# Questions tagged [binomial-coefficients]

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### Combined identity perturbation

I found the interesting inequality when I study hypergraph 2-coloring $$\sum_{i+j=k} \binom{r-1}{i}\binom{r-1}{j}(1-p)^i(1+p)^j \leq \binom{2r-2}{k}$$ $0\leq i, j < r$, $0\leq p \leq 1$. I want to ...
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### A conjecture on 'truncated joint moments' of binomial coefficients under binomial distribution

This is similar in spirit to Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution but gives some total estimates. Though the other ...
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### Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution

$\mu=1+\epsilon$ where $\epsilon>0$ holds. 1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$ This quantity can be ...
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### For human proofs of two novel combinatorial identities

For $n=0,1,2,\ldots$, let us define the polynomial $$S_n(x):=\sum_{k=0}^n\binom{x/2}k\binom{(x-1)/2}k\binom{-(x+1)/2}{n-k}\binom{-(x+2)/2}{n-k}.$$ Such polynomials occur in some series for $1/\pi$ ...
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### A combinatorics question: $\lim\limits_{n \to \infty} \frac1{2^{2n}} \sum\limits_{k=1}^n \sum\limits_{i=0}^{k-1} \binom nk \binom ni = \frac12$ [closed]

Am trying to show that $\lim_{n \rightarrow \infty} \frac{1}{2^{2n}} \sum_{k=1}^n \sum_{i=0}^{k-1} \binom{n}{k} \binom{n}{i} =0.5.$ I think that the above result is true but am not sure how to prove ...
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### Counting binary vectors that satisfy given distance constraints

Let's start with a warm up problem. Suppose I am given two binary vectors $x, y \in \{0, 1\}^n$ of length $n$ that differ in exactly $r$ places, i.e. $||x-y||_0=r$, i.e. their hamming distance is $r$...
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### Showing this formula counts these things

I'm writing an article, and I got stuck trying to prove that some numbers are positive. I have a relatively good intuition for guessing what an expression is counting, but in this case I'm not being ...
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### Positivity of a finite sum involving Stirling numbers of the first kind

Past days I've been trying to prove that certain polynomials have positive coefficients. After a lot of thinking, I came up with a formula for each coefficient individually, and they are not that ugly....
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### Showing equality of Eberlein polynomials

I have thought about the following question a long time and still got no progress. Currently I am writing my master thesis about association schemes in combinatorics and need an equality which seems ...
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### Q-analogue of an inequality

Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$. It is not super-difficult to prove the inequality $$\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.$$ This is actually quite a nice inequality that was ...
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### How to calculate$\sum \limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}$?

How to calculate $$\sum\limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}.$$
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### How to calculate: $\sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}$

How to calculate: $$\sum _{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$
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### A clean upper bound for the expectation of a function of a binomial random variable

I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
### Does the set $\{\binom x3+\binom y3+\binom z3:\ x,y,z\in\mathbb Z\}$ contain all integers?
The Gauss-Legendre theorem on sums of three squares states that $$\{x^2+y^2+z^2:\ x,y,z\in\mathbb Z\}=\mathbb N\setminus\{4^k(8m+7):\ k,m\in\mathbb N\},$$ where $\mathbb N=\{0,1,2,\ldots\}$. It is ...