# 初探卡特兰数及有关问题(On Cartland number and related problems)-其他

## 初探卡特兰数及有关问题(On Cartland number and related problems)

### 二、卡特兰数的发现与定义

————百度百科

There are many, many combinatorial definitions of the Catalan numbers, but the most common might be that Cn counts the number of lattice paths from (0,0) to (n,n) that only take unit step right and up, and never cross the diagonal line y=x (but they are allowed to touch the diagonal line). There is not a unique definition of Catalan numbers because all of the various combinatorial definitions are equivalent to each other, so which one you take as your definition is a stylistic preference.

*

————Math StackExchange论坛，源地址

————————

On Sunday, I took the first computer test of introduction to computing and programming since I went to college, but I didn’t have AC in the last question.

This problem gives a general term formula of Cartland number, which allows you to find the nth term of Cartland number.

After coming out of the examination room, I felt empty, not only because this question didn’t type out, which directly affected the whole exam, but also because I never seemed to have studied a mathematical problem completely out of interest

After this problem was solved by me, I checked the relevant knowledge of Cartland number on the Internet and found that Cartland number is also closely related to several types of problems.
Therefore, I think it is necessary to study the magical Cartland number here~

The first few terms of Cartland number and its recurrence formula are the most familiar places for us to study Cartland number. We might as well start to study it here.

### 1、 First knowledge of Cartland number

First, we give the first 10 terms of Cartland number:

Item 0: 1
Item 1: 1
Item 2: 2
Item 3: 5
Item 4: 14
Item 5: 42
Item 6: 132
Item 7: 429
Item 8: 1430
Item 9: 4862

Then, we give several general term formulas of Cartland number:

The first one: \ ({C {n + 1}} = \ frac {2 (2n + 1)} {{n + 2} {C} \)

The second type: \ ({c_n} = {c_0} {c_ {n – 1} + {c_1} {c_ {n – 2}} + \ cdots + {c_ {n – 2} {c_1} + {c_ {n – 1}} {c_0} (n \ Ge 2) \)

The third type: \ ({f _n} = \ frac {C _ {2n} ^ n} {n + 1} \) (use \ (f _n \) to distinguish it from the combinatorial number formula \ (C _ {2n} ^ n \)

The fourth type: \ ({C _n} = \ prod \ limits {k = 2} ^ n {\ frac {{n + K} {K} \)

### 2、 Discovery and definition of Cartland number

< strong > Catalan number (English: Catalan number), also known as Catalan number and mingantu number, is a number sequence that often appears in various counting problems in combinatorics. Named after the Belgian mathematician Oren Charlie Cartland. It was first discovered by Mongolian mathematician Ming Antu in the derivation of trigonometric function power series around 1730, and was published in the shortcut method of cutting circle density in 1774

————Baidu Encyclopedia

Definition of Cartland number:

There are many, many combinatorial definitions of the Catalan numbers, but the most common might be that Cn counts the number of lattice paths from (0,0) to (n,n) that only take unit step right and up, and never cross the diagonal line y=x (but they are allowed to touch the diagonal line). There is not a unique definition of Catalan numbers because all of the various combinatorial definitions are equivalent to each other, so which one you take as your definition is a stylistic preference.

*