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 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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50 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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38 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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66 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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60 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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32 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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61 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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26 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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120 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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95 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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105 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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97 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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90 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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83 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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32 views

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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72 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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187 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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47 views

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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181 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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479 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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159 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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107 views

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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110 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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289 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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192 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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63 views

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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292 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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51 views

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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149 views

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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192 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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25 views

Final step of a random walk proof

I am working through the last bit of a random walk proof to show that a 3-d random walk is transient. The result I am looking for states that: $\frac {1}{2}^{2s} {{2s}\choose{s}} \sum_{j+k\leq{n}} ...
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28 views

Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ It can be found ...
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22 views

Coefficients for the falling factorial

Hello fellow mathematicians, I am trying to find a generating function, or at least find some useful property from the coefficients of the falling factorial. Let $(x)_n$ denote a falling factorial, ...
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23 views

Binominial Theorem proving

As I was trying to understand the proof of Binomial Theorem by induction, I got stuck at this line. What formulas should be used to get from left to right part? Any explanations and answers ...
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17 views

Stirling or better binomial approximation

Is there a method to find a approximation to $\log_2 \dbinom{n}{n^a}$ with $a\in(0,1)$? Similar to Approximating the logarithm of the binomial coefficient however here argument scales as radical of ...
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45 views

Second question about a limit.

Is the following sequence converge? $$ \lim_{n\rightarrow\infty}\frac{1}{(1+M)^{2n}}\sum_{i=0}^{n}\left( \begin{array}{c} 2n ...
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45 views

Recursion Formula Euler Numbers

I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$ Where $E_k$ are the Euler Numbers. The approach that I have taken ...
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closed form for $\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$?

Does there exist any closed form for the following sum? $$\sum_{k=0}^{n-1}\frac1{\binom{2n-1}{k}}\sum_{r=0}^{k}\binom{2n-1}{r}$$ Edit: Then can we find an asymptotic nice approximation as $n\to ...
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23 views

How to compute $\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$

How to compute the following integral? $$\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$$ in which, $a$ is a positive real number, $n$ is positive integer and ...
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38 views

Prove by induction that the binomial coefficient equals the number of subsets of given size

Prove by induction on $n$ that the binomial coefficient $\begin{pmatrix}n\\m\end{pmatrix}$ is the number of subsets of $I_{n}$ having size equal to $m$. The solution is as follows: So far it can be ...
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68 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
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38 views

Summation with Binomial Coefficients, $\sum (-1)^k \binom{m_1}{k} \binom{m_2}{k} $

I have trouble doing this summation: $$ \sum_{k=0}^{\min(m_1,m_2)} (-1)^k \binom{m_1}{k} \binom{m_2}{k} $$ where $m_1$ and $m_2$ are positive integers. Can someone help?
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33 views

Closed form equation with binomial coefficients

I need a closed form for the sum $\sum\limits_{i=0}^{\infty}{n-iT-1 \choose i}x^i$ $n$, $T$ are constants and positive but may not be integers. However, they can take nearest integer values, if not ...
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34 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
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49 views

Closed Form for a Sequence

I have come across this sequence $$a_0 = -2, a_1 = 5, a_2 = -28, a_3 = 255$$ and, in general $$a_n = -\frac{1}{2}\bigg(\sum_{i=1}^n \binom{2n+4}{2i}a_{n-i} + \binom{2n+4}{2n+1}\bigg)$$ I've tried ...
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23 views

Sampling combinations (from a binomial coefficient) without replacement

The total number of combinations of $k$ items out of $n$ total is $n \choose k$, or a binomial coefficient. This can be a very large number even for pretty small $n$. The binomial coefficient ...
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89 views

n-th degree Bezier curve-bernstein polynomial

I want to make a code that will draw n-th degree Bezier curve which would be calculated through Bernstein polynomials.My problem is not related to code writing,but its math kind.Reason I have ...
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28 views

Sum of binomial coefficients multiplied

How to approximate the next one relative to $m$ $$ \sum_{k=0}^m \binom {n}{n-k} (n-k-\frac k{\sqrt{2}})^2? $$ Or for example the simplier sum $$ \sum_{k=0}^m \binom {n}{n-k} (n-k)^2? $$