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- factorial - Why does 0! = 1? - Mathematics Stack Exchange
$\begingroup$ The theorem that $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ already assumes $0!$ is defined to be $1$ $ Otherwise this would be restricted to $0 <k
- Factorial, but with addition - Mathematics Stack Exchange
Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the factorial $5!$ way EDIT: I know about the formula I want to know if there's a short notation
- Defining the factorial of a real number - Mathematics Stack Exchange
$\begingroup$ Some theorems that suggest that the Gamma Function is the "right" extension of the factorial to the complex plane are the Bohr–Mollerup theorem and the Wielandt theorem $\endgroup$ – kuzzooroo
- Factorial of zero is 1. Why? - Mathematics Stack Exchange
$\begingroup$ decrementing n down to 1-- which doesn't make sense for n = 0, so we seek another uniform property of factorial in order to extend the definition Similarly, why is 4 to the -1 power equal to 1 4? "multiply it by itself -1 times" doesn't make sense, so instead we use other properties to extend the definition to negative exponents
- Can the factorial function be written as a sum?
In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler (decomposition of primes as the sum of two squares), and, on the other hand, to modern combinatorics
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