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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Derivation of bound on expression involving binomial coefficient from Erdős and Rényi 1959

I'm in the process of working through Erdős and Rényi's 1959 article "On Random Graphs I". In the proof of the first Lemma, equation 14 gives a bound on an expression involving several binomial ...
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1answer
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$\binom{n}{k} : \binom{n}{k+1} : \binom{n}{k+2} = a : b : c$

It is a rather surprising fact (to me, at least) that $\displaystyle \binom{14}{4} = 1001$; $\displaystyle \binom{14}{5} = 2002$; $\displaystyle \binom{14}{6} = 3003$. Actually, this is the only ...
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Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
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Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
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Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$

Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ ...
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Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$

We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$
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The $n^{th}$ root of the geometric mean of binomial coefficients.

$\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean. Prove $$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$
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Prove that $\binom n2 + \binom {n-1}2$ is always a perfect square

Prove that if $n$ is a positive integer and $n >1$: $$\binom n2 + \binom {n-1}2$$ is always a perfect square. I know we need to turn that into a binomial, but I can't follow how. Please note I'm ...
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Derivation of binomial coefficient in binomial theorem.

How was the binomial coefficient of the binomial theorem derived? $$\frac{n!}{k!(n-k)!}$$
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Proof of a combinatorial identity: $\sum_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$ [duplicate]

Possible Duplicate: Identity involving binomial coefficients This was part of a homework assignment that I had, and I couldn't figure it out. Now it is bugging me. Can anyone help me? ...
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Hard elementary combinatorics problem

How does one compute (without brute force) the smallest integer $n$ such that $\binom{2n}{1}(-3)^0 + \binom{2n}{3}(-3)^1 + \binom{2n}{5}(-3)^2 + \cdots + \binom{2n}{2n-1}(-3)^{(n-1)} = 0$?
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Are there surprisingly identical binomial coefficients?

Suppose $\binom{n}{k}=\binom{n'}{k'}$ with $k \geq 2$, $k' \geq 2$, $n \geq 2k$ and $n' \geq 2k'$. Does it follow that $n=n'$ and $k=k'$? EDIT: Yup, ...
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Find the sum of this series :$ \frac{1}{{1!2009!}} + \frac{1}{{3!2007!}} + \cdots + \frac{1}{{1!2009!}}$

Find the sum of this series : $$\sum\limits_{\scriptstyle 1 \leqslant x \leqslant 2009 \atop {\scriptstyle x+y=2010 \atop \scriptstyle {\text{ }}x,y{\text{ odd}} }} {\frac{1}{{x!y!}}} = ...
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Summation Identity: $\sum_{i=1}^ni^3 = \left( \frac{n(n+1)}{2} \right)^2$

I have to prove: $$\sum\limits_{i = 1}^n i^3 = \Bigg( \frac{n(n+1)}{2}\Bigg)^2$$ Using the following: $$n^3 = 6 {n \choose 3} + 6 {n \choose 2} + n \quad \forall n \in \mathbb{N}$$ My work is that ...
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Asymptotics of $\sum_{k=0}^{n} {\binom n k}^a$

I need to estimate the asymptotics of $$\sum_{k=0}^{n} {\binom n k}^a, \quad a>2, \quad a \in \mathbb{N}$$ In particular, I'm pretty much interested in $a=4$ case, but if the general solution ...
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Proving $ { 2n \choose n } = 2^n \frac{ 1 \cdot 3 \cdot 5 \cdots (2n-1)}{n!} $

How to prove this binomial identity : $$ { 2n \choose n } = 2^n \frac{ 1 \cdot 3 \cdot 5 \cdots (2n-1)}{n!} $$ The left hand side arises while solving a standard binomial problem the right hand ...
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4answers
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Find closed form solution using generation function for the binomial coefficients

I don't have any idea how to start this problem. Could you give a hint? Find closed form solution using generation function for the binomial coefficients: $$a_n:=\sum_{k=0}^{n}\binom{n}{k}^2(-1)^k$$ ...
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Showing that $\lceil (\sqrt{3} + 1)^{2n} \rceil$ is divisible by $2^{n+1}$.

I have a question which has fluxommed me and my pals for the past few days. Any help or solution is welcome Show using Binomial theorem that the integer just after $(3^{1/2} + 1)^{2n}$ is divisble ...
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4answers
406 views

The binomial formula and the value of 0^0

Here is the text from Knuth's The Art of computer programming, 1.2.6 F formula 14: Knuth doesn't give the proof of the statement. So, I tried to write it myself. To make binomial formula equal to ...
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Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
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Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
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Find the Pascal's Limit [closed]

Let $P_{n}$ be the product of the numbers in row of Pascal's Triangle. Then evaluate $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}}$$
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Choice Problem: choose 5 days in a month, consecutive days are forbidden [duplicate]

I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. Example 3.19. A medical student has to work in a hospital for five days in January. ...
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Truncated alternating binomial sum

It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem. I'm trying to figure out what happens with the ...
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The sum $\sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j$

A recent answer of mine to a question on Math Overflow includes the sum $$S(n,k,x) = \sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j,$$ where $\left\{ j \atop k \right\}$ is a Stirling number ...
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partial sum involving factorials

Here is an interesting series I ran across. It is a binomial-type identity. $\displaystyle \sum_{k=0}^{n}\frac{(2n-k)!\cdot 2^{k}}{(n-k)!}=4^{n}\cdot n!$ I tried all sorts of playing around, ...
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Yet another sum involving binomial coefficients

Let $k,p$ be positive integers. Is there a closed form for the sums $$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$ $$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$ ...
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Combinatorics question: Show divisibility

Let $a\geq2$, $b\geq2$ be two prime numbers and k be a natural number with $k\leq min(a,b)$. How can one show that $z := \binom{a+b}{k} - \binom{a}{k} - \binom{b}{k}$ is divisible by the product ...
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Evaluate $\sum_{r=0}^n \binom{n}{r}\sin rx \cos (n-r)x$

Evaluate $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$ I tried to use binomial identities, but since there are trigonometric terms, I don't have the idea ...
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Squared binomial coefficient

I've got the following finite sum: $s_{n}=\sum\limits_{k=0}^{n}\binom{n}{k}^2p^k$ (esp. if $p$ is a function of $n$, like $p=\frac1{n}$), which can be rewritten as ...
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Expansion of $ (a_1 + a_2 + \cdots + a_k)^n $

Is there an expansion for the following summation? $$ (a_1 + a_2 + \cdots + a_k)^n $$
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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} + ...
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Calculate the binomial sum $ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i} $

I need any hint with calculating of the sum $$ I_n=\sum_{i=0}^n (-1)^i { 2n+1-i \choose i}. $$ Maple give the strange unsimplified result $$ I_n={\frac {1/12\,i\sqrt {3} \left( - \left( \left( ...
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How to prove the following limit: $\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0$?

How to prove the following limit: $$\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0?$$
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Show that $p \in \left[\frac{4^m}{2\sqrt{m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
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Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?

Wolframalpha tells me that $$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$ However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
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Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$

I am looking for the combinatorial argument for the identity: \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation} This is easy to show algebraically as: \begin{equation} \binom{n}{k} ...
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How to count matrices with rows and columns with an odd number of ones?

I proved that $\displaystyle \left(\sum_{k\, \rm odd}\binom{m}{k}\right)^{n-1}=\left(\sum_{k\;{\rm odd}}\binom{n}{k}\right)^{m-1}$ by counting matrices of size $n\times m$ with entries in $\{0,1\}$ ...
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About one generating function

Initially, I have the following problem: find $$\sum_{k=0}^{n+1}(−1)^{n−k}4^k{n+k+1 \choose 2k}.$$ I thought, if I found the function $g_n(x) = \sum_{k=0}^{n}{n+k \choose 2k}x^k$, the answer would be ...
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Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$

Show that $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$. In the LHS $\binom{n+r-1}{r}$ counts the number of ways of selecting $r$ objects from a set of size $n$ where ...
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Negative Exponents in Binomial Theorem

I'm looking at extensions of the binomial formula to negative powers. I've figured out how to do $n \choose k$ when $n < 0 $ and $k \geq 0$: $${n \choose k} = (-1)^k {-n + k - 1 \choose k}$$ So ...
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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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Cousin of the Vandermonde binomial identity

The Vandermonde binomial identity can be expressed as \begin{align*} \sum_{i+j=r} \binom{m}{i} \binom{n}{j} = \binom{m+n}{r} && r \leq m +n. \end{align*} While working on an algebra problem, I ...
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$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof

Is there a simple combinatorial proof for the following identity? $$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$ where $m,j \geq 0$, $k \geq n \geq 0$.
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Bounds on $\sum_{k=0}^{m} \binom{n}{k}x^k$ and $\sum_{k=0}^{m} \binom{n}{k}x^k(1-x)^{n-k}, m<n$

I've read this interesting article by Woersch (1994) dealing with approximation of binomial coefficients (rows of Pascal's triangle). I'm just wondering if similar bounds exist for partial binomial ...
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312 views

A binomial coefficient identity?

Suppose $p$, $k$ and $s$ are integers with $s,k \le p$. Consider the following polynomial in $x$ and $y$, $$ \sum_{\ell=0}^k \binom{s}{\ell} \binom{p-s}{k-\ell} x^\ell y^{p-\ell}$$ Does this ...
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1answer
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A finite sum involving the binomial coefficients and the harmonic numbers

Wikipedia has a proof of the identity $$ H_{n} =\sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$$ http://en.wikipedia.org/wiki/Harmonic_number#Calculation Curiously, there is also the identity ...
7
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1answer
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Determining when a certain binomial sum vanishes

Consider the following sum of signed binomial coefficients: $$S_{n,a,p} = \sum_{i \equiv a \mod p} \binom{n}{i}(-1)^i$$ ($n$ is a positive integer, $p$ is an odd prime, $a$ is between $0$ and ...
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3answers
282 views

Asymptotic difference between a function and its binomial average

The origin of this question is the identity $$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n - \sum_{k=1}^n \frac{1}{k 2^k}\right),$$ where $H_n$ is the $n$th harmonic number. Dividing by $2^n$, we ...
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251 views

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 ...