What is the value of $\displaystyle \sum_{n=0}^{\infty} \sum_{j=0}^{n} \sum_{k=0}^{n} \binom{n} {k} \binom{n} {j} (-1)^{j+k}$?
Unfortunately and to be honest, I have no idea to solve this problem. One might be progressed, but this is triple summation, you see. I have looked back at my textbook about this section, but found nothing about triple summation. Could you help me, at least give something brilliant to clear the trouble?
Notice that by expanding $(1-1)^n$ we obtain $ \sum_{k=0}^{n} \binom{n} {k} (-1)^{k}$. Moreover $$\sum_{j=0}^{n} \sum_{k=0}^{n} \binom{n} {k} \binom{n} {j} (-1)^{j+k}= \sum_{j=0}^{n}\binom{n} {j} (-1)^{j} \sum_{k=0}^{n} \binom{n} {k} (-1)^{k}.$$ Can you take it from here?