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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1answer
759 views

Counting subsets with r mod 5 elements

Some time ago Qiaochu Yuan asked about counting subsets of a set whose number of elements is divisible by 3 (or 4). The story becomes even more interesting if one asks about number of subsets of ...
12
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2answers
348 views

A binomial identity

I was wandering if someone knows an elementary proof of the following identity: $$ \frac{(a)_n (b)_n}{(n!)^2} = \sum_{k=0}^n (-1)^k {1-a-b \choose k} \frac{(1-a)_{n-k}(1-b)_{n-k}}{((n-k)!)^2}\ , $$ ...
12
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1answer
167 views

Asymptotic Behavior of a Sum with Binomial Coefficients

The Problem: Find the asymptotic behavior (with respect to $n$) of the following sum $$\sum\limits_{j = 3}^n \binom{n}{j} \frac{(j - 1)!}{2\cdot n^j}. $$ Where the Problem Comes From: If we ...
12
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1answer
169 views

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

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ which can also ...
12
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1answer
181 views

“Binomiable” numbers

Is there a nice criterion to determine whether a given natural $m$ can be written as a binomial number $\binom{n}{k}$ with $1 < k < n-1$? I've been thinking on this problem with a friend and ...
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10answers
715 views

Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$

I need a hand in showing that $$ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$$ Thanks in advance for any help.
11
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6answers
634 views

Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$ [duplicate]

I need to evaluate, for a certain worded question: If n is even $$\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots\binom{n}{n}$$ If n is odd ...
11
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5answers
443 views

How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
11
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3answers
739 views

Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$

I'm looking for a reference with the proof of the following binomial identity: $$\sum_{k=0}^n \binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$$ I've looked in a number of textbooks that have a ...
11
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3answers
796 views

How to get ${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$

I found this in my test book, any hints? Given $${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$$ Then find the value of x and y in n. According to the answer ...
11
votes
3answers
349 views

Computing: $\lim\limits_{n\to\infty}\left(\prod\limits_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$

I try to compute the following limit: $$\lim_{n\to\infty}\left(\prod_{k=1}^{n} \binom{n}{k}\right)^\frac{1}{n}$$ I'm interested in finding some reasonable ways of solving the limit. I don't find any ...
11
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7answers
543 views

Summation simplification $\sum_{k=0}^{n} \binom{2n}{k}^2$

$\sum_{k=0}^{n} \binom{2n}{k}^2$ So i'm trying to simplify this one and I'm stuck in nowhere. Some kind of tip would be appreciated. Thanks! :)
11
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3answers
216 views

Calculating $\sum_{0\le k\le n/2} \binom{n-k}{k}$

I would like to evaluate: $$\sum_{0\le k\le n/2}\binom{n-k}{k}$$ Any idea?
11
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4answers
465 views

Show that if $\prod\limits_{k=1}^{n}(x+a_k)=\sum\limits_{k=0}^{n} {n\choose k}a^k_kx^{n-k}$ then $a_1=a_2=a_3=…=a_{n-1}=a_n$

Let $a_0=1$. Prove that, if $$\prod_{k=1}^{n}(x+a_k)=\sum_{k=0}^{n} {n\choose k}a^k_kx^{n-k}=x^n+{n\choose 1}a_1x^{n-1}+{n\choose 2}a^2_2x^{n-2}+....+a^n_n,$$ then ...
11
votes
2answers
214 views

Complicated sum with binomial coefficients

I know how to prove, that $\frac{1}{2^{n}}\cdot\sum\limits_{k=0}^nC_n^k \cdot \sqrt{1+2^{2n}v^{2k}(1-v)^{2(n-k)}}$ tends to 2 if n tends to infinity for $v\in (0,\, 1),\ v\neq 1/2$. This can be proved ...
11
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4answers
305 views

Determine $\displaystyle \lim_{n \to \infty}{{n} \choose {\frac{n}{2}}}\frac{1}{2^n}$, where each $n$ is even

For each positive even integer $n$, set $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$ Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value. Here's ...
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5answers
257 views

Sum $\displaystyle \sum_{n=i}^{\infty} {2n \choose n-i}^{-1}$

$\displaystyle \sum_{n=i}^{\infty} {2n \choose n-i}^{-1}=\sum_{n=i}^{\infty} \frac {1}{{2n \choose n-i}}$ is a very interesting one. Here is what I have from WolframAlpha. $\displaystyle ...
11
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3answers
206 views

Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$

$$ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$$ Class themes are: Generating functions and formal power series.
11
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1answer
917 views

Sum of product of binomial coefficient

Is the following true? $$\sum_{x_1+x_2+...+x_n=n}\ \ \, \prod_{i=1}^{n}{k_i\choose x_i}={\sum_{i=1}^{n}k_i \choose n} .$$ I tried to use the multinomial theorem, but it doesn't seem applicable.
11
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1answer
312 views

Combinatorial Proof of $\binom{\binom{n}{2}}{2} = 3 \binom{n}{3}+ 3 \binom{n}{4}$ for $n \geq 4$

For $n \geq 4$, show that $\binom{\binom{n}{2}}{2} = 3 \binom{n}{3}+ 3 \binom{n}{4}$. LHS: So we have a set of $\binom{n}{2}$ elements, and we are choosing a $2$ element subset. RHS: We are ...
11
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2answers
290 views

The Fermat prime 257 and binomial sum $\sum_{n=0}^\infty \frac{(-1)^n}{\binom {8n}{4n}}$?

We have, $\begin{aligned} \sum_{n=0}^\infty \frac{(-1)^n}{\binom n{n/2}} &= \frac{4}{27}(9-\pi\sqrt{3}\,)\\[2.5mm] \sum_{n=0}^\infty \frac{(-1)^n}{\binom {2n}n} &= \frac{4}{5} - ...
11
votes
2answers
571 views

Closed form for a sum involving binomial coefficients

Let $n,k$ be positive integers. Is there a closed form of the sum $$\sum_{s=0}^{k} \binom{n}{s} \binom{s}{k-s}\text{?}$$ By that I mean a representation which is free of sums and hypergeometric ...
11
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2answers
119 views

What are the “numerator” and “denominator” of binomial coefficients called?

Do the numbers $n$ and $k$ in the binomial coefficient $\binom nk$ have a name? For the fraction $\frac nk$ we would use numerator and denominator. But I have not seen some terminology for binomial ...
11
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1answer
190 views

Interpretation of a combinatorial identity

I am trying to find an combinatorial interpretation for the following combinatorial identity involving iterated binomial coefficients, which appeared in the November 1980 edition of The American Math ...
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3answers
642 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
11
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1answer
170 views

Curious combinatorial summation

Let $\gamma$ denote a grid walk from the upper left corner $(1,k)$ to the lower right corner $(\ell,1)$ of the $k\times\ell$ rectangle $\{1,..,k\}\times\{1,..,\ell\}$. There are ...
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3answers
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Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$

Is it possible to write this in closed form: $$\sum_{k=0}^{n} k\binom{n}{k}\log\left(\vphantom{\Huge A}\binom{n}{k}\right)$$ Can you get something like $$n2^{n-1}\log(2^{n-1})$$
11
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1answer
757 views

Factorial canceling on expansion of binomial coefficients on Concrete Mathematics

On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as: \[ \frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z} \] where \[ ...
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10answers
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How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?

I am trying to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ but am not able to think something except induction,which is of-course not necessary (I think) here, so I am ...
10
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6answers
305 views

Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? [duplicate]

Why, conceptually, is it that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}?$$ I know how to prove that this is true, but I don't understand conceptually why it makes sense.
10
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5answers
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factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
10
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4answers
224 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
10
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2answers
311 views

Proving that $\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b$

Prove that for all positive integers $b$ that $$\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b.$$ My idea is induction, but I cannot figure stuff out on the inductive step.
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Good upper bound for $\sum\limits_{i=1}^{k}{n \choose i}$?

I want an upper bound on $$\sum_{i=1}^k \binom{n}{i}.$$ $O(n^k)$ seems to be an overkill -- could you suggest a tighter bound ?
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2answers
351 views

The integral $\int_0^1 \frac{(x+1)^n-1}{x} dx$

I know that the integral $\int_0^1 \frac{(x+1)^n-1}{x} dx,$ for $n \in \mathbb{Z}^+$, can be evaluated by expanding the numerator with the binomial theorem and integrating term by term. You get the ...
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3answers
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Proof of the identity $\sum_{k=0}^{\min[p,q]}{p\choose k}{q\choose k}{n+k\choose p+q}={n\choose p}{n\choose q}$

Prove the identity: $$\sum_{k=0}^{\min[p,q]}{p\choose k}{q\choose k}{n+k\choose p+q}={n\choose p}{n\choose q}.$$
10
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6answers
205 views

A limit on binomial coefficients

Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$. What I can do is just use Stolz formula. But I could not proceed.
10
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3answers
561 views

proof that $1 = \sum\limits_{k=0}^n (-1)^k { 2n \choose n,k,n-k } \frac{n}{n+k}$

I'm looking for a proof of this identity: $$ 1 = \sum_{k=0}^n (-1)^k { 2n \choose n,k,n-k } \frac{n}{n+k} $$ I'll take anything, but a combinatorial proof would be nice - all of the terms in the sum ...
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2answers
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Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
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3answers
434 views

Problem of limit with binomial coefficients

I thought that the following would made a nice exercise, but I am not sure how to evaluate its difficulty since I often miss elementary solutions. How about you try answering it? It would be great to ...
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1answer
227 views

Evaluating $\sum_{k=0}^n \binom{n}{k} 2^{k^2}$

Can someone please help me simplifying this sum $$\sum_{k=0}^n \binom{n}{k} 2^{k^2}$$ Wolframalpha fails (see here). Thanks in advance. The sum counts the number of (labelled) digraphs (with ...
10
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1answer
279 views

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Let $n$ be a positve integer. Prove that$$\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$$ for each non-negative ...
10
votes
1answer
163 views

Can this product be written so that symmetry is manifest?

Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression $$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$ apparently computes the ...
10
votes
1answer
129 views

How find all postive integer number such $(n+k)\nmid \binom{2n}{n}$

Question: Find the all integer $k$,such there are exist infinitely many $n$ such $$(n+k)\nmid \binom{2n}{n}$$ This is china 2014 (CMO problem 4),it's have been end exam three hours ago. I ...
10
votes
2answers
236 views

How prove binomial cofficients $\sum_{k=0}^{[\frac{n}{3}]}(-1)^k\binom{n+1}{k}\binom{2n-3k}{n}=\sum_{k=[\frac{n}{2}]}^n\binom{n+1}{k}\binom{k}{n-k}$

How prove this $$\sum_{k=0}^{[\frac{n}{3}]}(-1)^k\binom{n+1}{k}\binom{2n-3k}{n}=\sum_{k=[\frac{n}{2}]}^n\binom{n+1}{k}\binom{k}{n-k}$$ This equation How prove it? Thank you I want take this ...
10
votes
2answers
520 views

Techniques for summing ratio of binomial coefficients

There are several identities that involve the sum of the product of binomial coefficients. However what I am searching for is an identity that involves the ratio of binomial coefficients. ...
10
votes
1answer
373 views

Construction of generating function from identity

I am trying to solve identity involving binomials and Fibonacci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
9
votes
7answers
661 views

Proof of the Hockey-Stick Identity: $\sum_{t=0}^n \binom tk = \binom{n+1}{k+1}$

After reading this question, the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$ What's the name of this identity? Is it the identity of the Pascal's triangle ...
9
votes
4answers
574 views

Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$

The other day a friend of mine showed me this sum: $\sum_{k=0}^n\binom{3n}{3k}$. To find the explicit formula I plugged it into mathematica and got $\frac{8^n+2(-1)^n}{3}$. I am curious as to how one ...
9
votes
3answers
430 views

Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$