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).

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 $n$, $m$ and $k$ be three positive integers satisfying $m(k-1) < n$. Find the number of ways to select $k$ items from $\{1,\ 2,\ \cdots,\ n\}$ for form a strict increasing sequence and the difference between adjacent terms is no more than $m$.

As shown, an isosceles trapezoid is obtained by removing the top part of an equilateral triangle. The lengths of its two bases are $a$ and $b$, respectively, which are both integers. Both bases and sides are equally divided into unit-length parts. Their ending points are then connected to create several segments which are parallel to either two bases or one side. Find the number of equilateral triangles in this diagram.

Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.

Let $x_i\in\{+1,\ -1\}$, $i=1,\ 2,\ \cdots,\ 2n$. If their sum equals $0$ and the following inequality holds for any positive integer $k$ satisfying $1\le k < 2n$: $$x_1+x_2+\cdots + x_k\ge 0$$

Find the number of possible ordered sequence $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$.

Solve the recursion $$a_n=\sum^{n-1}_{k=0}a_{k}a_{n-k-1}=a_0a_{n-1}+a_1a_{n-2}+\cdots+a_{n-1}a_0$$

where $a_0=a_1=1$.