(right shifting) Find the generating function for the sequence: $$\underbrace{0, 0, \cdots, 0}_{k}, 1, 1, 1, 1, \cdots$$

Show that $$\frac{1}{1-x^2}=\sum_{k=0}^{\infty}x^{2k}$$

Show that $$\frac{1}{(1-ax)^n}=\sum_{k=0}^{\infty}a^k\binom{n-k}{n-1}x^k$$

Find the generating function for the sequence $1$, $2$, $3$, $4$, $\cdots$.

Find the generating function for the sequence $0$, $1$, $2$, $3$, $\cdots$.

Show that $$\frac{1}{(1-x)^2} = 1 + 2x + 3x^2 + 4x^3 + \cdots$$

Find the generating function for the sequence $2,\ 3,\ 4,\ 5,\ \cdots$.

Find the number of subsets of $\{1,\ 2,\ ,3\ ,\cdots,\ 2021\}$ the sum of whose elements is divisible by $101$. (Empty subset is permitted).

A partition of a positive integer $n$ is to write $n$ as a sum of some positive integers. Let $k$ be a positive integer. Show that the number of partitions of $n$ with exactly $k$ parts equals the number of partitions of $n$ whose largest part is exactly $k$.

A partition of a positive integer $n$ is to write $n$ as a sum of some positive integers.. Show that the number of partitions of a positive integer $n$ into distinct parts is equal to the number of partitions of $n$ where all parts are odd integers.

Find the number of parallelograms in the following equilateral triangle of side length $n$ which is made of some smaller unit equilateral triangles.

Let $n$ be a positive integer. Find the number of paths from $(0,0)$ to $(n,n)$ on a $n\times n$ grid using only unit up and right steps so that the path stays in the region $x\ge y$.

Find the number of expressions containing $n$ pairs of parentheses which are correctly matched. For example, when $n=3$, the answer is $5$ because all the valid expressions are $$((())),\ (()()),\ (())(),\ ()(()),\ ()()()$$

A tree is a structure which grows from a single root node. By convention, the root node is usually placed at the top. Then, new nodes, referred as children nodes, can be attached to an existing node, referred as parent node, by edges. The root node has no parent and all other nodes have exactly one parent. A node may have any number of children nodes, including no child at all. No looping chain of nodes is permitted in this structure. For example, there are exactly $5$ different types of trees with $4$ nodes. They are shown below. The goal is to find the number of different types of tree with $n$ nodes where $n$ is a positive integer greater than $1$.

Find the generating function for the sequence $\ 0,\ 1,\ -\frac{1}{2},\ \frac{1}{3},\ -\frac{1}{4},\ \cdots$

There are $30$ identical souvenirs to be distributed among $50$ students. Each student may receive more than one souvenir. How many different distribution plans are there?

Find the number of integer solutions to the equation $a+b+c=6$ where $-1 \le a < 2$ and $1\le b,\ c\le 4$.

Let $\mathbb{A}=\{a_1,\ a_2,\ \cdots,\ a_{100}\}$ be a set containing $100$ real numbers, $\mathbb{B}=\{b_1,\ b_2,\ \cdots,\ b_{50}\}$ be a set containing $50$ real numbers, and $\mathcal{F}$ be a mapping from $\mathbb{A}$ to $\mathbb{B}$. Find the number of possible $\mathcal{F}$ if  $\mathcal{F}(a_1) \le \mathcal{F}(a_2)\le\cdots\mathcal{F}(a_1)$, and for every $b_i\in\mathbb{B}$, there exists an element $a_i\in\mathbb{A}$ such that the $\mathcal{F}(a_i)=b_i$.

Let positive integers $n$ and $k$ satisfy $n\ge 2k$. How many $k$-sided convex polygons are there whose vertices are those of an $n$-sided convex polygon and edges are diagonals of the same $n$-polygon.

Show that for any positive integer $n$, the value of $\frac{(n^2)!}{(n!)^{n+1}}$ is an integer.

Assuming positive integer $n$ satisfies $n\equiv 1\pmod{4}$ and $n > 1$. Let $\mathbb{P}=\{a_1,\ a_2,\ \cdots,\ a_n\}$ be a permutation of $\{1,\ 2,\ \cdots,\ n\}$. If $k_p$ denotes the largest index $k$ associated with $\mathbb{P}$ such that the following inequality holds $$a_1 + a_2 + \cdots + a_k < a_{k+1}+a_{k+2}+\cdots + a_n$$

Find the sum of $k_p$ for all possible $\mathbb{P}$.

How many ways are there to arrange $8$ girls and $25$ boys to sit around a table so that there are at least $2$ boys between any pair of girls? If a sitting plan can be simply rotated to match another one, these two are treated as the same.

Let $\mathbb{S}=\{1,\ 2,\ \cdots,\ 1000\}$ and $\mathbb{A}$ be a subset of $\mathbb{S}$. If the number of elements in $\mathbb{A}$ is $201$ and their sum is a multiple of $5$, then $\mathbb{A}$ is called $\textit{good}$. How many good $\mathbb{A}$ are there?

There are $n \ge 6$ points on a circle, every two points are connected by a line segment. No three diagonals are concurrent. How many triangles are created by these sides and diagonals?