As you may have read in the title, I'm trying to find out whether my formula is true or not for this strange sum.
Here's my hypothesis: $$\forall n,t\in \mathbb N^2, \sum_{a_n=1}^{t+1}\left(\sum_{a_{n-1}=1}^{a_n}\left(\sum_{a_{n-2}=1}^{a_{n-1}}\left(...\left(\sum_{a_0=1}^{a_1}a_0\right)...\right)\right)\right) = \prod_{h=1}^{n+2}\frac{t+h}{h}=\frac{(t+n+2)!}{t!(n+2)!}$$
I don't really know how to prove my formula is right. I think my formula is true. I've tried to prove it by induction, but I don't know how to do it. Maybe, I am wrong. What's your thinking about it ?
Thank you for your help!