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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Proof of $\sum_{n=1}^\infty \frac{1}{n^4 \binom{2n}{n}}=\frac{17\pi^4}{3240}$

Recently, I was able to prove that $$\sum_{n=1}^\infty \frac{1}{n \binom{2n}{n}}= \frac{\pi}{3\sqrt{3}}$$ $$\sum_{n=1}^\infty \frac{1}{n^2 \binom{2n}{n}}= \frac{\pi^2}{18}$$ But does anybody know ...
16
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3answers
302 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial coefficient ...
7
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4answers
695 views

Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
5
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5answers
328 views

Proving $\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$

Prove that $$\sum_{m=0}^M \binom{m+k}{k} = \binom{k+M+1}{k+1}$$ by computing the coefficient of $z^M$ in the identity $$(1 + z + z^2 + \cdots ) \cdot \frac{1}{(1-z)^{k+1}} = \frac1{(1-z)^{k+2}}.$$ I ...
2
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0answers
34 views

Is this quantity divisible by $p$?

Let $p$ be a prime. Let $$x_{1} = \binom{2p-1}{p}-1$$ $$x_{2}=\binom{3p-1}{2p}-\binom{2p-1}{p}$$ $$x_{3} = \binom{4p-1}{3p}-\binom{3p-1}{2p}$$ and I observed that for small values of $p$ $x_{1}$, $x_{...
2
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2answers
87 views

Summation of Binomial Coefficient: $\sum\binom{n+k}{2k} \binom{2k}k \frac{(-1)^l}{k+1}$

I am trying to solve this summation problem . $$\sum\limits_{k = 0}^\infty {\left( {\begin{array}{*{20}{l}} {n + k}\\ {2k} \end{array}} \right)} \left( {\begin{array}{*{20}{l}} {2k}\\ k \end{array}} \...
1
vote
1answer
60 views

binomial coefficients difference? [on hold]

I need a difference of 2 binomial coefficients that would be equivalent to the following sum: $12\choose5$+$11\choose5$+$10\choose5$+$9\choose5$+$8\choose5$ How to answer this?
9
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5answers
170 views

Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$ I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find ...
1
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1answer
21 views

Trinomial Pascal's Triangle

I know that there's a trinomial theorem (and a multinomial theorem), but I was wondering if there was a similar structure for trinomials as there is for binomials, like Pascal's triangle. Thanks in ...
0
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1answer
43 views

Binomial coefficient paths?

Here's a problem and my attempt to answer it: We want to get a binomial coefficient identity depending on grid walking. Starting from the bottom left corner and going to the top right corner. You can ...
1
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1answer
51 views

Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$

Is there a closed form for the following? $$\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$$ how about a pretty limit for large $b$. I have tried using the binomial expansion for the $(b+ck)^...
3
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1answer
87 views

What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

Since both $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor$ and $\sum_{i=0}^n \binom{n}{i}$ have simple closed-form evaluations, it is natural to consider the evaluation of the binomial sum $\sum_{...
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1answer
70 views

Decomposition of $ \binom {n} {j-1}j^k $

It is easy to check that: $$ \binom {n} {j-1}j = \binom {n-1} {j-1}+\binom {n-1} {j-2}(n+1) $$ and $$ \binom {n} {j-1}j^2 = \binom {n-2} {j-1}+\binom {n-2} {j-2}(3n+2)+\binom {n-2} {j-3}(n+1)^2 $$ We ...
2
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2answers
42 views

Coefficient of $x^{103}$ in the following multinomial expansion.

What is the coefficient of $x^{103}$ in the expansion of $$(1+x+x^2+x^3+x^4)^{199}(x-1)^{201}$$ ?. The answer is an integer between $0-9$. So I wrote the given expression as $(x^5-1)^{199}(x-1)^{2}$. ...
4
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0answers
216 views

Research: Looking for a sequence that produce variation's of Pascal's triangle

Prologue I am an undergraduate so if my terminology or approach seem inappropriate/confusing please explain in the comments. I created a notation where $$F(0 \rightarrow n,x) = [\hspace{1mm}F(0 ,...
7
votes
4answers
171 views

Why does $(128)!$ equal the product of these binomial coefficients $128! = \binom{128}{64}\binom{64}{32}^2 \dots \binom21^{64}$?

I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of. It asks to prove the following: $$128! = \binom{128}{64}\binom{64}{32}^2\...
0
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2answers
39 views

binomial inequality with sums

Assume I have a series of numbers $a_1 \dots a_n$ where $0 \leq a_i \leq n-1$ and a positive integer $r$. how to show that the sum of number of ways to choose $r$ from $a_i$ is at least as $n$ times ...
4
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2answers
480 views

This question involves Pascal's Triangle & Binomial Theorem. Full Question

Could anyone please help me on the following problem: Factorize the expression $P(n)=n^x-n$ for $x=2,3,4,5$ Determine if the expression is always divisible by the corresponding $x$. If divisible use ...
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1answer
74 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 ...
4
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3answers
699 views

No closed form for the partial sum of ${n\choose k}$ for $k \le K$?

In Concrete Mathematics, the authors state that there is no closed form for $$\sum_{k\le K}{n\choose k}.$$ This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the book)....
2
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4answers
119 views

increasing sum of binomial coefficients

I've been working on a problem and got to a point where I need the closed form of $$\sum_{k=1}^nk\binom{m+k}{m+1}.$$ I wasn't making any headway so I figured I would see what Wolfram Alpha could do....
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1answer
18 views

What's the formula to map between multiindices and indices?

What is the formula to map between multiindices and indices? By multiindex, I mean a variable $I\in\mathbb{N}^d$ where $|I|=\sum\limits_{i=1}^d I_i=n$. Here, $d$ denotes the dimension. Basically, ...
4
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3answers
572 views

Give the combinatorial proof of the identity $\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$

Given the identity $$\sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$$ Need to give a combinatorial proof a) in terms of subsets b) by interpreting the parts in terms of compositions of ...
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0answers
13 views

Limit analysis for binomial series raised to 0 < fractional exponent < 1

I am trying to expand a series and trying to find a limit analysis: (1+x)^a where 0 < a < 1. I understand that a possible expansion is: 1 + ax + a(a-1)(x^2)/(2!) + a(a-1)(a-2)(x^3)/(3!) +... ...
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0answers
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Is there an identity related to $\binom{n-j-1}{k}+\binom{k+j}{k}\pmod{n}$?

I noticed that when $n$ is an odd prime, the following congruence $$\binom{n-j-1}{k}+\binom{k+j}{k} \equiv 0 \pmod{n}$$ holds for $0 \le j \le \frac{(n-k)}2$ and odd values of $k$ such that $0 < k ...
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5answers
105 views

For all positive integer $n$ prove the equality: $\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}^2}{k+1}=\frac{\binom{2n}{n}}{2n}$

For all positive integer $n$ prove the equality: $$\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}^2}{k+1}=\frac{\binom{2n}{n}}{2n}$$ My work so far: $$\frac{n\binom{n-1}{k}}{k+1}=\frac{n(n-1)!}{(k+1)k!(n-k-...
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0answers
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If the last term of $(2^{1/3}-\frac{1}{\sqrt{2}})^n$ is $(\frac{1}{3^{5/3}})^{\log(\frac{8}{3})}$, what is the $5\rm{th}$ term from the beginning? [closed]

If the last term of $$\left(2^{1/3}-\frac{1}{\sqrt{2}}\right)^n$$ is $$\left(\frac{1}{3^{5/3}}\right)^{\log(\frac{8}{3})}$$ then the value of $5th$ term from beginning is ?. So I simplified $(243)^{1/...
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1answer
144 views

Combinatorial proof for $\sum_{i=r}^{n}(2i-r)\binom{i-1}{r-1}^2=r\binom{n}{r}^2$

I cfound the following identity, but I'm having trouble finding a combinatorial interpretation. Can someone help me? $$\sum_{i=r}^{n}(2i-r)\binom{i-1}{r-1}^2=r\binom{n}{r}^2$$
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2answers
86 views

How can I calculate $\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}$

I tried Google and various ways, including walking the list questions in the chronological order as far. How can I show $$\sum_{i=0}^{n}\sum_{j=i}^{n} \binom{n+1}{j+1}\binom{n}{i}=2^{2n}$$ ...
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1answer
1k views

binomial calculation method

I want solve this probability: For $p= 0.4$ $q=0.8$ $n= 20$ $1-P(5<x<11)$ = $1-\sum_{k=6}^{10} \binom{20}{k}(0.4)^k(0.6)^{20-k}. Wolfram Alpha -> = 0,2531$ Is calculation method ...
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1answer
24 views

Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
2
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3answers
84 views

Formula for $\sum_{i\geq 0} i{n \choose 2i}$?

So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone ...
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2answers
38 views

Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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1answer
24 views

# of bit strings of length n (even>2), with n/2-1 zeros and n/2+1 ones, zero followed by one

case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one Answer: 3 case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, ...
0
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1answer
127 views

Prime Factors in Pascal's Triangle

The question about reversing n choose k made me look a little further into Pascal's triangle, but my curiosity is not satiated. I am now curious of the following: Given $ n > k > 1 $, show ...
3
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2answers
116 views

Binomial theorem incomplete expansion

I got this question and I am little bit confused, whether the question is correct or not. $n\choose0$-$ n\choose1 $+$ n\choose2$-$ n\choose3$+.........+$(-1)^r$$ n\choose r$=$28$ Now we are ...
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2answers
64 views

Understanding steps in a proof of $\sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1}$

So, the task is to prove that: $$\sum_{k=1}^n\binom{n-1}{k-1}\binom{2n-1}{k}^{-1}=\frac{2}{n+1}$$ I tried different methods but none led me to the solution. I looked up the solution and I can't even ...
0
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0answers
31 views

How to fast compute the coefficient of $x^{2^{n-1}}$ in polynomial $[\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]]^{2^{n-r+1}}$?

How to fast compute the coefficient of $x^{2^{n-1}}$ in polynomial $$\left(\frac{1}{2}[(1+x)^{2^{r-1}}+(1-x)^{2^{r-1}}]\right)^{2^{n-r+1}}$$ or compute the coefficient of any term $x^k$ for $k\geq 0$? ...
2
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3answers
68 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
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2answers
91 views

Sum of binomial coefficients $\sum_{k=n}^{r}\binom{k}{n}$

Is is possible to find the sum of a binomial coefficient series like: $\sum_{k=n}^{r}\dbinom{k}{n}$? Just a random thought.
3
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2answers
129 views

A combinatorial identity $\sum_{i=0}^k \binom ni \binom{-n}{k-i} =0$

Can anyone prove the following identity for me? $\sum_{i=0}^k \begin{pmatrix} n\\ i \end{pmatrix} \begin{pmatrix} -n\\ k-i \end{pmatrix}=0$ for any positive integers $n,k$. I'm pretty sure this is ...
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2answers
51 views

Binomial Coefficient Identity Involving Summation

Prove that $$\sum_{j=0}^n (-1)^j \binom{n+j-1}{j}\binom{N+n}{n-j} = \binom{N}{n} $$ I tried to prove this via binomial expansions of $(1-x)^N (1+x)^{-m}$, and equating the coefficients of $x$, ...
16
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6answers
3k views

Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates?
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1answer
57 views

A sum of squared binomial coefficients

I've been wondering how to work out the compact form of the following. $$\sum^{50}_{k=1}\binom{101}{2k+1}^{2}$$
2
votes
3answers
38 views

Multiple objects, Number of combinations

A bag contains colored balls: 8 red balls 4 white balls 4 blue balls 4 green balls 2 purple balls 2 orange balls 1 yellow ball 1 black ball Total of 26 balls. I'd like to determine the number of ...
0
votes
2answers
52 views

Triangular numbers and pascal's trangle

The following are the triangular numbers. rank = 1 2 3 4 5 6 term = 1 3 6 10 15 21 A rule for triangular numbers is: ...
1
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2answers
92 views

What is the probability that when flipping a fair coin ten heads will be thrown in a row

I just cant figure this out.. Should I be using binomial distribution? Chance of getting ten tail long series in 100 coin throws
5
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1answer
649 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
0
votes
3answers
110 views

Finding the coefficient of $x^7$ in the expansion $(1 + x)^{23}$

By definition, the Binomial Theorem states: $$(x+y)^n = {n\choose 0}x^n + {n\choose 1}x^{n-1}y + {n\choose 2}x^{n-2}y^2 + \cdots + {n \choose {n-1}}xy^{n-1} + {n \choose n}y^n$$ For any $x,y\in\...
4
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2answers
86 views

Binomial coefficients mod p

I want to find the following sum mod $p$ (prime number): Let $i\geq \frac{p-1}{2}$, $ \sum_{k=i}^{p-1} \binom{k}{i}\binom{k}{p-i-1} \pmod{p} $ OK, I succeeded in simplyfying this argument to the ...