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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Closed form for binomial coefficient series

If $6|n$, is there a closed form for $$\sum_{t=\frac{n}{2}}^n\binom{\frac{n^2}{3}}{t}\binom{\frac{2n^2}{3}}{n-t}?$$
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Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
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27 views

How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_{2},\cdots,k_{d}}^2$

Question: Assmue that $d$ is give postive integer numbers,and $$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots ...
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Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
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28 views

Algorithm for determining whether certain numbers appear in Pascal's triangle

Is there any easy characterisation for the numbers which appear in Pascal's triangle that ARE NOT $\dbinom{n}{1}$, $\dbinom{n}{n-1}$? Is there a fast way to determine if some number (given its prime ...
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58 views

The distribution of sum of two binomials with complement success probabilities

It is well-known that if $X$ and $Y$ are independent binomial random variables with parameters $(n_1,p)$ and $(n_2,p)$ respectively, $Z = X+Y$ has a binomial distribution with with parameters ...
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How to get a nice approximation of $f(N,s)=\sum_{k=0}^{N}{N \choose k}{k \choose s-k+N}$ when $N>>1$ and $|s|<<N$?

I need to approximate the above sum in order to calculate $\mathbb{E}(s^2)$, which is the expectation value determined by the probability density function $f$ and the position $s$. Any idea?
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106 views

A sum with binomial coefficients in the numerator and denominator.

I am struggling with a combinatorial sum as apart of a long statistical-mechanics derivation. I would appreciate any help. I seek the result of the following summation, for integer $\ell,n$, and ...
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59 views

Is there a sharper bound than exponential for $\sum_{k\ge0}\frac{m!(k+n-m)!}{(k+n)!}\frac{s^k}{k!}$?

I am trying find a bound for an expression and I am getting something not quite as convenient as I hoped. Going through my calculations again I think that the only place I use a non sharp bound is ...
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30 views

Equation for those level curves?

Related to this question Let $n\in\mathbb{N}^*,\alpha\in\mathbb{R}$. What would be the equation $y=f(x)$ for the curve defined by $\ln\binom{N-y}{x}=\alpha$ That's how they look : TL;DR : What ...
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Number of draws that contain at least $k$ red coins

Assume there are $n$ coins in an urn from which $r$ are read. What is the number of draws of $r$ coins that contain at least $k$ red coins? It is obvious that there are ...
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57 views

Closed form expression sum-product of binomials

Is it possible to find a closed form expression for $$\sum_{j=1}^a\sum_{i=1}^{b} {i+j-1\choose j} {i+j-1\choose i},$$ where $a \geq 1$, and $b \geq 1$ are integers. I couldn't apply any type of ...
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52 views

Approximation of the following mathematical formula

I have the following mathematical expression which I need to simplify: $$\mu^2\sum_{x=0}^{n}\left(\frac{\theta}{\mu}\right)^x\frac{1}{H_x}{n+a\choose x}$$ $\mu$, $\theta$, $D$, and $a$ are ...
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43 views

How find the sum $2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\frac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$

Find the sum $$2\sum_{k=1}^{\infty}\sum_{i=0}^{2k-1}\dfrac{\binom{2k}{i}\cdot B_{i}\cdot(m-1)^{2k-i}}{2k(2k-1)m^{2k-1}}$$ where $B_{i}$ is Bernoulli numbers. my idea: since ...
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72 views

the number of $m$-divisible subsets of an $n$-set

Let $\omega$ be a primitive $m^{th}$ root of unity. How can we use the binomial expansion of $(1+\omega)^n$ to find out the number of $m$-divisible subsets of an $n$-set. Actually, I mean, to find a ...
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67 views

Chu-Vandermonde-like combinatorial identity

I am looking for a simple combinatorial proof of the binomial identity: $$\sum_{j=0}^n \binom{2j}{j}\binom{2n-2j}{n-j} = 4^n.\tag{1}$$ The standard way I know is to exploit the generating function: ...
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34 views

binomial coefficient divisibility properties

I'm looking for a (possibly large) list of divisibility properties of binomial coefficients. Does anyone know a good reference? For example, Graham, Knuth, and Patashnik, Discrete Combinatorics, ...
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64 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
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29 views

Shortest ways in a grid above the angle bisector

Suppose, you have a grid with the side lengths n and m and the angle bisector from the upper left corner to the bottom side. To walk along the lines from A to B, there are $\binom{n + m}{n}$ shortest ...
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130 views

Is there a closed form formula for the Bernoulli numbers?

A while ago I found this algorithm. Today I read in wikipedia that Euler zig zag numbers can be used for computing the Bernoulli numbers. This Mathematica program computes the Euler zig zag numbers ...
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98 views

Feller vol. I, probability of 'r' successes in binomial process

This question is about the calculation of probability of at least 'r' successes in a binomial process given in the page 151 of feller:intro to probability:vol-1. Before deriving equation 3.5, text ...
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54 views

Smallest $r$ such that $\sum_{k=0,\,\, i+kr = qm}^{\lfloor (n-i)/r \rfloor} \binom{n}{i + kr} = 0 \pmod n$

I want to find the smallest positive integer $r$ such that $$\sum_{k=0,\,\, i+kr = qm}^{\lfloor (n-i)/r \rfloor} \binom{n}{i + kr} = 0 \pmod n$$ where $n=pq$, and every $i+kr = qm$ for some $m$ is ...
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111 views

Simplifying product of binomial coefficients

Let $j,k,l,n$ be positive integers such that $j,k,l \leq n$, $j-l \leq k$ and $l \leq k$. Is there any way to simplify the product $$ \binom{n-j-k-l}{k-j-l}\binom{n-j-k+l}{l}^2, $$ perhaps as a ...
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99 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
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96 views

Expectation of functions of binomial random variable involving logarithms

Let $X\sim\text{Binomial}(n,p)$ where $n$ is the number of trials and $p$ the probability of success of each trial. I am trying to evaluate the expected value of the following functions of $X$: ...
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94 views

solving equation from factorial/binomial coefficient

I'd like to find the sample size for which, in a combination with replacement, the probability of having at least one object of each k class is greater than $p$. Each object can take $k$ levels. I ...
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About bounding the “proportion” of a binomial coefficient

Is there some explicit examples of functions $f,g:(0,1/2]\to\mathbb R$ such that $f(\delta)\leq\dbinom{n}{\lfloor\delta n\rfloor}\leq g(\delta)$ for all natural $n$ (or at least for all $n\geq N_0$ ...
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81 views

Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
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222 views

Compute the value of a sum (as an expression involving one or two binomial coefficients)

I've been asked to compute the value of a sum. The answer should be an expression involving one or two binomial coefficients. The initial expression: $$ \sum_{k} \binom{80}{k} \binom{k+1}{31} $$ ...
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is this binomial distribution correct?

I am trying to work out what is the chances of getting the following marks $(100\%, 70\%, 60\%, 50\%)$ in a paper containing $50$ questions, each question containing yes/no options. Using Binomial ...
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185 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
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506 views

Binomial coefficient modulo prime power without generalized Lucas theorem

I've been working on this problem computing $n \choose r$ for large $n$ and $r$, modulo a composite. I could implement the generalized lucas theorem to handle the prime power case, but I want to ...
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169 views

Binomial coefficient (mod m)

$n\ge k $ $ m>0$ How to find? $$\binom{n}{k}\mod m$$ without counting $n,k!$
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Recognizing binomial mixtures

I'd like to know a procedure to recognize whether a given probability distribution over outcomes $\{0, \dots, n\}$ can be expressed as a mixture of $n$-trial binomial distributions with different ...
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32 views

For $X \sim \mathrm{Binomial}(n,\frac{1}{2})$ does there exist $a,b,c,Y$ s.t. $\Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$?

I need to upper bound some complicated expressions involving binomial distributions: Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$. I want to find $a,b,c,m$ such that for $Y \sim ...
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111 views

Binomial coefficient intervals (inequality)

For given $N$, $x$ and $k$ such that $0\leq x<N$ and $2\leq k\leq \left\lfloor \frac{N+1-2x}{2}\right\rfloor $, does it exist $p,$ $2\leq p\leq \left\lfloor \frac{N+1}{2}\right\rfloor $ such that ...
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299 views

The Lucas Theorem and facts

I have studied the Lucas theorem and I encountered the following facts. How to deduce the following facts from The Lucas theorem? (1) If d, q > 1 are integers such that , $$\binom{nd}{md}$$ ...
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201 views

Lucas' theorem Consequence

Lucas' theorem consequence $$\binom {m}{n}=\ \prod_{i=0}^k\;\ \binom {m_i}{n_i} \pmod{p},$$ $$m=m_k\;p^k+m_{k-1}\;p^{k-1}+\cdots +m_1\; p+m_0,$$ $$n=n_k\;p^k+n_{k-1}\;p^{k-1}+\cdots +n_1\;p+n_0$$ ...
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calculation of the sum using idea of one answer

I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ? ...
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293 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
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Inexpressibility of certain coefficients and discrete versions of Hölder's theorem

In an answer to a recent question, I noted that there were probably no explicit formulas for Stirling numbers (of the first kind, specifically) and speculated that this might be coupled to a sort of ...
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n-th self discrete convolution

Lets define discrete $ f_N(i) = 1,\space i = 1...N $ I need to find $ G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m} $ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
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201 views

Interdependent constraints combination problem

I am trying to solve the following combination problem. You have 4 knobs or levers that have maximum values, such as 0-20, 0-30, 0-50 and 0-100. Their total values must equal an amount, say 47. Their ...
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34 views

Binomial-like distribution

Starting with $1$, for $n$ trials multiply by either $1+p$ or $1-p$, with $0 \le p< \le 1$. Does this distribution have a name? What are its properties, such as density (PDF)? It is like a skewed ...
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20 views

Dividors of binomialcoefficient

Is it true, that $ \prod\limits_{\frac{n}{2}<p\le \frac{6n}{7}} p$ divides $ \binom{3n}{n} $? Thank you in advance. I have no idea how to prove it.
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A complicated summation of binomial coefficients

I am trying to evaluate this sum. I think closed form of this sum is not possible, but there might be some bound or approximate result. So far I was unable to find any approximation. Any help will be ...
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24 views

Proving that binomial coefficients are integers without induction, combinatorics, or formula for exponent of prime

I know of at least three ways to prove that binomial coefficients are integers. One is combinatorial--binomial coefficients count subsets, and thus are integers. Another is inductive, for example ...
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35 views

Exponentiation of Pascal's Triangle(in matrix form)

I want to find a pattern in subsequent exponentiations of the pascal triangle shown in the form below Matrix P[K+1][K+1]: $$ \begin{matrix} \binom{0}{0} & 0 & 0 & 0\cdots ...
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30 views

Calculating $\binom{n}{r}$ modulo $p^a$ where $p$ is prime

I need to calculate the value of $\binom{n}{r}$ modulo $p^a$ where $p$ is a prime number. If $a$ is equal to $1$, this is solved by Lucas' Theorem. Can anyone help me in this case by taking a ...
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23 views

Finding the term number and coefficient of a term containing $a^0$

Find the term number and the coefficient of the term containing $a^0$. $(a + a^{-2})^{12}$ My thought process so far is: That the binomial expansion to the $12$th power will have $13$ terms, ...