Q:

Sum of product of binomial coefficients.

Let $n\geq 1$ and define
$$
S(n) = \binom{n}{0}\binom{n}{0} - \binom{n+1}{1}\binom{n}{1} + \binom{n+2}{2}\binom{n}{2} - \cdots + (-1)^n\binom{n+n}{n}\binom{n}{n}
$$
In other words
$$
S(n) = \sum_{i=0}^n (-1)^i\binom{n+i}{i}\binom{n}{i}
$$

Then $S(n)=(-1)^n$

I am unable to make any progress.

A:

$$S_n=\sum_{k=0}^{n} (-1)^k {n+k \choose k} {n \choose k}=\sum_{k=0}^{n} (-1)^k {n+k \choose n} {n \choose k}.$$
$$\implies S_n= [x^n]\sum_{k=0}^{n} (-1)^k (1+x)^{n+k} {n \choose k}.$$
$$\implies S_n=[x^n] (1+x)^n(1-1-x)^n= (-1)^n,$$

