Factorial

Graphico: factoriales del numeros 1-5

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

Le factorial 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).

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

Exemplos

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

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

Duple factorial

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

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

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

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