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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Convolution of binomial coefficients

As part of a (SE) problem I've been working on, I came up with this expression: $$ \sum_{i=0}^M\binom{M-1+i}{i}\binom{M+i}{i} $$ I'd like to get a closed form for this, but after a considerable amount ...
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3answers
56 views

How to expand $\sqrt{x^6+1}$ using Maclaurin's series

The expansion would be $\sum_{n=0}^\infty$$\frac{1}{2}\choose n $$x^{6n}$ How to evaluate binomial coefficient with rational numbers? If $\frac{1}{2}\choose n $=$2n\choose n $$\times ...
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1answer
71 views

Sum with binomial coefficients and integer powers

I would like to have an analytic expression for the following sum $$ G_{n,a} = \sum_{p=1}^n \frac{(-1)^p p^{2(a+n)}}{(n-p)! (n+p)!} \;. $$ I am not sure it has a closed form, but I would at least ...
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1answer
65 views

Summation of a series of binomial coefficients

What will be summation of this Series: $$ {}_nC_1 + {}_nC_2 + {}_nC_3+\cdots +{}_nC_k$$ for $k \leq n$?
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2answers
68 views

$\frac{1}{{9\choose r}} -\frac{1}{{10\choose r}} = \frac{11}{6{11\choose r}}$. Is there a way to find $r$ without using algebra?

$$\frac{1}{\dbinom 9r} -\frac{1}{{\dbinom{10}r}} = \frac{11}{6\times \dbinom{11}r}$$ I guess directly applying algebra for this problem would be enough. But are there any simpler and prettier ...
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0answers
33 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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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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2answers
56 views

Find sum with binomial coefficients and powers of 2

Find this sum for positive $n$ and $m$: $$S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}.$$ Obviosly, $S(n,m)=S(m,n)$. Therefore I've tried ...
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4answers
261 views

How to simplify this triple summation containing binomial coefficients?

$$ \large\sum_{i=0}^{n} \sum_{j=i}^{n} \sum_{k=j}^{n} \binom{i+m-1}{m-1}\binom{j+m-1}{m-1}\binom{k+m-1}{m-1} $$ How to solve it when this involve more than thousand summation ?
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Category of binomial rings

A binomial ring is a commutative ring $R$ such that (1) the additive group of $R$ is torsionfree and (2) $n!$ divides $x(x-1)\dotsc(x-n+1)$ for all $n \in \mathbb{N}$ and $x \in R$. We may then define ...
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196 views

Prove a combinatorial identity

Prove the combinatorial identity $$ \sum_{n_1+\ldots+n_m=n} \;\; \prod_{i=1}^m \frac{1}{n_i}\binom{2n_i}{n_i-1}=\frac{m}{n}\binom{2n}{n-m}, \enspace n_i>0,i=1,\ldots,m $$ I "discovered" this ...
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2answers
47 views

Proof by counting two ways

Proof by counting two ways: \begin{equation}\sum_{k_1+k_2+...+k_m=n}{k_1\choose a_1}{k_2\choose a_2}...{k_m\choose a_m}={n+m-1\choose a_1+a_2+...+a_m+m-1}\end{equation} I have a proof by algebra for ...
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binomial expression of a powered term [duplicate]

One answer to a previous question of mine asserted that $$k^2=\binom k2+\binom {k+1}2.$$ I checked that the formula is true. However, it intrigued me. Is there a similar expression for $k^3$? How ...
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1answer
30 views

Limit of consecutive sum of binomials

Let $a$ be a positive integer. What is the limit $$\lim_{n\rightarrow\infty}\frac{\dbinom{an}{0}+\dbinom{an}{1}+\cdots+\dbinom{an}{n-1}}{2^{an}}$$ where $n$ takes on integer values? Since the ...
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129 views

Closed form of partial hypergeometric sum

Can we get closed form for $$\sum_{k=0}^m \left(-\frac12\right)^k \binom{2m}{m-k}k^p,\quad p\in\mathbb{N}\,?$$ In Concrete Mathematics Knuth describes Gosper's algorithm and its Zeilberger's ...
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Rough bound for sum $\binom{3n}{0}+\binom{3n}{1}+\cdots+\binom{3n}{n-1}$

Is it true that $$\frac{\dbinom{3n}{0}+\dbinom{3n}{1}+\cdots+\dbinom{3n}{n-1}}{2^{3n}}<\frac13$$ for all positive integers $n$? I've plotted the first few values of $n$ and noticed that the ...
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4answers
34 views

Determine the coefficient

I've been stuck on this question for a little while, could someone point me in the right direction? I'm supposed to determine the coefficient of $x^8$. $$x^8\quad in \quad \frac{x}{(1-x)(1-2x)} $$ ...
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1answer
94 views

Express $\binom{n+k-1}{ k}$ as a sum $\Sigma_{i=0}^k a_i$

I am stucked at this problem: It is known that the number of ways to arrange $k$ non-distinguished balls into $n$ cells is $\binom{n+k-1}{ k}$, Now by partitioning the cells into disjoint subsets, ...
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4answers
297 views

How to prove $ \sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
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1answer
42 views

Sum of binomial coefficients from $\binom{2m+1}{0}$ to $ \binom{2m+1}{m} $

Prove that: $\displaystyle 4^m = \binom{2m+1}{0}+\binom{2m+1}{1}+\binom{2m+1}{2}+\ldots + \binom{2m+1}{m} $ From the Binomial Theorem: $\displaystyle (a+b)^n = ...
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2answers
32 views

Proof that repeated sum equals binomial formula

Let $s, d$ be positive integers. Can you prove the following general formula for the repeated sum? I developed this problem on my own, but is it a well known result? $$\sum_{i_1 = 0}^s \sum_{i_2 = ...
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2answers
65 views

Nested sum $\sum_{i<j< \cdots < k} ij \cdots k$

I am wondering if there is any known closed form for the following nested sum? : $$ \sum_{i<j<\cdots <k} ij\cdots k $$ where each $i,j,\cdots,k =1, \cdots, n$ I tried the first one: $$ ...
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3answers
48 views

Calculate sum $S=\sum_{k=0}^{n}\binom{k}m$

Calculate sum $$S=\sum_{k=0}^{n}\begin{pmatrix} k \\ m\end{pmatrix}$$ My solution if $n<m$, $S=0$ else $$S=\sum_{k=0}^{n}\binom{k}m=\sum_{k=m}^{n}\begin{pmatrix} k \\ ...
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3answers
68 views

The value of the sum $\binom {20}0 -\binom {20}2+ \binom{20}4-…-\binom{20}{18}+\binom{20}{20}$

$\binom {20}0 -\binom {20}2+ \binom{20}4-...-\binom{20}{18}+\binom{20}{20}$ The question specifically gives intervals in which the answer is, but it's probably assumed that you should calculate the ...
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Multiplication Principle and Inclusion-Exclusion: $2^n = \sum_{i = 0}^n (-1)^i \binom{n}{i} \binom{2n - 2i}{n - 2i}$

I began to compose an unnecessarily complicated answer to this question: If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want ...
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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, ...
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1answer
56 views

prove $\sum \limits_{k=1}^n A(n,k){x+k-1 \choose n}=x^n$

A descent in the permutation $\sigma = a_1 \cdots a_n \in S_n$ is an index $i\in[n-1$] for which $a_i > a_{i+1}$. Let A(n, k) be the number of permutations of $[n]$ with $k-1$ descents where $n ...
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Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
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1answer
34 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
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5answers
136 views

How to prove $\sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0$ [closed]

I would like to prove that: \begin{equation*} \sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0;~k\geq0 ; n\geq1. \end{equation*} Can any one help me how to do that? Thanks
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Find a formula for the binomial coefficients of the Macluarin series for $\frac{1}{(1+x)^{1/2}}$

The Maclaurin series for $\frac{1}{(1+x)^{1/2}}$ is \begin{equation*} 1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}...~. \end{equation*} I can't figure much out other than it ...
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1answer
26 views

The biggest binomial coefficient in $(n+\frac 1n)^n$ if the product of the fourth member in the expansion and the fourth from back member is 14400

I'm stuck on this one. I am not expected to know how to solve cubic equations so this gets even more confusing, as i get $\binom n 3 = 120$. So I can't even calculate $n$. Is there a way to go around ...
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Beta functions and Pascal Pyramids

I've looked over this paper and there was something about the limits presented that looked rather familiar. Then I thought, hey, those look like beta functions: \begin{align} ...
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1answer
27 views

Are binomial series multiplicative in their bases?

In $ℚ[Z]$, by $Z \choose k$ denote the polynomial $${Z \choose k} = \frac{1}{k!}·\prod_{i=0}^{k-1} (Z-i),$$ so that ${Z \choose k}(n) = {n \choose k} = \frac{n!}{k!·(n-k)!}$ in $ℚ$. Now, in the ring ...
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Enough Information? (Linear Algebra over Finite Fields)

This problem works over the field $\mathbb{F}_p$. Suppose $p$ is a prime and the $i\in I$ index the set of $p$ vectors $v_i$. Fix $k$ an integer with $1\leq k<p$. Let $v_i$ have the following ...
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1answer
48 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
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Finding the Value of Ties

THE CHALLENGE If "Winning all the time = 1" "Losing all the time = -1" "Tie all the time = 0" What value would "Winning HALF the time" be? Is there an error in my original variables that would help ...
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1answer
30 views

q-binomial Identity

Unfortunately I am not able to solve the following problem: I tried finding a bijection similar to the prove of this binomial identity: $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$ ...
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1answer
29 views

Probability and Statictic / Binomial

The cost of a trial conducted in the research and development center of an industrial establishment is known to be 1 million dolars. If the test is negative, in addition to this a new trial is ...
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1answer
42 views

Generating the Nth combination of a binomial coefficient

I'm designing a protocol, and need a bit of help. I am able to neatly condense the problem I am having into a allegory, I hope it doesn't sound too contrived. Alice has flipped a coin ...
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1answer
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Questions on integer-valued polynomials

An integer-valued polynomial or numerical polynomial is a polynomial $f \in \mathbb Q[x]$ with the property that $f(\mathbb Z)\subseteq \mathbb Z$. The set of numerical polynomials forms a subring ...
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Proving a sequence of numbers in binomial

Consider the set $P_r={n\choose r}p^r(1-p)^{n-r}$ Prove that: $$\sum_{r=1}^nrP_r=np$$ By far I attempted: $$\sum_{r=1}^nr{n\choose r}p^r(1-p)^{n-r}=\sum_{r=1}^nn{n-1\choose r-1}p^r(1-p)^{n-r}$$ ...
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Proving that $r{n \choose r}=n{n-1\choose r-1}$

For proving that: $r{n \choose r}=n{n-1\choose r-1}$ I attempted it with: $r{n\choose r}=\frac{rn!}{r!(n-r)!}=\frac{n!}{(r-1)!(n-r)!}$ $n{n-1\choose ...
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110 views

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
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Formal power series coefficient problem

Find the coefficient of: $[x^{33}](x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}$ I have figured out that I need to use this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ But I ...
0
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2answers
65 views

In how many ways can you split six persons in two groups?

In how many ways can you split six persons in two groups? I think that I should use the binomial coefficient to calculate this but I dont know how. If the two groups has to have equal size, then ...
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2answers
93 views

Roots of a polynomial whose coefficients are ratios of binomial coefficients

Prove that $\left\{\cot^2\left(\dfrac{k\pi}{2n+1}\right)\right\}_{k=1}^{n}$ are the roots of the equation $$x^n-\dfrac{\dbinom{2n+1}{3}}{\dbinom{2n+1}{1}}x^{n-1} + ...
4
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1answer
119 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...
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1answer
25 views

Show that $(m-1)^{n-1}*n^{m-2}=\binom{n+m-1}{n}$ [closed]

I need help proving that $(m-1)^{n-1}*n^{m-2}=\binom{n+m-1}{n}$.
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1answer
46 views

Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that ...