... 0answers 197 views ### Sum problem involving Factorials got the following problem to prove for$n \in \mathbb{N}$and$1 \leq i \leq n$: \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! 2^{... 0answers 87 views ### An inequality concerning non-negative integer matrices with constant row and column sums I'd appreciate any suggestions for how to prove (or disprove) the inequality described below. Some notation first: for positive integers$k$and$M$, let${\mathcal D}_{k,M}$denote the set of all$k \...
A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a_1, a_2, \ldots, a_k)$ such that $\sum_{i=1}^k a_i = n$. I am interested in the case ...