As a kid I was bugged by the fact that Stirling formula doesn't give exact integer result, so I set to find my own formula. I failed, but discovered that sum from 1 to N is (n*n+n)/2. Surely if perfect formula for sum exists, for multiplication should exist too.
For those who are curious how we can construct such a function, and why it looks so funky on the negative side, I strongly recommend this video https://www.youtube.com/watch?v=v_HeaeUUOnc
I think they mean a (traditionally) closed form expression, for which I don't know if there is a particularly simple explanation of why there isn't such a form but there is for summing integers to n instead of multiplying them.
