$$2 n! (n-1)! \sum_{r=0}^{n-3} \frac{(r+2)(r+1)}{(n+r+2)!(n-r-2)!} =n!\left(2^{2n-2}+1-\binom{2n-1}{n-1}-n\right).$$
T. Amdeberhan
- 43.2k
- 5
- 57
- 217