一些常见的二元生成函数

组合数相关:

形式化的:

\[G(x,y) = \sum_{i = 0}\sum_{j = 0}(\frac{i}{j})x^iy^j\\ =\sum_{i = 0}(x+xy)^i\\ =\frac{1}{1 - x - xy}\\ \]

则称\(G(x,y)\)为二元组\((x,y)\)在组合数意义下的一个二元生成函数

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具体用途----处理一些比较诡异的组合数上的连续求和

一般是通过将\(y\)带换成\(x^k\),\(x\)也相应变形的方法来处理

\[G(x , x^k) = \sum_{i = 0}\sum_{j = 0}(\frac{i}{j})x^{i + kj}\\ [x^n] = \sum_{i + kj = n}(\frac{i}{j}) \]

具体参见例题:

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\[\sum_{i = 0}^m(\frac{m}{i})\\ 可以写成求和式:\\ \sum_{i + 0\times j = m}(\frac{i}{j})\\ 即G(x , x^0 = 1) = \frac{1}{1 - 2x}\\ 根据二项式定理有[x^n] = 2^n\\ \]

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\[\sum_{i = 0}^m(\frac{i + m}{2i})\\ 可以写成求和式:\\ \sum_{i + m - (2 * i / 2) = m}\\ x = i + m , y = 2 * i \\ 即x - y/2 = m \]

则令\(y = x^{-\frac{1}{2}}\)

\[G(x^1 , x^{-\frac{1}{2}}) = \frac{1}{1 - x-\sqrt x}\\ [x^n] = \frac{1}{1 - x - \sqrt x},[z^{2n}] = \frac{1}{1 - z^2-z} = F_{2n + 1} \]

即第\((2n + 1)\)个斐波拉契数

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\[S_1(x,y) = \sum_{i = 0}\sum_{j = 0}[\frac{i}{j}]\frac{x^iy^j}{i!}\\ 一个环的EGF为F = \sum_{i = 0}\frac{x^i}{i} = -ln(1-x)\\ 引入变量y记录成环次数,则有:G = exp(yF) = exp(ln(\frac{1}{1-x})^y) = (1-x)^{-y}\\ 即有S_1(x,y) = (1-x)^{-y}\\ 则有一行的OFG如下\\ =n!\sum_{k = 0}(\frac{-y}{k})(-1)^{k}x^k =n!\sum_{k = 0}(\frac{k + y - 1}{k})x^k\\ =[x^n]n!(\frac{n + y - 1}{n}) = y^{\overline{n}} \]

一列的EGF：

\[\frac{(-ln(1-x))^k}{k!},可以较简单得到 \]

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第二类斯特林数

\[S_2(x,y) = \sum_{i = 0}\sum_{j = 0}\{\frac{i}{j}\}\frac{x^iy^j}{i!} = exp(y(e^x-1))\\ = \sum_{i=0}\frac{y^i(e^x-1)^i}{i!} = \sum_{i = 0}\frac{y^i}{i!}\sum_{k = 0}e^{kx}(-1)^k(\frac{i}{k})\\ 随便化一下大概就是\\ [x^n] = \sum_{i = 0}\frac{y^i}{i!}\sum_{k = 0}^i(\frac{i}{k})k^n(-1)^{i-k} \]