（再问一条）求生成函数的封闭形式推导

学术版

zhiyangfan @ 2021-07-19 20:43:36

rt。对于等差数列 \langle \alpha,\alpha+\beta,\alpha+2\beta,\cdot\cdot\cdot,\alpha+n\beta \rangle 的OGF F(x)=\sum_{n\ge 0}(\alpha+n\beta)x^n，它的封闭形式咋推导啊/kel。

百度不到，问了一个队爷，他说用 \frac{1}{(1-x)^k} 的线性组合，但我没推出来。/kk

by 小粉兔 @ 2021-07-19 20:45:25

这不就 \frac{\alpha}{1-x}+\frac{\beta}{{(1-x)}^2}

by zhiyangfan @ 2021-07-19 20:46:53

@小粉兔 CCCCCOrz 谢谢谢谢

by Rui_R @ 2021-07-19 20:49:15

F(x)\cdot (1-x)=\alpha + \beta\sum_{i=1} x^i

= \alpha + \frac{\beta x}{1-x}

F(x) = \frac{\alpha(1-x)+\beta x}{(1-x)^2}

错了的话骂轻一点

by zhiyangfan @ 2021-07-19 20:51:06

@Rui_R CCCCOst 十分感谢

by Rui_R @ 2021-07-19 20:51:32

艹为啥不一样

by zhiyangfan @ 2021-07-19 20:53:24

@Rui_R 还没太看懂您的推导，但您的结果跟小粉兔的不太一样/kk

by zhiyangfan @ 2021-07-19 20:53:42

草刚刷新，我的锅

by Spasmodic @ 2021-07-19 20:54:09

@Rui_R 目测兔有问题令 \alpha=0,\beta=1 就起飞了吧

by zhiyangfan @ 2021-07-19 20:57:23

好像兔确实锅了/fad

OI wiki上给了 \alpha=1,\beta=1 的情况，跟兔给的不太一样/fad

by x义x @ 2021-07-19 20:57:46

@happyChristmas 无伤，\dfrac{\alpha}{1-x}+\dfrac{\beta x}{(1-x)^2} 就好了

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