It should be in the sub-forum "Interesting Math Problem" but there is no such place for me to post It is not very hard but the answer is extremely elegant !!
Show that
and
let be an integer. let we have: call this
now let then and therefore integration by parts gives us:
put this in to get:
that's what i got and i think it's good enough! of course this recurrence relation will also give us a general formula for but i'm not going to do it!