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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On a theorem of Hensel

In the paper Binomial coefficients modulo prime powers, Andrew Granville state the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
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15 views

Multinomial coefficients modulo a prime

Let $p$ be a prime and let $m \geq 1$. Lucas' theorem implies that the binomial coefficient ${p^m-1 \choose k}$ is not divisible by $p$ for any $0 \leq k \leq p^m-1$. I wonder if something similar ...
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72 views

How to prove this equality about Eulerian numbers?

I want to prove the following equality where $A(k,m)$ is the Eulerian number : $$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$ I previously proved ...
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40 views

A theorem about binomial coefficient module prime

For any integer $r$ and prime $p$, there is a integer $n$ which $\binom{2n}{n}\equiv r \pmod{p}$. I tried Lucas's theorem, but I was stuck. Suppose $r\neq 0$, otherwise we can let $n=p$. Let ...
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42 views

Ho can I find the sum of this sequence?

There is a sequence $$ F_n= \begin{cases} aF_{n-1}+q^{n-2}F_{n-2},& n \text{ is even}\\ bF_{n-1}+q^{n-2}F_{n-2},& n \text{ is odd} \end{cases} $$ with the initial conditions $F_0 = 0 ...
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Pascal triangle with non equiprobable events

As many of you will know when flipping n times a coin where each side is equally probable we can calculate the probability of getting x times heads with the triangle of pascal, that would be ${n ...
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41 views

For any given $k$, show that an integer $n$ can be represented as: $n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$

For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$. My ...
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39 views

Does the property ${n\choose r}={n\choose n-r}$ have a name?

Due to the relation between Pascal's Triangle and the choose function in probability theory, we can deduce that $${n\choose r}={n\choose n-r}$$ because Pascal's Triangle is symmetric. This can also be ...
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44 views

Problem with induction of binomial coefficiency

(Sorry for making up math language, I am roughly translating math terms here) This is part of some of the induction exercises in the book "Otto Forster: Analysis 1" (1.2): ...
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For what values of $n$ do more than two thirds of subsets of $[n]$ have size beteen $\frac{n}{3}$ and $\frac{2n}{3}$

My actual problem is slightly different. For what values of $n$ do we have: $2\sum_{i=1}^{\lfloor n/3 \rfloor}\binom{n}{i}\geq \sum\limits_{i=\lfloor n/3 \rfloor+1}^{\lceil n/2 \rceil-1} ...
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29 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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28 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
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36 views

Very odd binomial coefficients

The number of odd binomial coefficients in each row of Pascal's triangle is always a power of two although their sum rarely is. One of these rare occasions occurs for numbers of the form $\,$$n = 2^m ...
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28 views

Determine maximal addend in Newton Binomial Expansion.

Determine the maximal addend in Newton Binomial Expansion of the expression $$\left ( 2n+\frac{1}{2n} \right )^{4n+1},\quad \left ( \forall n \in \mathbb{N} \setminus \left \{ 1 \right \} \right )$$ ...
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36 views

Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ ...
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53 views

Recurrence relationships and a “weighted Pascal's triangle”

I was thinking about this problem a few days ago and in the process I came up with what I can best describe as a two-dimensional recurrence relationship. It seemed obvious to me that this was ...
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41 views

Prove the following result on binomial coefficients

If $(1+x)^n=^n\!\!C_0+^n\!\!C_1x+^n\!\!C_2x^2+\cdots+^n\!\!C_nx^n$, then show that $$(^n\!\!C_0-^n\!\!C_2+^n\!\!C_4-^n\!\!C_6+\cdots)^2+(^n\!C_1-^n\!\!C_3+^n\!\!C_5-\cdots)^2\\ ...
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43 views

Simple binomial coefficients sum question

I know $\sum_\limits{i=0}^n \dbinom{n}{i}=2^n$, and some others, but is there a simple expression when the lower and upper limits are arbitrary, i.e.; $$\sum_\limits{i=\alpha}^\beta ...
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48 views

A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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55 views

Binomial Sum Formula

I can't find a good closed form expression for this, $\sum_{k=0}^n\left[\binom{n}{k}\binom{m}{k}\right]$, where n is the variable, and m is a fixed constant, to be included in the formula. :( Can ...
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68 views

Inequality with power function and binomial coefficients

Any suggestion on how to proceed to show: $$\frac{2(m+1)^m -1 }{(m+1)m} - \sum_{k=0}^{m} {{m}\choose{k}} \frac{m^k}{(k+1)^2} >0 $$ where $m\geq 2$ is of course an integer. Numerical results ...
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60 views

How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $

Let $a\in\mathbb N$, and $b\in\mathbb R, b\geq 1$ How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $
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78 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
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15 views

Binomial random variable/z-score question?

I was given the problem: In a restaurant called ”Allegory”, on average 1 in 10 people order a bottle of white wine. Out of a sample of 50 people 11 chose a bottle of white wine. Has this wine become ...
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49 views

Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

For a given k, ${n\choose k}$ is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding base p digit of k (consider the p-ary notation for n ...
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39 views

Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
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61 views

How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
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40 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
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how to represent a binomial coefficient in terms of a series?

I have to find the power series for (n+m C m) or (n+m C m ) - 1 i.e representing it in some sort of power . Is it even possible ? P.S. :- Thanks in advance .
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Weighted sum of binomial coefficients by powers of lower value.

I am trying to calculate $\sum_{i=0}^n i^K {n \choose i}$ for $K \in \mathbb{N}$. Clearly, the case $K=0$ is trivially $2^n$ by the binomial theorem. For higher $K$ I am stumped. I know I can use: ...
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56 views

Truncated Binomial Series

Can the truncated binomial series be expressed as a closed form \begin{align} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{k} x^{k} \end{align}
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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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51 views

How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the ...
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53 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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173 views

Sum concerning Catalan triangle and binomials

I am trying to prove the following relation. For $k \in \mathbb{N},k \geq 2$ and $i=1,\ldots,k-1$ \begin{equation} {2k-2-i \choose k-1-i}=\sum \limits_{l=0}^{k-1-i} 2^{2l} T(k-2-l,k-1-i-l)-2 \sum ...
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63 views

Maximizing expected value when distribution is binomial

Consider the following problem: $$\max_{n\in\mathbb N}\;f(n)= \frac12 \left[v_0 \sum_{i=\lceil k_n \rceil}^n \binom{n}{i}p^i (1-p)^{n-i} + v_1\sum_{i=1}^{\lfloor k_n ...
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General solution for a combinatorial problem

I want to find a general solution for a problem. I explain the problem with an example. $\underline{Problem}:$We have a matrix $A$ of size $M \times N$, where $M <N$. We choose sub-matrices of ...
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61 views

$\mathbb{E}(\alpha^Y)$, where $Y$ is negative binomial

It is given that $\alpha>0$ and that \begin{equation} \mathbb{P}(Y=y)=\begin{pmatrix} y+k-1\\ y \end{pmatrix} (1-p)^kp^y \end{equation} are there any ideas how to calculate expected value of ...
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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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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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31 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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116 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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29 views

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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144 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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61 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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53 views

A sum of powers of binomials

For $n$ and $k$ non-negative integers, let $$F(n,k) = \sum_{i=0}^{n}\binom{n}{i}^k.$$ For example, $F(n,0)=n+1$, $F(n,1)=2^n$ and $F(n,2)=\binom{2n}{n}$. Does there exist a general formula for ...
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38 views

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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70 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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54 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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49 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 ...