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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85 views

Show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$

In this question we are asked to show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$ What I did: $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \sum_{k=2012}^{n} 2^k*1^{n-k}\binom{n}{k} \leq ...
6
votes
3answers
106 views

Prove that $\prod_{k=1}^{\infty} \big\{(1+\frac1{k})^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$

This result, $$\prod_{k=1}^{\infty} \big\{\big(1+\frac1{k}\big)^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$$ is in a paper by Hirschhorn in the current issue of the Fibonacci Quarterly (vol. ...
6
votes
1answer
174 views

An enlightening proof of a specific combinatorial identity

Concerns about the arithmetic genus of projective hypersurfaces led me to make the following combinatorial conjecture: $${d-1\choose n+1} =\sum_{i=0}^{n+1} (-1)^{n+i+1} {d\choose i}$$ for $d \geq 1$, ...
6
votes
1answer
175 views

$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ 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 $m,j \geq 0$, $k \geq n \geq 0$.
6
votes
1answer
894 views

Partial sum of rows of Pascal's triangle

I'm interested in finding $$\sum_{k=0}^m \binom{n}{k}, \quad m<n$$ which form rows of Pascal's triangle. Surely $\sum\limits_{k=0}^n \binom{k}{m}$ using addition formula, but the one above ...
6
votes
2answers
394 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
6
votes
2answers
186 views

A resemblance between 2 binomial identities - why?

Let $F$ be any field (or even ring). The following formal power series identity (i.e., equality in $F[[x]]$) holds for any $j \ge 0$: $$(1-x)^{-j} = \sum_{i \ge 0} \binom{i +j -1}{i} x^i $$ The ...
6
votes
1answer
135 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 ...
6
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1answer
334 views

Vandermonde's-like identity

The Vandermonde's identity gives $$\sum_{k=0}^r \binom{m}{k}\binom{n}{r-k}=\binom{m+n}{r}.$$ Here is an example of Vandermonde's-like identity: For all $0 \le m \le n$, $$\sum_{k=0}^{2m} ...
6
votes
1answer
758 views

Relation between different ways of accessing bernoulli numbers with matrices

First Variant: Bernoulli numbers can easily be expressed by linear algebra equations. For example just using the recursion formula $$\sum_{k=0}^{n-1}{n\choose k}B_k=0$$ which is equation (34) from ...
6
votes
1answer
222 views

Lower bound on binomial coefficient

I encountered the following claim $$\frac{1}{n+1}2^{nH_2(k/n)} \le \binom{n}{k} \le 2^{nH_2(k/n)}$$ where $H_2$ is the binary entropy function. The upper bound is rather well known but how does one ...
6
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0answers
139 views

Sum with binomial coefficients and a square root

I encountered this sum from working on an integral: $$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sqrt{k}$$ I don't think it can be written as a hypergeometric function, because of this square root. Does ...
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0answers
109 views

Binomial formula for $(x+1)^{1/3}$ (related to Newton's binomial theorem)

I know that $$\displaystyle \sqrt{1+x} = \sum_{j=0}^{\infty}\left( \frac{(-1)^{(j-1)}}{2^{2j-1}\cdot(2j-1)}\binom{2j-1}{j}x^j\right). $$ Now, I want to evaluate $\sqrt[3]{1+x}$ but stuck at some ...
6
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0answers
121 views

Characterization of Sequences with Integral Binomial Coefficients

For any sequence of positive integers $\{ a_i \}_{i \ge 1}$ we can define the generalized binomial coefficients $\binom{n}{k}_{a}$ as follows: $$m!_a = a_1 a_2 \cdots a_m, \binom{n}{k}_a = ...
6
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0answers
143 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
6
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0answers
269 views

Construction of generating function from identity

I am trying to solve identity involving binomials and fibbonaci 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 ...
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7answers
294 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
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3answers
288 views

Fraction of two binomial coefficients

In an exercise I was asked to simplify a term containing the following fraction: $${\binom{m}{k}\over\binom{n}{k}}$$ The solution does assume the following is true in the first step, without ...
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2answers
201 views

What's the intuition behind this equality involving combinatorics? [duplicate]

What is the intuition behind $$ \binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k} $$ ? I can't grasp why picking a group of $k$ out of $n$ bijects to first picking a group of $k-1$ out of $n-1$ ...
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218 views

A question about limit

My question is: What is the result of this limit: $\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?
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2answers
89 views

Evaluate: $ \sum _ {k=1}^{n}{\frac{k}{n}\binom{n}{k}t^k(1-t)^{n-k}}$

Evaluate: $$ \sum _ {k=1}^{n}{\frac{k}{n}\dbinom{n}{k}t^k(1-t)^{n-k}} $$ $\dbinom{n}{k}$stands for the usual binomial coefficient giving the number of ways of choosing $k$ objects from n objects. ...
5
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4answers
178 views

Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$

I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of ...
5
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3answers
233 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
5
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3answers
191 views

Calculate $\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $

Is any formula which can help me to calculate directly the following sum : $$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$ Or can you help me for an approximation? Thanks :)
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4answers
228 views

Identity for $\sum\limits_{j = a}^{N} \binom{N}{j} \binom{j}{a} d^{-j}$?

I have run across the following multinomial series: $$ \sum_{j = a}^{N} \binom{N}{j} \binom{j}{a} d^{-j} $$ Here, $d>1$. This seems like a formula which has either a well-known identity, ...
5
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2answers
98 views

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is? I tried this: ...
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5answers
1k views

Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $

[Corrected question] I'm struggling at proving the following combinatorical identity: $$\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $$ I would like to see a combinatorical (logical) solution, ...
5
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2answers
61 views

Partial fraction expansion of $\frac{1}{x(x+1)(x+2)\cdots(x+n)}$

I try to find a partial fraction expansion of $\dfrac{1}{\prod_{k=0}^n (x+k)}$ (to calculate its integral). After checking some values of $n$, I noticed that it seems to be true that ...
5
votes
4answers
476 views

How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$ [duplicate]

I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do ...
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427 views

A Binomial Coefficient Sum

In my work on $f$-vectors in polytopes, I ran across an interesting sum which has resisted all attempts of algebraic simplification. Does the following binomial coefficient sum simplify? \begin{align} ...
5
votes
2answers
231 views

Proof of inequality $\sum\limits_{k=0}^{n}\binom n k\frac{5^k}{5^k+1}\ge\frac{2^n\cdot 5^n}{3^n+5^n}$

Show that $$\sum_{k=0}^{n}\binom n k\frac{5^k}{5^k+1}\ge\frac{2^n\cdot 5^n}{3^n+5^n}$$ where $$\binom n k=\frac{n!}{k!(n-k)!}$$
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2answers
101 views

Binomial coefficients equal to a prime squared

I am looking for some reading on when binomial coefficients are equal to $p^2$ for $p$ a prime. In general I imagine this is rare, as there are simply too many factors. Concretely, I am looking for ...
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3answers
670 views

Proving that $\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$

I have to prove that: $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$ I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 ...
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2answers
3k views

A fair coin is tossed $n$ times by two people. What is the probability that they get same number of heads?

Say we have Tom and John, each tosses a fair coin $n$ times. What is the probability that they get same number of heads? I tried to do it this way: individually, the probability of getting $k$ ...
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589 views

How can I simplify this expression involving binomial coefficients?

How can I simplify the following expression? $$\sum_{k=1}^n \binom{n}{k}^2$$
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306 views

What does the notation $\binom{n}{i}$ mean?

What do the parentheses next to the summation involving the binomial coefficients mean? Like this: $$\sum _{i=0}^{n} \binom{n}{i}a^{(n-i)}b^i=\left(a+b\right)^n $$
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3answers
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How to simplify or calculate a formula with very big factorials

I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
5
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3answers
661 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
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1answer
148 views

Limit of $ \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right)$

I can see numerically that $$\lim_{n \to \infty} \sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right) = 1$$ but how can you prove this? Using Stirling's approximation ...
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213 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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3answers
504 views

Bounding ${n \choose k}$

Let $k > 0$, and $n > 2k$. Why is it necessarily true that $${n \choose k} > \frac{n^k}{2^k k!}$$ And is the condition $n > 2k$ necessary for this inequality to hold?
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6answers
220 views

Asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k$

I am looking to show that $$\lim_{n \rightarrow \infty}\frac{1}{e^n}\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k = 0. $$ In my application, $c = (e+1)/2 \approx 1.85914\ldots$. I have been ...
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2answers
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Prove that $\lim_{n \to \infty} \binom{n}{k}a^n = 0$

I'm working with this problem but I have no idea how to solve it. Here $k$ is fixed and $0<a<1$. I was trying to use that $\lim_{n \to \infty} a^n =0$ and that $\binom{n}{k}\leq\frac{n^k}{k!}$ ...
5
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2answers
770 views

Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$

How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in ...
5
votes
1answer
273 views

Is this binomial coefficient identity known?

I stumbled across this identity involving binomial coefficients this morning: If $n$, $k$, $a$, and $b$ are positive integers and $n=a+b$, then $ \displaystyle \binom{n}{k} =\sum_{i=0}^k ...
5
votes
2answers
172 views

Show that the $k$th forward difference of $x^n$ is divisible by $k!$

Define the forward difference operator $$\Delta f(x) = f(x+1) - f(x)$$ I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices ...
5
votes
3answers
212 views

Two series relations, each one implies the other - from Andrews' partition book

That's my first question here, and i was encouraged to post because my question in MathOverflow (HERE) was beautifully and fast answered. But my questions in not at research level... As i said there, ...
5
votes
1answer
181 views

Summation of weighted squares of binomial coefficients

Show that $$\sum_{k=0}^n \left[ \frac{n-2k}{n} {n\choose k}\right]^2=\frac{2}{n}{2n-2 \choose n-1}.$$
5
votes
4answers
217 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 ...
5
votes
3answers
104 views

Closed formula for linear binomial identity

I have the following identity: \begin{equation} m^4 = Z{m\choose 4}+Y{m\choose 3}+X{m\choose 2}+W{m\choose 1} \end{equation} I solved for the values and learned of the interpretation of W, X, Y, and ...