Preface
I found that I now learn a new algorithm always done it related topics to write 233
The inversion unit root in general is not a very difficult position, but it does solve some of the problems of a necessary prerequisite
It may be (O (k) \) \ find a number of columns (or generation function) of all the time subscripts \ (K \) point values and a multiple of
Some of the following basic unit root posture such as the nature and requirements of the following method and the like omitted
Formula
First on the unit root inversion formula:
\[[k|n]=\frac{1}{k}\sum_{i=0}^{k-1}\omega_k^{ni}\]
Let's consider the proof of this formula, Category talk:
If \ (k | the n-\) , then:
\[\frac{1}{k}\sum_{i=0}^{k-1}\omega_k^{ni}=\frac{1}{k}\sum_{i=0}^{k-1}(\omega_k^n)^i=\frac{1}{k}\sum_{i=0}^{k-1}\omega_k^0=1\]
If \ (k \ not | the n-\) , then in accordance with geometric series summation are:
\[\frac{1}{k}\sum_{i=0}^{k-1}\omega_k^{ni}=\frac{1}{k}(\omega_k^0\cdot \frac{\omega_k^0-\omega_k^{kn}}{1-\omega_k^n})\]
Because of its molecules \ (1-1 = 0 \) , so that the equation holds
Others
Sometimes we just know that \ (k | n \) point value and is not enough, for example, we want to know the subscript \ (\ mod k = r \ ) of point values and
Considering to solve the problem by translation function, we are at this time if the multiplied sequence generation function \ (x ^ {- r} \) and then apply the above process can get answers
Example
Simply give a few examples of hand and thought it
- LOJ # 6485. LJJ Science binomial generating function inversion unit root + sol
- BZOJ 3328: PYXFIB generating function inversion unit root + sol
- UOJ # 450. [2018] Team repeater job generating unit root function inversion + sol
Postscript
Recently we came to realize that the generating function of beauty, so do question the recent math direction are thinking right direction near 233