Let $p$, $q$, and $n$ be three positive integers, show that $$\sum_{k=0}^n\binom{p+k}{p}\binom{q+n-k}{q} = \binom{p+q+n+1}{p+q+1}$$

Calculate the value of $$\displaystyle\sum_{k=0}^{n}\frac{1}{2^k}\binom{n}{k}$$

Compute the value of $$\sum_{k=0}^{n}(-1)^k\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}-\frac{1}{2}\binom{n}{1}+\frac{1}{3}\binom{n}{2} -\cdots+ (-1)^n\frac{1}{n+1}\binom{n}{n}$$

Show that $$\displaystyle\sum_{k=0}^{n}\binom{2n}{k} = 2^{2n-1}+\frac{1}{2}\binom{2n}{n}$$

Show that for any positive integer $n$, the value of $\displaystyle\sum_{k=0}^{n}2^{3k}\binom{2n+1}{2k+1}$ is not a multiple of $5$.

Let $\lfloor{x}\rfloor$ be the largest integer not exceeding real number $x$. Show that $$\sum_{k=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\left(\left(1-\frac{2k}{n}\right)\binom{n}{k}\right)^2=\frac{1}{n}\binom{2n-2}{n-1}$$

Let $n$, $r$, and $m$ all be positive integers, $r\le m$, and $\omega_k=e^{\frac{2k\pi}{m}i}$ be a complex root to the equation $x^m=1$. Show $$\sum_{k=0}^{\lfloor{\frac{n-r}{m}}\rfloor}\binom{n}{r+km}x^{r+km}=\frac{1}{m}\sum_{k=0}^{m-1}\omega^{-r}(1+x\omega_k)^n$$

where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding real number $x$.

Let $n$ be a positive integer and $N=\displaystyle\sum_{k=0}^{n}(-1)^k\binom{n}{k}^2$. Show that $N=0$ if $n$ is odd, and $N=(-1)^{\frac{n}{2}}\displaystyle\binom{n}{\frac{n}{2}}$ if $n$ is even.

Let $n$ be a positive integer and function $\lfloor{x}\rfloor$ return the largest integer not exceeding $x$. Compute the value of $$\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n-k}{k}$$

Let $m$ and $n$ be two positive integers satisfying $m\le n$. Find the value of $$S_{m,n} = \displaystyle\sum_{k=0}^{m}(-1)^k\binom{m}{n}$$

Let $m$ and $n$ be two positive integers satisfying $m < n$. Show that $$S_{m,n}=\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0$$

Show that $$\sum_{k=0}^{n}(-1)^k\frac{m}{m+k}\binom{n}{k}=\frac{1}{\binom{m+n}{n}}$$

Show that $$\sum_{k=0}^{n}(-1)^k2^{2n-2k}\binom{2n-k+1}{k}=n+1$$

Let the binary representation of positive integer $n$ be $b_tb_{t-1}\cdots b_1b_0$. Show that $$\binom{n}{2^j} \equiv b_j \pmod{2}$$

where $j$ is a non-negative integer. Note that $\binom{n}{m} = 0$ if $m > n$.

Let $n$ be a positive integer and $k$ be the number of $1$s in $n$'s binary representation. Show there are $2^k$ odd integers in $\binom{n}{0}$, $\binom{n}{1}$, $\cdots$, $\binom{n}{n}$.

Show that $$\frac{2n+2}{2n+1}\cdot\frac{1}{\binom{2n}{k}}=\frac{1}{\binom{2n+1}{k}}+\frac{1}{\binom{2n+1}{k+1}}$$

Calculate the value of $$\sum_{k=1}^{2n-1}(-1)^{k-1}\binom{2n}{k}^{-1}$$

Let $n$ be a positive integer. Show that $$\sum_{k=0}^{n}k^2\binom{n}{k}^2=n^2\binom{2n-2}{n-1}$$

Find the value of $$\sum_{k=0}^{n-1}\binom{2n-1}{k}$$

Calculate the value of $$\displaystyle\sum_{k=0}^{n}(-1)^{k}k\binom{n}{k}$$

Show that $$\left(\sum_{k=0}^{\infty}x^k\right)^2=\sum_{k=0}^{\infty}(k+1)x^k$$

Let $f(x)$ be the generating function for $a_0$, $a_1$, $a_2$, $\cdots$. Find the generating function for $$a_0, a_0 + a_1, a_0+a_1+a_2, \cdots$$

Show that $$\sqrt{1+x}=1+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\cdot 2^{2n-1}}\binom{2n-2}{n-1}x^n$$

Let $n$ be a positive integer greater than $1$. Show $$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}\binom{2(k-1)}{k-1}\binom{2(n-k-1)}{n-k-1}=\frac{1}{n}\binom{2(n-1)}{n-1}$$

Show that $$\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty}\binom{2n}{n}x^n$$