Intereting Posts

An application of the Inverse function theorem
Sudoku puzzle with exactly 3 solutions
A sum including binomial coefficients
Prove that the normalisation of $A=k/(Y^2-X^2-X^3)$ is $k$ where $t=Y/X$ (Reid, Exercise 4.5)
What is the difference between a function and a distribution?
Probability of making it across a path of $n$ tiles through random walk
Show that there is a invertible continuous function $h: \mathbb{Q} → \mathbb{Q}$ such that $h(−1) = 0$, $h(0) = 1$, $h(1) = −1$.
Ideal of the pullback of a closed subscheme
Plane less a finite number of points is connected
In a noetherian integral domain every non invertible element is a product of irreducible elements
Complex Analysis, Entire functions
$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?
Probability of cycle in random graph
How to find the sine of an angle
Why is every map to an indiscrete space continuous?

This is a conjecture:

How can I prove that

\begin{equation}

\left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| \leq \binom{n}{r}

\end{equation}

- How to show $\binom{2p}{p} \equiv 2\pmod p$?
- Infinite Series $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$
- Eigenvectors of harmonic series matrix
- Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
- How is $\lim_{x \to a}\left(\frac{x^n - a^n}{x - a}\right) = n\times a^{n-1}$?
- Evaluating 'combinatorial' sum
for $0\leq a \leq n$, $0\leq r \leq n$ and $n,r,a \in \mathbb{N}$ ?

- Minimal cut edges number in connected Eulerian graph.
- Asymptotics of binomial coefficients and the entropy function
- Finding an MST among all spanning trees with maximum of white edges
- At least one monochromatic triangle from $p_n=\lfloor{en!}\rfloor+1$ points
- why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
- Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
- Maximum number of edges in a simple graph?
- How can I prove this combinatorial identity $\sum_{j=0}^n j\binom{2n}{n+j}\binom{m+j-1}{2m-1}=m\cdot4^{n-m}\cdot\binom{n}{m}$?
- Hamiltonian and non-Hamiltonian connected graph using the same degree sequence
- In graph theory, what is linked list?

We have $$\left|\sum_{i=0}^{r}\left(-1\right)^{r}\dbinom{a}{i}\dbinom{n-a}{r-i}\right|\leq\sum_{i=0}^{r}\dbinom{a}{i}\dbinom{n-a}{r-i}=\dbinom{n}{r}

$$ where the last identity follows from the Chu-Vandermonde identity.

- Large $n$ asymptotic of $\int_0^\infty \left( 1 + x/n\right)^{n-1} \exp(-x) \, \mathrm{d} x$
- ${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$
- Why is CH true if it cannot be proved?
- Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$
- Show that $u_1^3+u_2^3+\cdots+u_n^3$ is a multiple of $u_1+u_2+\cdots+u_n$
- Finding Rotation Axis and Angle to Align Two “Oriented Vectors”
- How much area in a unit square is not covered by $k$ disks of area $1/k$ centered at random points within the square?
- Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple
- Solving equations of form $3^n – 1 \bmod{k} = 0$, $k$ prime
- Find the number of irreducible polynomials in any given degree
- Estimating a probability of head of a biased coin
- Outer measure is countably subadditive
- $\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$
- Finite State Markov Chain Stationary Distribution
- Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?