Question Number 161234 by mathlove last updated on 14/Dec/21
$$\mathrm{1}:!{x}=? \\ $$$$\mathrm{2}:!!{x}=? \\ $$$$\mathrm{3}:{x}!!=?\:\:\:\:{when}\:\:\:{X}\:\:{is}\:\:{odd} \\ $$$$\mathrm{4}:{x}!!=?\:\:\:\:{when}\:\:{X}\:\:{is}\:\:{evan} \\ $$$$\mathrm{5}:{x}!!!=?\:\:\:\:\:\:\:{when}\:{X}\:{is}\:{odd} \\ $$$$\mathrm{6}:\mathrm{x}!!!=?\:\:\:\:\:\:\:{when}\:{X}\:\:{is}\:{evan} \\ $$
Commented by mr W last updated on 14/Dec/21
$$!{x}\:{is}\:{defined}\:{as}\:{number}\:{of}\:{derangements}. \\ $$$${i}\:{think}\:!!{x}\:{is}\:{not}\:{defined}.\:{you}\:{could} \\ $$$${mean}\:!\left(!{x}\right). \\ $$$${x}!!\:{is}\:{defined}\:{as}\:{x}×\left({x}−\mathrm{2}\right)×\left({x}−\mathrm{4}\right)×...×{k}\: \\ $$$$\left({k}\geqslant\mathrm{1}\right) \\ $$$${similarly}\:{you}\:{could}\:{define}\:{x}!!!\:{as} \\ $$$${x}!!!={x}×\left({x}−\mathrm{3}\right)×\left({x}−\mathrm{6}\right)×...×{k} \\ $$$$\left({k}\geqslant\mathrm{1}\right) \\ $$