\[ f(a) = \frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)}{z-a}dz \]

Beautiful and accessible math in all browsers - MathJax

This post is used to check out whether the mathjax plugin (hexo-math) still works.

Inline Formula

Consider the sequence of \(n\) distinct positive integers: \(c_1, c_2, \cdots, c_n\). The child calls a vertex-weighted rooted binary tree good if and only if for every vertex \(v\), the weight of \(v\) is in the set \(\{c_1, c_2, \cdots, c_n\}\) . Also our child thinks that the weight of a vertex-weighted tree is the sum of all vertices' weights.

Given an integer \(m\), can you for all $s (1  s ≤ m) $ calculate the number of good vertex-weighted rooted binary trees with weight \(s\) ? Please, check the samples for better understanding what trees are considered different.

We only want to know the answer modulo \(998244353\) ( \(7 × 17 × 2^{23} + 1\) , a prime number).

Block Formula

\[ \int \frac{dx}{cos^2x}=\int sec^2xdx=tan x+C\ \]

\[ \int \frac{dx}{sin^2x}=\int csc^2xdx=-cot x+C\ \]

\[ \sum\limits_{i = 0}^{E} w[i]{M \choose i} {N \choose iS} \frac{(iS)!}{(S!)^{i}} \sum\limits_{j = 0}^{E - i} (-1)^{j} {M - i \choose j} {N - iS \choose jS} \frac{(jS)!}{(S!)^{j}} (M - i - j)^{N - iS - jS} \]

\[ = \sum\limits_{i = 0}^{E} w[i]{M \choose i} {N \choose iS} \frac{(iS)!}{(S!)^{i}} \sum\limits_{j = i}^{E} (-1)^{j - i} {M - i \choose j - i} {N - iS \choose jS - iS} \frac{(jS - iS)!}{(S!)^{j - i}} (M - j)^{N - jS} \]

Matrix

\[ \begin{bmatrix} (\omega_n^0)^0 & (\omega_n^0)^1 & \cdots & (\omega_n^0)^{n-1}\\\\ (\omega_n^1)^0 & (\omega_n^1)^1 & \cdots & (\omega_n^1)^{n-1}\\\\ \vdots & \vdots & \ddots & \vdots \\\\ (\omega_n^{n-1})^0 & (\omega_n^{n-1})^1 & \cdots & (\omega_n^{n-1})^{n-1} \end{bmatrix} \begin{bmatrix} a_0 \\\\ a_1 \\\\ \vdots \\\\ a_{n-1} \end{bmatrix} = \begin{bmatrix} A(\omega_n^0) \\\\ A(\omega_n^1) \\\\ \vdots \\\\ A(\omega_n^{n-1}) \end{bmatrix} \]

\[ \begin{bmatrix} a_0 \\\\ a_1 \\\\ \vdots \\\\ a_{n-1} \end{bmatrix} = \frac{1}{n} \begin{bmatrix} (\omega_n^{-0})^0 & (\omega_n^{-0})^1 & \cdots & (\omega_n^{-0})^{n-1} \\\\ (\omega_n^{-1})^0 & (\omega_n^{-1})^1 & \cdots & (\omega_n^{-1})^{n-1} \\\\ \vdots & \vdots & \ddots & \vdots \\\\ (\omega_n^{-(n-1)})^0 & (\omega_n^{-(n-1)})^1 & \cdots & (\omega_n^{-(n-1)})^{n-1} \end{bmatrix} \begin{bmatrix} A(\omega_n^0) \\\\ A(\omega_n^1) \\\\ \vdots \\\\ A(\omega_n^{n-1}) \end{bmatrix} \]

To prevent escaping, you need to use \\\\ to make a new line rather than \\ .

For more about the Mathjax symbols, refer to ：The Comprehensive LaTeX Symbol List .