Factorial

Le factorial^[1] es un function que a un numero natural attribue le producto de omne numeros de 1 usque iste numero incluse. Iste producto es representate per un puncto de exclamation postponite. Iste notation es originari del mathematico alsatian Christian Kramp (1760–1826).

Graphico: factoriales del numeros 1-5

Factorial
instantia de: integer-valued function[*], integer sequence[*]
Commons: Factorial (function)

Su definition es ${\displaystyle n!:=\prod _{i=1}^{n}i}$.

Exemplos

${\displaystyle {\begin{array}{rll}0!&:=1&\\1!&=1&=1\\2!&=1\cdot 2&=2\\3!&=1\cdot 2\cdot 3&=6\\4!&=1\cdot 2\cdot 3\cdot 4&=24\\5!&=1\cdot 2\cdot 3\cdot 4\cdot 5&=120\\6!&=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6&=720\\7!&=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7&=5'040\\8!&=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8&=40'320\\9!&=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9&=362'880\end{array}}}$

Le plus grande parte del calculatores electronic pote indicar maximalmente ${\displaystyle 69!}$, quia ${\displaystyle 69!}$ es le ultime factorial escrivente se con minus de 100 cifras.

Duple factorial

Le duple factorial o semifactorial se nota ${\displaystyle n!!}$ de ${\displaystyle n}$ e es definite recursivemente secundo

${\displaystyle n!!={\begin{cases}1&{\text{si }}n=0{\text{ o }}n=1,\\n\cdot (n-2)!!&{\text{si }}n\geq 2.\end{cases}}}$

per exemplo ${\displaystyle 8!!=8\cdot 6\cdot 4\cdot 2=384}$ e ${\displaystyle 9!!=9\cdot 7\cdot 5\cdot 3\cdot 1=945}$. Le sequentia del duple factorial per ${\displaystyle n=0,1,2,\dots }$ es ^[2]:

${\displaystyle 1,1,2,3,8,15,48,105,384,945,3'840,\dots }$

Referentias

1. Derivation (in ordine alphabetic): (ca) Factorial || (de) Fakultät (Mathematik) || (en) Factorial || (es) Factorial || (fr) Factorielle || (it) Fattoriale || (pt) Fatorial || (ro) Factorial || (ru) Факториал
2. Patrono:OEIS