All Questions

Is it true that $$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$ for all natural $n$ and all natural $p\ge2n$, where $$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)! (p-r+i)! (n-r+i)! ...