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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The $k$_th coefficient of the polynomial :$f_n(z)f_m(jz)+f_m(z)f_n(jz) $

Let $j=e^{2i\pi/3}$ ( $i$ is the complex number $i^2=-1$), and let : $$f_n(z)=(1+z)^n$$ Question Is there an expression (without using sums) of the $k$_th coefficient of the following polynomial ...
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Binomial Theorem Help

In my assignment I have the following question: Alan
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61 views

Sequence closed expression or others

What are closed expression or any other expression (involving integrals, specials functions...) for $\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$ where $t>0$ integer Thank you
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2answers
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pascal triangle, getting to the sum of it

So we know that $$\binom{n}0+\binom{n}1+\ldots+\binom{n}n=2^n\;.$$ What about the following sum? $$\binom{n}n+\binom{n+1}n+\ldots+\binom{n+m}n\;?$$ (a) Identify several examples of this sum on ...
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How to show $u_n=\binom{n}{l}\Rightarrow s_n=\binom{n+1}{l+1}$?

Hello respected everyone. Before I ask my query, let us first define binomial coeffcients as follows: For $n, r\in \mathbb N$, we define $$\binom{n}{r}=\begin{cases} \frac{n!}{r!(n-r)!}~~~ ...
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1answer
29 views

Dealing with floor function in binomial coefficients

I'm trying to estimate $\binom{n}{\left \lfloor{\alpha n}\right \rfloor }$ asymptotically using Stirling's formula. However, I'm a little lost with what to do about the floor function here. In the ...
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2answers
92 views

Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases:

Find the number of solutions of the equation $x+y +z +w = 15$ in the following cases: (a) $x,y,z,w \geq 0$ (b) $x,y,z,w > 0$ (c) $x>2, y>-2, z>0, w>-3$ I think I have an idea on how ...
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1answer
24 views

How is the Binomial coefficient simplified to a falling factorial?

I'm learning how to take the derivative of the binomial coefficient and found a blog post that was quite useful. However I am unclear as to how the first step bellow was simplified to the second step ...
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1answer
37 views

Trailing zeroes in binomial coefficient

I have a doubt regarding trailing zeroes in binomial coefficients... Question: How would you calculate the number of trailing zeroes in the binomial coefficient of ${n\choose r}$ upto values of ...
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1answer
39 views

Number of ways to combine two sets of values

Good day. I have an algorithm which iterates over a set of values S =[1 -1 0.5 -0.5] for a set of parameters in vector ...
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1answer
21 views

Asymptotic for binomial coefficients

Does there exist any asymptotic formula for binomial coefficients ${n \choose k}$ for large $n$ when $k$ is fixed?
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27 views

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$

Show $\sum_{n=0}^{\infty}\binom{n+k}{k}z^{n}=\frac{1}{(1-z)^{k+1}}$ where $|z|<1$ and $k \geq 0$. I know The right hand side: \begin{align*} \frac{1}{(1-z)^{k+1}} & = ...
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1answer
41 views

Remove first element of sequence, compute cumulative sum, iterate

(This is related to my previous question General formula for iterated cumulative sum.) Consider the sequence $S_0$ consisting of ones: $$ 1,1,1,1,1,1,\ldots $$ Now remove the first element and ...
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1answer
26 views

Integer Series Expansion

For any two $p,q \in \mathbb{Z}$ and $n \in \mathbb{Z}^+$, can one prove that $a_n = \frac{p(-p)^n - q(p-2q)^n}{(p-q)}$ is an integer with recursion relation $a_0 = 1,$ $a_n = ...
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4answers
36 views

How to computer the summation of a binomial coefficient/ show the following is true

$\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$. I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make ...
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33 views

Show that $\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$

The Problem Show that $$\binom{1/2}{k} = \frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k}$$ My Work $$\begin{align*}\frac{(-1)^{k+1}}{4^k(2k-1)}\binom{2k}{k} &= ...
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How to show C(n,k)= C(n,n-k)

I am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the ...
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1answer
20 views

Binomial series after using binomial identity

Following http://en.wikipedia.org/wiki/Binomial_coefficient#Newton.27s_binomial_series , I am trying to prove that $$ \sum_{\kappa=0}^\infty \binom{\eta + \kappa}{\kappa}x^\kappa = (1 - ...
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2answers
108 views

Even/Odd Binomial Coefficients

I was wondering if there's a nice general solution for the following problem: How many numbers in the $n^\text{th}$ row of Pascal's triangle are even? How many are odd?
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212 views

Counting elements in cartesian power with plurality + pattern constraints

Problem: I have an alphabet with n=8 letters (say $X=\{A, B, C, D, E, F, G, H\}$). I'm looking for words with m=24 letters, with three constraints: letter $A$ is the relative majority (like in ...
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Prove Binom Sum $\sum_{k=0}^n(-1)^k \binom{n}{k} = 0$ [duplicate]

Let: $$ (-1)^0=1 $$ I need to prove that: $$ \sum_{k=0}^n(-1)^k \binom{n}{k} = 0 $$ Thanks!
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35 views

Differentiation of a binomial-like sum with respect to m (total number of trials)?

I want to differentiate a function with respect to m, and I know that I have to first find a representation for the whole sum to do that since m is discrete. However, I could not find one yet. Any ...
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36 views

Recovering the coefficients $b_r$ of the binomial sum $\sum_{r=0}^n\binom{n}rb_r$ [closed]

Suppose that the sequences of real numbers $a_0,a_1,a_2,a_3,\ldots$ and $b_0,b_1,b_2,b_3,\ldots$ satisfy the relation $$a_n=\sum_{r=0}^n\binom{n}rb_r\;.$$ Then prove that ...
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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} ...
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34 views

Negative binomial theorem

I have been supplied with a combinatorical proof based on the n'th power, however I am trying to prove this by induction. I have no problem with the base case, or assuming that n=N. However, for ...
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21 views

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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1answer
42 views

Sum of binomial coefficients and powers

The following identity is true for $n\geq1$: $$ n!=\sum_{k=1}^n (-1)^{n-k} {n\choose k} k^{n} $$ You can obtain it from the equation in this question by setting the variables equal to 1. I was ...
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52 views

Induction proof: sum of binomial coefficients

Prove by mathematical induction for all natural numbers $n$: $$\sum_{k=0}^{n} \binom{n}{k}=2^n$$ thus is it sufficient to show that (but I think I made a mistake) thus how to do it properly? ...
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59 views

$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$

I want to examine the convergence of the series $$\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{2^{2n}}$$ In case it converges I want to evaluate it. I tried the D' Alembert theorem but it was ...
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1answer
54 views

Simplify the sum

Consider the sum of two polynomials $$ \sum_{k=0}^{n-1} {{{n-1} \choose {k}}^2 z^{2k}}+\sum_{k=0}^{n-2} {{n-2} \choose {k}} {{n} \choose {k+1}} z^{2k+1}=\sum_{i=0}^{2n-1}a_i z^i. $$ I want to find the ...
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39 views

Issue to identify binomial distribution

I'm trying to find a general formula for the following case : A production process has one of three possible states: (1) in control, (2) out of control with a type 1 problem, or (3) out of control ...
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43 views

How to compute $\displaystyle\sum_{k\equiv 1\!\!\pmod{\!4}}\!\!\binom{2014}{k}$?

I have to compute: $S=\binom{2014}{1}+\binom{2014}{5}+\binom{2014}{9}+...+\binom{2014}{2009}+\binom{2014}{2013}$ Could someone help me ?
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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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109 views

How to expand $(x+y)^{-n}$?

How to expand $(x+y)^{-n}$ by binomial theorem, where $n$ is a positive integer? Is there any limitation for $x$ and $y$? If it can be expanded, how to compute the coefficients? Many thanks in ...
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1answer
18 views

Approximating a binomial sum over a simplex

For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with. Can we do something ...
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23 views

Compute convolution

Let $a>0$ and $a_1>0$ be real numbers. Using the convolution theorem I have shown a following identity: \begin{equation} \sum\limits_{l=1}^n \binom{a_1}{n-l} \frac{1}{l} \binom{a}{l} = -\gamma ...
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1answer
30 views

Express $\binom{n+2}{k}$ according to $\binom{n}{k}$

I've just begun studying binomial coefficient and I'm trying to express $\dbinom{n+2}{k}$ according to $ \dbinom{n}{k}$. With this result I have to conclude that $\dbinom{2n}{2k}, \dbinom{2n+1}{2k}$ ...
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1answer
32 views

To prove for every integer $k >1 $ , $2+2^k \choose r$ is even for all $2 < r \le n=1+2^{k-1}$ , without Lucas' theorem

Can we prove by induction that for every integer $k >1 $ , $2+2^k \choose r$ is even for all $2 < r \le n=1+2^{k-1}$ ? Or by some divisibility properties of Binomial co-efficients ? I wanted to ...
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53 views

Use of Multinomial theorem.

I have the next identity which I want to prove. $$(\sum_{j}k_j^2)^{s} = \sum_{b_1+\ldots+ b_n =s} \prod_j k_j^{2b_j}$$ Obviously I need to use the Multinomial theorem, but how to procceed from ...
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How can I compute $\sum\limits_{k = 1}^n \binom{n - 1}{k - 1}$?

I know what $n \choose k$ equals, but I don't see how that would help me solve the sum of $n - 1 \choose k - 1$ from $k = 1$ to $n$. Is there any special trick I should know?
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304 views

How to deal with this double summation?

I'm stuck with the proof of this result: $$2^n = \sum_{t=-\frac{n-1}{2}}^{\frac{n-1}{2}} \binom{n+1}{\frac{n+1}{2} + t} \sum_{k=\vert t \vert}^{\frac{n-1}{2}} \binom{\frac{n-1}{2}+k}{k} ...
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1answer
48 views

To prove ${2p - 1 \choose p } \equiv 1 \pmod{p^2}$ without using Wolstenholme's theorem

How to prove that ${2p - 1 \choose p} \equiv 1 \pmod{p^2}$ ? I don't want to use Wolstenholme's theorem; but one might use $p|{p \choose k} , 1 \le k \le p - 1$ , and $(p - 1)! \sum_{k = 1}^{p - 1} ...
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75 views

What are all the even positive integers $n$ such that for $1 \le r \le n-1 $ , $2n \choose r$ is odd for exactly one $r$ ?

Motivated from this To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$ ; what are all the even positive integers $n$ such that for $1 \le r ...
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Production process testing - binomial probabilities

A process produces faulty items with probability 0.04. Samples of 100 items are taken from batches at random and if there are less than 5 faulty items in the sample then the batch is accepted; ...
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165 views

Show that there are infinitely many integers such that $ \binom{m}{n-1} = \binom{m-1}{n} $

This question comes from the 1st Brazilian's IMO TST of 2004. I have found no solutions of it online, though I have developed one. After getting to $ mn = (m-n)(m-n+1) $, my solution relies on the ...
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30 views

Simpler expression for binomial sum

Is there any closed expression for the following sum: $$\sum_{i=0}^{l-k} \binom{n-l}{i} \binom{l-k}{i} \binom{l-i}{k}$$ where $ k<l < n/2$?
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What does 9P2 mean, and how would one solve it?

We are studying "Sequences, Series, and Probability" and it likely related to binomial theorem and pascals triangle. I've a test tomorrow morning, and if I can't figure this out soon, I'm likely to ...
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1answer
86 views

Mean value of minimum of binomial variables

Let $X,Y$ be two independent random variables with binomial distribution: $B[4n,p]$ with $p=0.5$. Let $M=\min(X,Y)$. What is the expectation of $M$? This question seems related but it has no answer: ...
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2answers
33 views

Some binomial equality

I am trying to prove the following equality $$ \sum_{r=k}^{n}\binom{2n+1}{2r+1}\binom{r}{k}=\binom{2n-k}{k}2^{2n-2k}~~;~k\le n. $$ I noticed that for $k=0$ it becomes $$ ...
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1answer
41 views

$k$ divides $\binom{kn}{n}$

For positive integers $k,n$ , is it true that $k$ divides $\binom{kn}{n}$? I can write $$\binom{kn}{n}=\frac{(kn)(kn-1)\cdots(kn-n+1)}{n(n-1)\cdots 1}$$ but must the $k$ at the top remain after ...