test02 - 浪店

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test02

2026-03-18

test

$f(x)=\sum_{i=0}^{n}a_{i}x^{i}$

$f(x)=\sum_{i=0}^{n}f(i)\prod_{j=0,j\le n,j\neq i}\frac{x-j}{i-j}$

\begin{aligned} f(x+m)&=\sum_{i=0}^{n}f(i)\prod_{j=0,j\le n,j\neq i}\frac{x+m-j}{i-j}\\ &=\sum_{i=0}^{n}\frac{f(i)(x+m)!}{(x+m-n-1)!(x+m-i)i!(-1)^{n-i}(n-i)!}\\ d_{i}&=\frac{f(i)}{i!(-1)^{n-i}(n-i)!}\\ e_{i}&=\frac{1}{m-n+i}\\ A(x)&=\sum_{i=0}^{n}d_{i}x^{i}\\ B(x)&=\sum_{i=0}^{n}e_{i}x^{i}\\ [x^{t}]A(x)B(x)&=\sum_{i=0}^{t}d_{i}e_{t-i}=\sum_{i=0}^{t}\frac{f(i)}{i!(-1)^{n-i}(n-i)!(m-n+t-i))}\\ [x^{t+n}]A(x)B(x)&=\sum_{i=0}^{t}d_{i}e_{t-i}=\sum_{i=0}^{t}\frac{f(i)}{i!(-1)^{n-i}(n-i)!(m+t-i))}=\frac{(x+t-n-1)!}{(x+t)!}f(t+m)\\ f(t+m)&=\frac{(t+m)!}{(t+m-n-1)!}[x^{t+n}]A(x)B(x)=[x^{t+n}]A(x)B(x)\prod_{i=t+m-n}^{t+m}i\\ \end{aligned}

\begin{align} g(x)&=\frac{1}{1-x}\\ [x^{i}][g(x)]^{k}&=\binom{k+i-1}{i}\\ [x^{i}][g^{-1}(x)]^{k}&=\binom{k}{i}(-1)^{i}\\ \binom{k}{i}&=\frac{k!}{i!(k-i)!}=\binom{k}{i-1}\frac{k-i+1}{i}\\ \binom{k+i-1}{i}&=\frac{(k+i-1)!}{i!(k-1)!}=\binom{k+i-2}{i-1}\frac{k+i-1}{i} \end{align}

$\binom{k}{i}=\frac{k!}{i!(k-i)!}=\frac{k(k-1)...(k-i+1)}{i!}\equiv\frac{k'(k'-1)...(k'-i+1)}{i!}=\frac{k'!}{i!(k'-i)!}\equiv\binom{k'}{i}$

\begin{align} \binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\\ g(x)&=-x+1\\ f(x)&=a_{0}+a_{1}x+a_{2}x^{2}\\ \end{align}

f(x)=\sum_{i=0}^{n}

\begin{align} \ln\frac{1}{1-x^{V}}&=-\ln(1-x^{V})\\ &=-\sum_{i=1}\frac{x^{Vi}}{i}\\ &=\sum_{i=1}\frac{x^{Vi}}{i}\\ \end{align}

$Bell\_Number$

B_{n+1}=\sum_{i=0}^{n}\binom{n}{i}B_{i}

\begin{align} F(x)&=\sum_{i=0}\frac{B_{i}}{i!}x^{i}\\ \frac{B_{n+1}}{n!}&=\sum_{i=0}^{n}\frac{1}{(n-i)!}\frac{B_{i}}{i!}\\ e^{x}\cdot F(x)&=F'(x)\\ e^{x}&=\frac{F'(x)}{F(x)}\\ e^{x}+c&=\ln(F(x))\\ F(x)&=e^{e^{x}+c}\\ F(x)&=e^{e^{x}-1} \end{align}

\begin{align} 1011 1001 0111 1010 1000 0101 1000 0110 1011 1001 + 1000 0110 1011 1111 1 00111 110 - 7 6 -1+6/8 \times 2^-8 -7/2^10 \end{align}

$f_{i}$ 表示 $i$ 个点联通的方案数, $g_{i}$ 表示任意一图

\begin{align} g_{n}&=2^{\binom{n}{2}}\\ g_{n}&=\sum_{i=1}^{n}\binom{n-1}{i-1}f_{i}g_{n-i}\\ g_{n}&=\sum_{i=1}^{n}\frac{(n-1)!}{(i-1)!(n-i)!}f_{i}g_{n-i}\\ \frac{g_{n}}{(n-1)!}&=\sum_{i=1}^{n}\frac{f_{i}}{(i-1)!}\frac{g_{n-i}}{(n-i)!}\\ G'(x)&=F'(x)G(x)\\ F'(x)&=\frac{G'(x)}{G(x)}\\ F(x)&=\ln G(x)+c\\ \end{align}

\begin{align} \sum_{S\subset T}\mu(\prod_{i\in S}i)\varphi(\prod_{i\in S}i)&=0&,k=0\ \sum_{S\subset T}\mu(\prod_{i\in S}i)\varphi(\prod_{i\in S}i)&=\sum_{S\subset T}(-1)^{|S|}\prod_{i\in S}(i-1)&,k=1\ &=\sum_{S\subset T}\prod_{i\in S}(-(i-1))\ f_{p}(x)&=1-(p-1)x^{p}\ ans&=\prod_{p}f_{p}(x)|{x=1}-1\ \sum{S\subset T}\mu(\prod_{i\in S}i)\varphi(\prod_{i\in S}i)&=\sum_{S\subset T}g_{S}\prod_{i\in S}(-(i-1)) &,k=2\

\end{align}

工科数学分析-3.19

工科高等代数3.17