# Tagged Questions

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### How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$ Question: $P_m(x)$ always ...
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### Alternating sum of square roots of binomial coefficients

Let $$c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}.$$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
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### Prove that expression is integer

Numerical experiments suggest that $\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$ is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, ...
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### Binomial supercongruences: is there any reason for them?

One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots,$$ of the Apéry ...
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### Are there good bounds on binomial coefficients?

Motivated by the central limit theorem, one expects that $$\binom{n}{k} \approx \frac{2^n}{\sqrt{\pi n/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right).$$ Computations suggest that the ratio of the two ...
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### A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$H_n=\sum_{k=1}^{n}\frac{1}{k},$$ and the generalized harmonic numbers are defined by $$H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}.$$ Recently, I have found ...
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### Maximum value of the binomial coefficient as a polynomial

What is the maximum (absolute) value of the binomial coefficient $\begin{pmatrix}x \\ k\end{pmatrix} = \frac{1}{k!}x(x-1)(x-2)\dotsb(x-k+1)$ for real $x$ in the interval $0 \leq x \leq k-1$?
### Divisibility of a binomial coefficient by $p^2$ — current status
While skimming the book Concrete Mathematics, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96) Is ${2n \choose n}$ ...