# Tagged Questions

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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### An interesting Sum involving Binomial Coefficients

How would you evaluate $$\sum _{ k=1 }^{ n } k\left( \begin{matrix} 2n \\ n+k \end{matrix} \right)$$ I tried using Vandermonde identity but I can't seem to nail it down.
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### Combinatorial proof for two identities [duplicate]

Does exist a combinatorial proof for the following two identities ? $\sum_{k = 0}^{n} \binom{x+k}{k} = \binom{x+n+1}{n}$ $\sum_{k = 0}^{n} k\binom{n}{k} = n2^{n-1}$ I know how to derive the ...
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### How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}}$

Can we find $$\sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion?
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### Polynomial in $\mathbb{Q}[x]$ sending integers to integers?

We can view the binomial coefficient $\binom{x}{k}$ has a polynomial in $x$ with degree $k$. So taking some $f\in\mathbb{Q}[x]$, why is $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$, precisely when the ...
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### Inequality with central binomial coefficients

For every even positive number $N$ we have: $${2N \choose N } < 2^N {N \choose N/2 } < 2 {2N \choose N }$$ (Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2}$ for ...
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### Proving a binomial sum with simple result

In deriving a formula I've come up with an expression that I need to prove, specifically: $$(-1)^n = \sum_{j=1}^n (-1)^j \binom{n+1}{j+1} j^n$$ where $n$ is a positive integer. This seems ...
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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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### Smallest constant in exponent so that limit of sum is $0$

I am trying to work out the smallest constant $c>0$ so that $$\lim_{n \to \infty} \sum_{a=1}^n \sum_{b=0}^n {n \choose a} {n-a \choose b} \left({a+b \choose a} 2^{-a-b}\right)^{c n/\ln{n}} =0 .$$ ...
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### A binomial identity from Mathematical Reflections

Here is the problem: Let $m,n$ be positive integers with $n>m$. Prove that $\displaystyle\sum_{k=0}^{\lfloor\frac{n+m}2\rfloor} (-1)^{k}\binom{n}{k}\binom{m+n-2k}{n-1}=\binom{n}{m+1}$ This ...
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### Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$

Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$ I don't understand where to start. Please help.
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### Combinatorial proof that $\sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{(k+1)^2} = \frac{H_{n+1}}{n+1}$

This recent question contains two proofs that $$\sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{(k+1)^2} = \frac{H_{n+1}}{n+1}.$$ One, by Antonio Vargas, uses a double integral. The other, by me, uses the ...
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### Product of binomial coefficient as a basis

I am stuck with the following problem. Every polynomial of degree $d$ can be expressed as $$p(x) = p_d \binom{x}{d}+ p_{d-1}\binom{x}{d-1} + \cdots + p_0 \binom{x}{0}$$ What is the ...
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### $\binom{2p-1}{p-1}\equiv 1\pmod{\! p^2}$ implies $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{\! p^2}$; where $p>3$ is a prime?

From $\binom{2p-1}{p-1}\equiv 1\pmod{\! p^2}$ how does one get $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{\! p^2},\,\forall a,b \in \mathbb N,\, a>b$; where $p>3$ is a prime ?
Short version: can we prove that $$\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \frac{k!}{n^{2k}} \to \frac1e$$ as $n \to \infty$? Long version: First, consider $$a_n = \sum_{k=0}^n \frac{(-1)^k}{k!}$$ It is ...