Render Math Equations in Hexo

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

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 \le  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 .


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