阶乘 Factorial

（重定向自Hyperfactorial）

一个正整数的阶乘/层（英语：factorial）是所有小于及等于该数的正整数的积，并且有0的阶乘为1。自然数n的阶乘写作n!。1808年，基斯顿·卡曼引进这个表示法。

n!=\prod_{k=1}^n k \quad\forall n\ge1

亦即n!=1×2×3×...×n。阶乘亦可以递归方式定义：0!=1，n!=(n-1)!×n。

阶乘亦可定义于整个实数（负整数除外），其与伽玛函数的关系为：

z!=\Gamma(z+1)=\int_{0}^{\infty} t^z e^{-t}\, dt

n!可质因子分解为 $\prod_{p \le n} p^{\sum_{r=1}^n [\frac{n}{p^r}]}$ ，如6!=2×3×5。

单词	Hyperfactorial
释义	Hyperfactorial 中文百科阶乘 Factorial （重定向自Hyperfactorial）一个正整数的阶乘/层（英语：factorial）是所有小于及等于该数的正整数的积，并且有0的阶乘为1。自然数n的阶乘写作n!。1808年，基斯顿·卡曼引进这个表示法。 $n!=\prod_{k=1}^n k \quad\forall n\ge1$ 亦即n!=1×2×3×...×n。阶乘亦可以递归方式定义：0!=1，n!=(n-1)!×n。阶乘亦可定义于整个实数（负整数除外），其与伽玛函数的关系为： $z!=\Gamma(z+1)=\int_{0}^{\infty} t^z e^{-t}\, dt$ n!可质因子分解为 $\prod_{p \le n} p^{\sum_{r=1}^n [\frac{n}{p^r}]}$ ，如6!=2×3×5。英语百科 Factorial 阶乘（重定向自Hyperfactorial） In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). This fact was known at least as early as the 12th century, to Indian scholars. Fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):
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Hyperfactorial

阶乘 Factorial

Factorial 阶乘