# 生成函数的性质速查

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# 普通生成函数 (OGF)

## OGF 的性质

OGF的导数(数列乘下标) $A’(z) = \sum\limits_{n=0}\limits^{\infty}(n+1)a_{n+1}z^{n}$
OGF的积分(数列除下标) $\int_{0}^{z}A(t)\mathrm{d}t = \sum\limits_{n=1}\limits^{\infty}\frac{a_{n-1}}{n-1}z^{n}$
OGF自变量的比例因子 $A(\lambda z) = \sum\limits_{n=0}\limits^{\infty}\lambda^{n}a_{n}z^{n}$
OGF的复合 $A(B(z)) = \sum\limits_{n=0}\limits^{\infty}a_{n}(B(z))^{n}$，要求 $b_{0} = 0$

## 常见数列的 OGF

$a_{n} = 1$ $A(z) = \frac{1}{1-z}$
$a_{n} = n$ $A(z) = \frac{z}{(1-z)^{2}}$
$a_{n} = \binom{n}{2}$ $A(z) = \frac{z^{2}}{(1-z)^{3}}$
$a_{n} = \binom{n}{m}$ $A(z) = \frac{z^{m}}{(1-z)^{m+1}}$
$a_{n} = \binom{m}{n}$ $A(z) = (1+z)^{m}$
$a_{2k}=1, a_{2k+1}=0$ $A(z) = \frac{1}{1-z^{2}}$
$a_{n} = c^{n}$ $A(z) = \frac{1}{1-cz}$
$a_{n} = \frac{1}{n!}$ $A(z) = e^{z}$
$a_{n} = \frac{1}{n}$ $A(z) = -\ln(1-z)$
$a_{n} = H_{n}$ $A(z) = \frac{1}{1-z}\ln\frac{1}{1-z}$
$a_{n} = n(H_{n} - 1)$ $A(z) = \frac{z}{(1-z)^{2}}\ln\frac{1}{1-z}$

# 指数生成函数 (EGF)

## EGF 的性质

EGF乘自变量(数列乘下标) $zA(z) = \sum\limits_{n=0}\limits^{\infty}na_{n-1}\frac{z^{n}}{n!}$
EGF除自变量(数列除下标) $\frac{A(z)-A(0)}{z} = \sum\limits_{n=1}\limits^{\infty}\frac{a_{n+1}}{n+1}\frac{z^{n}}{n!}$

## 常见数列的 EGF

$a_{n} = 1$ $A(z) = e^{z}$
$a_{n} = n$ $A(z) = ze^{z}$
$a_{n} = \binom{n}{2}$ $A(z) = \frac{1}{2}z^{2}e^{z}$
$a_{n} = \binom{n}{m}$ $A(z) = \frac{1}{m!}z^{m}e^{z}$
$a_{2k}=1, a_{2k+1}=0$ $A(z) = \frac{1}{2}(e^{z} + e^{-z})$
$a_{n} = c^{n}$ $A(z) = e^{cz}$
$a_{n} = \frac{1}{n}$ $A(z) = \frac{e^{z}-1}{z}$
$a_{n} = n!$ $A(z) = \frac{1}{1-z}$

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